probability, statistics, random walks, and Brownian motion
in connection with number theory
thoughtproving quotes
historical background  H. Cramér's contribution to the field
(approximately) chronological bibliography
papers by Khrennikov, et al. on padic probability, etc.
number theoretic applications of random matrix theory
thoughtprovoking quotes
"It is evident that the primes are randomly distributed but,
unfortunately, we don't know what 'random' means."
R.C. Vaughan (February 1990)
"Consider the integers divisible by both p and q
[p and q both prime]. To be divisible by p and
q is equivalent to being divisible by pq and
consequently the density of the new set is 1/pq. Now,
1/pq = 1/p * 1/q, and we can interpret this by
saying that the "events" of being divisible by p and q
are independent. This holds, of course, for any number of primes,
and we can say using a picturesque but not very precise language,
that the primes play a game of chance! This simple, nearly trivial,
observation is the beginning of a new development which links in a
significant way number theory on the one hand and probability theory
on the other."
M. Kac,
Statistical Independence in Probability, Analysis and
Number Theory. (Wiley, 1959)
"Tables of prime numbers display a chaotic aspect whose apparent
disorder somewhat resembles classical random models arising, for
example, from physical phenomena. And here is exactly the purpose
of this little book: to describe, and then try to understand, how
a sequence so precisely determined as that of prime numbers can
incorporate so great a share of randomness.
Let us take this point a little further. Total randomness,
chaos, is infinite complexity. Besides, the complexity of an
integer grows with its size: is the number 2^{6972593}1
prime? In the neighbourhood of infinity, the sequence of integers,
and therefore that of prime numbers, contains randomness. Current
research in modern analytic number theory tries to account for this
aspect."
G. Tenenbaum and
M. Mendès France, from
The Prime
Numbers and Their Distribution (AMS, 2000) page xii
"Given the sporadic, randomlike quality of the primes, it is quite surprising how
much can be proved about them. Interestingly, theorems about the primes are usually
proved by exploiting this seeming randomness...
Much research on prime numbers has this sort of flavour. Your first devise a
probabilistic model for the primes  that is, you pretend to yourself that they have
been selected according to some random procedure. Next, you work out what would be
true if the primes really were generated randomly. That allows you to guess the answers
to many questions. Finally, you try to show that the model is realistic enough for you
guesses to be approximately correct...
It is interesting that the probabilistic model is a model not of a physical
phenomenon, but of another piece of mathematics. Although the prime numbers are rigidly
determined, they somehow feel like experimental data. Once we regard them that way,
it becomes tempting to devise simplified models that allow us to predict what the answers
to certain probabilistic questions are likely to be. And such models have indeed
sometimes led people to proofs valid for the primes themselves."
T. Gowers, Mathematics:
A Very Short Introduction (Oxford Univ. Press, 2002)
p.120121
"I think that it's just a completely outrageous conjecture to make in the first place that probability should have anything to do with the primes."
T. Gowers, from his keynote talk "The Importance of Mathematics" at The Clay Mathematics Institute Millenium Meeting (2000) [see here, at about 1:40]
"Prime numbers present mathematicians with one of the strangest tensions in their subject. On the one hand a
number is either prime or it isn't. No flip of a coin will suddenly make a number divisible by some smaller number. Yet
there is no denying that the list of primes looks like a randomly chosen sequence of numbers. Physicists have grown
used to the idea that a quantum die decides the fate of the universe, randomly choosing at each throw where scientists
will find matter. But it is something of an embarrassment to have to admit that these fundamental numbers on which
mathematics is based appear to have been laid out by Nature flipping a coin, deciding at each toss the fate of each number.
Randomness and chaos are anathema to the mathematician."
M. du Sautoy, The Music of the Primes
(Fourth Estate, 2003) p.6
"Probability is not a notion of pure mathematics,
but of physics or philosophy."
G.H Hardy
and J.E. Littlewood,
"Some problems of 'partitio numerorum:' III: on the expression
of a number as a sum of primes" Acta Mathematica 44 (1922)
"One of the remarkable aspects of the distribution of prime numbers
is their tendency to exhibit global regularity and local irregularity. The
prime numbers behave like the 'ideal
gases' which physicists are so fond of.
Considered from an external point of view, the distribution is  in broad
terms  deterministic, but as soon as we try to describe the situation at a
given point, statistical fluctuations occur as in a game of chance where it
is known that on average the heads will match the tail but where, at any one
moment, the next throw cannot be predicted. Prime numbers try to occupy all
the room available (meaning that they behave as randomly as possible), given
that they need to be compatible with the drastic constraint imposed on them,
namely to generate the ultraregular sequence of integers.
This idea underpins the majority of conjectures concerning prime numbers:
everything which is not trivially forbidden should actually happen..."
G. Tenenbaum and M. Mendès France, from
The Prime
Numbers and Their Distribution (AMS, 2000) page 51
"In investigations concerning the asymptotic properties of
arithmetic functions, it is often possible to make an interesting use
of probability arguments. If, e.g., we are interested in the
distribution of a given sequence S of integers, we then consider S as
a member of an infinite class C of sequences, which may be concretely
interpreted as the possible realizations of some game of chance. It
is then in many cases possible to prove that, with a probability
equal to 1, a certain relation R holds in C, i.e., that in a definite
mathematical sense, 'almost all' sequences of C satisfy R. Of course
we cannot, in general, conclude that R holds for the particular
sequence S, but results suggested in this way may sometimes afterwards
be rigorously proved by other methods."
H.
Cramér, "On the order of magnitude of the difference between
consecutive prime numbers", Acta Arithmetica 2 (1937) 2346.
In an homage to the revered mathematician Paul Erdös, who died
September 20, 1996, D. Mackenzie mentioned a theory Erdös published in
1940 with M.
Kac. This theory states that a plot of the number of
prime factors of very large numbers forms a bell curve  almost as if
these numbers were "choosing" their prime factors at random. Alluding
to an assertion Einstein is said to have made, Carl Pomerance has suggested that
Erdös might have said something like this:
"God may not play dice with the universe, but something strange is
going on with the prime numbers."
D. Mackenzie, "Homage to an Itinerant Master", Science 275 (1997) 759 (this
mistakenly attributes the quotes to Erdös himself)
historical background  H.
Cramér's contribution to the field
Although Erdös, Kac, Wintner
and Kubilius are generally credited with
the founding of probabilistic number theory, the following (earlier) work
of Harald
Cramér should not be overlooked:
H.
Cramér, "Prime numbers and probability", Skand. Mat.Kongr.
8 (1935) 107115.
A. Granville,
"Harald Cramér and the distribution of prime
numbers", Scandanavian Actuarial Journal 1 (1995),
1228. [an excellent survey of Cramér's work in this area]
The following is an excerpt from the chapter "Stochastic
Distribution of Prime Numbers" of G. Tenenbaum and M. Mendès France's
wonderful little book
The Prime
Numbers and Their Distribution (AMS, 2000):
"The most natural questions about prime numbers are often quite
hard to answer. Furthermore, it is even rather difficult, in certain
cases, to make a reasonable guess for what the exact answer might be.
Having studied the distribution of prime numbers since the twenties,
Cramér
proposed at the end of the thirties a simple and fascinating
method for producing conjectures concerning primes: the sequence
of prime numbers behaves like a random sequence with the same growth
constraint.
Let us formalize a little. The prime number
theorem suggests that the probability that an integer of size
n be a prime is close to 1/log n. Denote by
{X_{n}} (where n = 2,3,4,...) a sequence of
independent random variables taking values 0 and 1 with the
probability(X_{n} = 1) = 1/log n
(n > 2)
Then the random sequence
S := {n > 1: X_{n} = 1}
constitutes, according to Cramér, a stochastic model of the sequence
of primes.
In more intuitive terms, we may consider the following mechanism.
Arrange, in an infinite (or very long) sequence, urns U_{3},
U_{4}, etc., with the property that the urn
U_{n} contains one white ball and about log n
black balls. Now choose a ball at random from each urn and assign the
integer n to the sequence S if the ball taken from
U_{n} is white. The sequence S models the
sequence of primes. Of course, for each particular drawing the sequence
S will possess specific properties which will distinguish it
significantly from the sequence P of all prime numbers. However,
if a property turns up sufficiently frequently to occur almost
surely when the drawings are repeated, we conjecture with Cramér
that it is also shared by P."
A clearlywritten elementary introduction to Cramér's
ideas is contained in the following article:
D.L. Snell, "Chance in the Primes, Part II",
Chance
News 11.02
C. Bonanno and M.S. Mega, "Toward a dynamical model for prime numbers"
Chaos, Solitons and Fractals 20 (2004) 107118
[abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers. Our
approach is based on two complexity methods, the Computable Information Content and the Entropy Information
Gain, looking for analogies between the prime numbers and intermittency."
The main idea here is that the Manneville map T_{z} exhibits a phase
transition at z = 2, at which point the mean Algorithmic Information Content
of the associated symbolic dynamics is n/log n. n is a kind of iteration number.
For this to work, the domain of T_{z} [0,1] must be partitioned as
[0,0.618...] U [0.618...,1] where 1.618... is the golden mean.
The authors attempt to exploit the resemblance to the approximating function in the Prime
Number Theorem, and in some sense model the distribution of primes in dynamical terms,
i.e. relate the prime number series (as a binary string) to the orbits of the Manneville
map T_{2}. Certain refinements of this are then explored.
"We remark that this approach to study prime numbers is similar to the probabilistic
approach introduced by Cramér...that is we assume that the [binary] string [generated
by the sequence of primes]...is one of a family of strings on which there is a probability measure..."
(approximately) chronological bibliography
P. Doyle's English
translation of von Sternach's 1896 treatise on number theoretical random walks, associated with this
introductory article on the subject (scroll down to the section "Probability and the Riemann Hypothesis").
P. Erdös and A. Wintner, "Additive arithmetic functions and
statistical independence", American Journal of Mathematics
61 (1939) 713722.
P. Erdös and M. Kac, "The Gaussian law of errors in the theory of additive number
theoretic functions", American Journal of Mathematics 62 (1940) 738742.
notes on the ErdösKac theorem
H. Riesel, "The ErdösKac Theorem", in
Prime Numbers and Computer Methods for Factorization, 2nd
ed. (Birkhäuser, 1994) pp. 158159
"The Prime Number Theorem obtained by
statistical methods"  a heuristic argument from What is
Mathematics? by R. Courant and H. Robbins (1941)
M. Kac, "Probability methods in some problems of analysis and number
theory", Bulletin of the AMS (1949) 641665.
A. Rényi, "On the density of certain sequences of integers", Publ.
Inst. Math. Belgrade 8 (1955) 157162.
I.P. Kubilius, "Probability methods in number theory" (in Russian),
Usp. Mat. Nauk 68 (1956) 3166.
A. Rényi and P. Turán, "On a theorem of ErdösKac", Acta. Arith.
4 (1958) 7184.
D. Hawkins, "The random sieve",
Mathematics Magazine 31(1958) 13.
D. Hawkins, "Random sieves, II", Journal of Number Theory 6
(1974) 192200.
A. Denjoy, "Probabilites confirmant l'hypothese de Riemann sur les zeros
de zeta(s), C.R. Acad. Sci. Paris 259 (1964)
31433145.
Notes on Denjoy's probabilistic interpretation
of the Riemann Hypothesis from H. Edwards book
Riemann's Zeta Function
I.J. Good and R.F. Churchhouse, "The Riemann hypothesis and pseudorandom features of the Möbius
sequence", Mathematics of Computation 22 (1968) 857864
[abstract:] "A study of the cumulative sums of the Möbius function on the
Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a
number of conjectures. One of these is that any conjecture of the Mertens type, viz.
$M(N) = \sum_{n=1}^{N}\mu(n) \lt k(\sqrt(N))$
where $k$ is a positive constant, is false, and indeed, the authors conjecture that
$\Lim \sup {M(x)(x \log \log x)^{1/2}} = \sqrt{(12)/\pi}$"
S. Golomb, "A class of probability distributions on the integers", Journal
of Number Theory 2 (1970) 189192.
P. Billingsley, "Prime numbers and Brownian motion", American Mathematical
Monthly 80 (1973) 1099.
W. Neudecker and D. Williams, "The 'Riemann Hypothesis' for the Hawkins random
sieve" Compositio Mathematica 29 (1974) 197200.
P. Nanopoulos, "Loi de Dirichlet sur N^{*} et pseudoprobabilites",
C.R. Acad. Sci. Paris Ser. AB 280 (22) (1975) A1543A1546.
C.C. Heyde, "On asymptotic behavior
for the Hawkins random sieve", Proc. AMS 56 (1976) 277280
[abstract:] "This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve
of Eratosthenes. Analogues of the prime number theorem and Mertens' theorem have previously been obtained for this
sieve by classical probabilistic methods. In the present paper, sharper results akin to the Riemann hypothesis are
obtained by a more elegant martingale approach."
P. Diaconis, F. Mosteller and H. Onishi, "Second order terms for
the variances and covariances of the number of prime factors 
including the square free case", Journal of Number Theory
9 (1977) 187202.
C.C. Heyde, "A log log improvement to the
Riemann Hypothesis for the Hawkins random sieve" Ann. Prob. 6 no.5 (1978) 870875
[abstract:] "This paper is concerned with the Hawkins random sieve which is a probabilistic analouge of the sieve of
Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been
established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results
for both mantingales and tail sums of martingale differences to deduce a considerably improved log log replacement for the
Riemann hypothesis result."
J. Kubilius,
Probabilistic Methods in the Theory of Numbers (AMS, 1978)
P.D.T.A. Elliot,
Probabilistic Number Theory I: Meanvalue Theorems, Grundlehren der
Mathematischen Wissenschaften 239 (Springer, 1979)
P.D.T.A. Elliot, Probabilistic Number Theory II: Central Limit
Theorems, Grundlehren der Mathematischen Wissenschaften 240
(Springer, 1980)
R.O. Rabin, "Probabilistic algorithm for testing primality", Journal of Number Theory
12 (1980) 128138.
M. Shlesinger, "On the Riemann hypothesis: a fractal random walk approach", Physica A
138 (1986) 310319
[abstract:] "In his investigation of the distribution of prime numbers Riemann,
in 1859, introduced the zeta function with a complex argument. His
analysis led him to hypothesize that all the complex zeros of the
zeta function lie on a vertical line in the complex plane. The proof
or disproof of this hypothesis has been a famous outstanding problem
in mathematics. We are able to recast Riemann's Hypothesis into a
probabilistic framework connected to the fractal behavior of a lattice
random walk. Fractal random walks were introduced by P. Levy, and in
the continuum are called Levy flights. For one particular lattice
version of a Levy flight we show the connection to Weierstrass'
continuous but nowhere differentiable function. For a different
lattice version, using a Mellin transform analysis, we show how the
zeroes of the zeta function become the singularities of a complex
integrand which governs the behavior of a fractal random walk. The
laws of probability place restrictions on the locations of the zeroes
of the zeta function. No inconsistencies with probability theory are found if the Riemann Hypothesis is false."
V.K. Murty and M.R. Murty,
"An analog of the ErdösKac theorem for Fourier coefficients of modular
forms", Indian Journal of Pure and Applied Mathematics 15
(10) (1984) 10901101.
L. Smith and P. Diaconis, "Honest Bernoulli excursions", Journal of
Applied Probability 25 (1988) 464477
J.V. Armitage, "The Riemann
Hypothesis and the Hamiltonian of a quantum mechanical system" (section
5: "A random walk approximation to the Riemann Hypothesis"), from Number
Theory and Dynamical Systems, eds. M.M. Dodson and J.A.G. Vickers (LMS
Lecture Notes, series 134, Cambridge University Press), 153172.
"The connection between random walks and Brownian motion is wellknown
and so also the connection with the Schrödinger equation, on replacing
'time' with 'imaginary time'. In this section we use a random walk approach
to the OrnsteinUhlenbeck process (or the FokkerPlanck equation) to exhibit
a polynomial whose zeros, under a suitable limiting process, ought to be
the zeros of the Riemann zetafunction."
D. Williams, "Brownian motion and the Riemann zetafunction", from Disorder in Physical Systems (Clarendon Press, 1990) 361372.
J.L. Lucio and Y. Meurice, "Asymptotic properties of random walks
on padic spaces", University of Iowa Preprint 9033, 15 (1990)
S.W. Golomb, "Probability,
information theory, and prime number theory", Discrete
Mathematics 106107 (1992) 219229
[abstract:] "For any probability distribution D =
{\alpha(n)} on Z^{+}, we define. . . the
probability in D that a 'random' integer is a multiple of
m; and . . . the probability in D that a 'random'
integer is relatively prime to k. We specialize this general
situation to three important families of distributions . . . Several
basic results and concepts from analytic prime number theory are
revisited from the perspective of these families of probability
distributions, and the Shannon entropy for each of these families is
determined."
K.S. Alexander, K. Baclawski and G.C. Rota, "A stochastic
interpretation of the Riemann zeta function", Proceedings of the National Academy of Sciences USA
90 (1993) 697699
[abstract:] "We give a stochastic process for which the terms of the Riemann zeta function occur as the probability distributions of
the elementary random variables of the process."
S. Albeverio and W. Karwowski,
"A Random Walk on pAdics  the Generator and Its Spectrum",
Stochastic Processes and their Applications 53 (1994) 122.
G. Tenenbaum, Introduction to Analytic and Probabilistic Number
Theory, Cambridge Studies in Advanced Mathematics 46
(C.U.P., 1995)
N. Boston,
"A probabilistic generalization of the Riemann zeta function",
Analytic Number Theory, Vol. 1, Progr. Math. 138,
(Birkhauser, 1996) 155162.
P. Joshi and S. Chakraborty, "Moments of Cauchy order statistics via
Riemann zeta functions", from Statistical Theory and Applications
(editors H.N. Nagaraja, et al.) 111127 (Springer 1996)
C. Calude,
P. Hertling
and B. Khoussainov, "Do the zeros of Riemann's zetafunction form a random sequence?",
Bulletin of the EATCS 62
(1997) 199207
J. Chang and
Y. Peres, "Ladder Heights, Gaussian random walks, and the Riemann zeta function", Annals of Probability,
25 (1997) 787802.
[abstract:] "Let $\{S_n:n\ge 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and
let $\tau$ be the time at which the random walk first takes a positive value, so that $S_\tau$ is the first ladder height. Then the expected value
$E_\theta S_\tau$, originally defined for $\theta>0$, may be extended to be an analytic function of the complex variable $\theta$ throughout the
complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about $\theta=0$ may be
written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was
known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge
of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting
expected overshoot."
P. Biane, J. Pitman, and M. Yor,
"Probability
laws related to the Jacobi theta and Riemann zeta functions, and
Brownian excursions", Bulletin of the AMS 38 (2001) 435465.
J. Pitman and M. Yor, "Infinitely
divisible laws associated with hyperbolic functions", Canadian Journal of Mathematics 55 no.2 (2003) 292330
[abstract:] "The infinitely divisible distributions of positive random variables
C_{t}, S_{t} and T_{t} with Laplace transforms in x
(1/(cosh(2x)^{1/2}))^{t}, ((2x)^{1/2}/sinh((2x)^{1/2}))^{t},
and (tanh((2x)^{1/2})/(2x)^{1/2})^{t}
respectively are characterized for various t > 0 in a number of different ways:
by simple relations between their moments and cumulants, by corresponding relations between
the distributions and their Levy measures, by recursions for their Mellin transforms, and by
differential equaitons satisfied by their Laplace transforms.
Some of these results are interpreted probabilistically via known appearances of these distributions for
t = 1 or 2 in the description of the laws of various functionals of Brownian
motion and Bessel processes, such as the heights and lengths of excursions of a onedimensional
brownian motion. The distributions of C_{1} and S_{2} are also
known to appear in the Mellin representations of two important functions in analytic number
theory, the Riemann zeta function and the Dirichlet Lfunction associated with the
quadratic character modulo 4.
Related families of infinitely divisible laws, including the gamma, logistic and
generalised hyperbolic secant distributions, are derived from S_{t} and
C_{t} by operations such as Brownian subordinations, exponential tilting,
and weak limits, and characterized in various ways."
Marc Yor is acknowledged for "fascinating discussions on the connection between
[a theorem in the following paper] and planar Brownian motion" here:
C. Hughes, J. Keating, and N. O'Connell,
"On the
Characteristic Polynomial of a Random Unitary Matrix", Communications in Mathematical Physics
220 (2001) 429451
S. Asmussen, P. Glynn and J. Pitman, "Discretization error in simulation
of onedimensional reflecting Brownian motion", Annals of Applied Probability
5 (1995) 875896
Analytic and Probabilistic Methods in Number Theory 
New Trends in Probability and Statistics, Volume 4 (VSP, 1997)
C. M. Bender, S. Boettcher, L. R. Mead,
"Random walks in
noninteger dimension", Journal of Mathematical Physics 35 (1994) 368388
"One can define a random walk on a hypercubic lattice in a space of
integer dimension D... In this paper we propose a random walk which
gives acceptable probabilities for all real values of D. This Ddimensional
random walk is defined on a rotationallysymmetric geometry consisting
of concentric spheres. We give the exact result for the probability of
returning to the origin for all values of D in terms of the Riemann
zeta function. This result has a numbertheoretic interpretation."
M. Mystkowski, "Random
walk on padics with nonzero killing part", Reports on Mathematical Physics 34 (1994) 133141
S. Albeverio and W. Karwowski, "A
random walk on padics  the generator and its spectrum", Stochastic Processes and their Applications, 53
(1994) 122
S. Albeverio, W. Karwowski and X. Zhao,
"Asymptotics
and spectral results for random walks on padics", Stochastic Processes and their
Applications, 83 (1999)
M. Wolf, "Random walk on the prime numbers",
Physica A 250
(1998) 335.
I. Vardi,
"Deterministic percolation",
Communications in Mathematical Physics 207 (1999) 4366
[excerpt from introduction:] "...percolation theory has been of great interest in physics,
as it is one of the simplest models to exhibit phase transitions. In this paper, I will examine how questions of percolation
theory can be posed in a deterministic setting. Thus deterministic percolation is
the study of unbounded walks on a single subset of a graph, e.g., defined by number
theoretic conditions. This might be of interest in physics and probability theory as
it studies percolation in a deterministic setting and in number theory where it can be
interpreted as studying the disorder inherent in the natural numbers."
I. Vardi, "Prime
percolation", Experimental Mathematics 7 (1998) 275288
[abstract:] "This paper examines the question of whether there is an unbounded walk of
bounded step size along Gaussian primes. Percolation theory predicts that for a low enough
density of random Gaussian integers no walk exists, which suggests that no such walk exists
along prime numbers, since they have arbitrarily small density over large enough regions. In
analogy with the Cramér conjecture, I construct a random model of Gaussian primes and show
that an unbounded walk of step size $k\sqrt{\logz}$ at $z$ exists with probability 1 if
$k \gt \sqrt{2\pi\lambda_{c}}$ and does not exist with probability 1 if
$k \lt \sqrt{2\pi\lambda_{c}}$ where $\lambda_{c}$ ~ 0.35 is a constant in continuum
percolation, and so conjecture that the critical step size for Gaussian primes is also
$\sqrt{2\pi\lambda_{c}\logz}$.
J.F. Burnol,
"The Explicit Formula and a
propagator" (preprint, 09/98)
[abstract:] "I give a new derivation of the Explicit
Formula for the general number field K, which treats all primes in exactly the same way, whether they are
discrete or archimedean, and also ramified or not. In another token, I
advance a probabilistic interpretation of Weil's positivity criterion, as
opposed to the usual geometrical analogies or goals. But in the end, I
argue that the new formulation of the Explicit Formula signals a specific
link with Quantum Fields, as opposed to the
HilbertPólya operator idea (which leads rather to Quantum Mechanics)."
[excerpt:] "To express his positivity criterion Weil uses conventions slightly
distinct from ours. He moves the local terms to be together with the poles,
and makes a shift of 1/2 in the Mellin transform. In this way he gets a distribution
C and translates the Riemann Hypothesis into a positivity criterion:
$C(F \star F^{\tau}) \geq 0$
for an arbitrary testfunction F on C...In the function field case
$C(F \star F^{\tau})$ can be given a geometric interpretation as an intersection
number of cycles on an algebraic surface, and the positivity follows from the
Hodge Index Theorem...
But another interpretation is possible that does not seem to have been pushed
forward so far. To prove that a number is nonnegative it is enough to exhibit it
as the variance of a random variable. In our case this means taht there should be
a generalized, stationary, zero mean, stochastic process with C as "time"
whose covariance would be C. That is we have a probability measure $mu$ on
the distribgutions on the classes of ideles...
If such a probability measure $mu$ could be constructed, corresponding to an
'arithmetic stochastic process', then the Riemann Hypothesis would follow of occurs."
C. DonatiMartin, Zhan Shi and M. Yor, "The joint law of the last
zeros of Brownian motion and of its Lévy transform",
Ergodic Theory and Dynamical Systems 20 (2000) 709725.
C. Castro (Perelman),
"pAdic stochastic
dynamics, supersymmetry and the Riemann conjecture" (preprint 01/01)
"Supersymmetry, padic stochastic dynamics, Brownian motion,
FokkerPlanck equation, Langevin equation, prime number random distribution,
random matrices, padic fractal strings, the adelic condition, etc.
..are all deeply interconnected in this paper."
C. Castro and J. Mahecha, "Fractal supersymmetric quantum
mechanics, geometric probability and the Riemann Hypothesis", International Journal of
Geometric Methods in Modern Physics 1 no. 6 (2004) 751793
[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann
zetafunction are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on
Supersymmetric QM, whose potential was related to the GaussJacobi theta series, allows to
provide the proper framework to construct the well defined algorithm to compute the probability
to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes
the answer to the very difficult question whether the probability that the RH is true is
indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework
to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function
sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced
in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an
infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal
supersymmetric quantum mechanical (SUSYQM) model to implement the HilbertPólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s
for its spectrum. Quantum inverse scattering methods related to a fractal potential
given by a Weierstrass function (continuous but nowhere differentiable) are applied to the
analog of the fractal analog of the CBC (ComtetBandraukCampbell) formula in SUSY QM. It
requires using suitable fractal derivatives and integrals of irrational order whose parameter
$\beta$ is onehalf the fractal dimension (D = 1.5) of the Weierstrass function. An
ordinary SUSYQM oscillator is also constructed whose eigenvalues are of the form
$\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the
function sinh(s). Finally, we discuss the relationship to
the theory of 1/f noise."
P.F. Kelly and T. Pilling,
"Physically inspired
analysis of prime number constellations" (preprint, 08/01)
"We adopt a physically motivated empirical approach to the characterisation of the distributions
of twin and triplet primes within the set of primes, rather than in the set of all natural numbers.
Remarkably, the occurrences of twins or triplets in any finite sequence of primes are like fixedprobability
random events. The respective probabilities are not constant, but instead depend on the length of the
sequence in ways that we have been able to parameterise. For twins the "decay constant'' decreases as
the reciprocal of the logarithm of the length of the sequence, whereas for triplets the falloff is faster: decreasing
as the square of the reciprocal of the logarithm of the number of primes. The manner of the decrease is consistent
with the HardyLittlewood Conjectures, developed using purely number theoretic tools of analysis."
A.N. Kochubei, Pseudodifferential
Equations and Stochastics over nonArchimedian Fields
(Marcel Dekker, 2001)
(from publisher's description) "This stateoftheart reference
provides comprehensive coverage of the most recent developments in the
theory of nonArchimedean pseudodifferential equations and its
application to stochastics and mathematical physics offering current
methods of construction for stochastic processes on the field of
padic numbers and related structures. Develops a new theory for
parabolic equations over nonArchimedean fields in relation to
Markov processes!"
D.L. Snell, "Chance in the Primes, Part II",
Chance
News 11.02
This clearlywritten elementary article provides examples of "how chance has been
used in the study of prime numbers". Probabilistic 'proof' of the Prime Number Theorem, Harald
Cramér's 'random primes', primebased random walks, sieving, cryptography and the
Möbius
function are all discussed.
notes on
the probabilities associated with "visible point" lattice problems and
the Riemann zeta function
F. Todor, "A probability interpretation for the Zeta function of Riemann and some
technical regularization for spectral representation" (paper presented to
the 11th Conference on Applied and Industrial Mathematics, Romania, May 2003)
G. Chaitin, "Some
thoughts on the Riemann Hypothesis (preprint, 06/03)
[excerpt:] "I have always had an interest in probabilistic methods
in elementary number theory. This was one of the things that inspired
me to come up with my definition of algorithmic randomness
and to find algorithmic randomness in arithmetic in connection with
diophantine equation."
Chaitin goes on to recommend the following:
G. Pólya, "Heuristic reasoning in the theory of numbers",
American Mathematical Monthly 66 (1957) 375384
B.M. Hambly and
M.L. Lapidus, "Random fractal strings: their
zeta functions, complex dimensions and spectral asymptotics" (preprint 02/04)
[abstract:] "In this paper a string is a sequence of positive nonincreasing real numbers
which sums to one. For our purposes a fractal string is a string formed from the lengths of
removed subintervals created by a recursive decomposition of the unit interval. By using
the so called complex dimensions of the string, the poles of an associated zeta function, it
is possible to obtain detailed information about the behaviour of the asymptotic properties
of the string. We consider random versions of fractal strings. We show that using a random
recursive selfsimilar construction it is possible to obtain similar results to those for
deterministic selfsimilar strings. In the case of strings generated by the excursions of
stable subordinators, we show that the complex dimensions can only lie on the real line.
The results allow us to discuss the geometric and spectral asymptotics of onedimensional
domains with random fractal boundary."
S.N. Evangelou, D.E. Katsanos, "Quantum
correlations from Brownian diffusion of chaotic levelspacings" (preprint 04/04)
[abstract:] "Quantum chaos is linked to Brownian diffusion of the underlying quantum energy levelspacing
sequences. The levelspacings viewed as functions of their order execute random walks which imply uncorrelated
random increments of the levelspacings while the integrability to chaos transition becomes a change from Poisson to
Gauss statistics for the levelspacing increments. This universal nature of quantum chaotic spectral correlations is
numerically demonstrated for eigenvalues from random tight binding lattices and for zeros of the Riemann zeta function."
G. Corso, "Families
and clustering in a natural numbers network", Phys. Rev. E 69 (2004)
"We develop a network in which the natural numbers are the vertices. The decomposition
of natural numbers by prime numbers is used to establish the connections. We perform data
collapse and show that the degree distribution of these networks scales linearly with the
number of vertices. We explore the families of vertices in connection with prime numbers
decomposition. We compare the average distance of the network and the clustering
coefficient with the distance and clustering coefficient of the corresponding random
graph. In case we set connections among vertices each time the numbers share a common
prime number the network has properties similar to a random graph. If the criterion for
establishing links becomes more selective, only prime numbers greater than p_{l} are used to
establish links, where the network has high clustering coefficient."
B. Green and T. Tao, "The primes
contain arbitrarily long arithmetic progressions" (preprint 04/04)
[abstract:] "We prove that there are arbitrarily long arithmetic progressions of primes. There
are three major ingredients. The first is Szemerédi's theorem, which asserts that any subset of the
integers of positive density contains progressions of arbitrary length. The second, which is the main
new ingredient of this paper, is a certain transference principle. This allows us to deduce from
Szemerédi's theorem that any subset of a sufficiently pseudorandom set of positive relative density
contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and
Yildirim. Using this, one may place the primes inside a pseudorandom set of 'almost primes' with
positive relative density."
[from proof outline, p.4] "Perhaps surprisingly for a result about primes, our paper has at least
as much in common with the ergodictheoretic approach as it does with the harmonic analysis approach
of Gowers. We will use a language which suggests this close connection, without actually relying
explicitly on any ergodic theoretical concepts".
T. Tao, "Obstructions to uniformity, and arithmetic
patterns in the primes" (preprint 05/05, submitted for special edition of
Quarterly J. Pure Appl. Math. in honour of John Coates)
[abstract:] "In this expository article, we describe the recent approach, motivated by
ergodic theory, towards detecting arithmetic patterns in the primes, and in particular
establishing that the primes contain arbitrarily long arithmetic progressions. One of the
driving philosophies is to identify precisely what the obstructions could be that prevent
the primes (or any other set) from behaving 'randomly', and then either show that the
obstructions do not actually occur, or else convert the obstructions into usable
structural information on the primes."
B. Kra, "The GreenTao
Theorem on arithmetic progressions in the primes: an ergodic point of view", Bull. Amer. Math. Soc. 43
(2006), 323
[abstract:] "A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic
progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that
there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic
number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist
infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some
of the ideas in the proof, concentrating on the connections to ergodic theory."
Z. Rudnick, "A Central Limit Theorem for the spectrum of the
modular domain" (to appear in Annales Henri Poincaré)
[abstract:] "We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth
counting functions. We show that in a certain regime, these have Gaussian fluctuations."
K.H. Knuth, "Deriving laws from ordering
relations", In: G.J. Erickson, Y. Zhai (eds.), Bayesian Inference and
Maximum Entropy Methods in Science and Engineering, AIP Conference Proceedings 707
(2003) 204235.
[abstract:] "The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication
among logical statements to degrees of implication, which are manipulated using rules derived from consistency with
Boolean algebra. These rules are known as the sum rule, the product rule and Bayes' Theorem, and the measure
resulting from this generalization is probability. In this paper, I will describe how Cox's technique can be further
generalized to include other algebras and hence other problems in science and mathematics. The result is a
methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to
derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in
probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend
these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be
demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum
mechanics."
K.H. Knuth, "Lattice duality: The origin of probability
and entropy", Neurocomputing 67 C (2005) 245274
[author's description:] "This paper shows how a straightforward generalization of the zeta
function of a distributive lattice gives rise to bivaluations that
represent degrees of belief in Boolean lattices of assertions and degrees
of relevance in the distributive lattice of questions. The distributive
lattice of questions originates from Richard T. Cox's definition of a
question as the set of all possible answers, which I show is equivalent to
the ordered set of downsets of assertions. Thus the Boolean lattice of
assertionns is shown to be dual to the distributive lattice of questions
in the sense of Birkhoff's Representation Theorem. A straightforward
correspondence between bivaluations generalized from the zeta functions
of each lattice give rise to bivaluations that represent probabilities in
the lattice of assertions and bivaluations that represent entropies and
higherorder informations in the lattice of questions."
Probability and Number Theory 
Kanazawa 2005, eds. S. Akiyama, et al. (Mathematical Society of Japan, 2008)
[publisher's description:] "This volume is the proceedings of the International Conference on Probability and
Number Theory held at Kanazawa, Japan, in June 2005 and includes several survey articles on probabilistic number
theory and research papers on various recent topics around the border area between probability theory and number
theory. This volume is useful for all researchers and graduate students who are interested in probability theory
and number theory."
A. Kubasiak, J. Korbicz, J. Zakrzewski, M. Lewenstein, "FermiDirac statistics and the number theory" (preprint 07/05)
[abstract:] "We relate the FermiDirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given
integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic
expressions for cumulants of the probability distribution of the number of different partitions."
C.S. Calude and M. Stay, "Natural
halting probabilities, partial randomness, and zeta functions" (preprint 01/06)
[abstract:] "We introduce the natural halting probability and the natural complexity of a Turing machine
and we relate them to programsize complexity and Chaitin's halting probability. A classification of Turing machines
according to their natural (Omega) halting probabilities is proposed: divergent, convergent and tuatara. We prove the
existence of universal convergent and tuatara machines. Various results on randomness and partial randomness are proved.
For example, we show that the natural halting probability of a universal tuatara machine is c.e. and random. A new type
of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic)
randomness  which cannot be characterised in terms of plain complexity  various types of partial randomness admit such
characterisations."
O. Shanker, "Zeroes of Riemann zeta function
and Hurst exponent" (preprint 01/06)
[abstract:] "The theory underlying the location of the zeros of the Riemann zeta function is one of the most intriguing unsolved problems.
It is interesting to physicists because of the HilbertPólya Conjecture, that the nontrivial zeros of the zeta function correspond to
the eigenvalues of some positive operator. Since there is no proof yet for this conjecture, it is important to study the properties of the
locations of the zeroes empirically using a variety of methods. In this paper we use the rescaled range analysis to study the spacings
between successive zeroes. We find that for large orders of the zeroes the spacings seem to have a Hurst exponent of about 0.095. This implies
that the distribution has a high fractal dimension, and shows a lot of detailed structure.
The distribution appears to be of the antipersistent fractional Brownian motion type, with a
significant degree of antipersistence."
Wang Liang and Huang Yan, "Pseudo Random test of prime
numbers" (preprint 03/2006)
[abstract:] "The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict
theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for
randomness. In this paper, we just apply these methods to study the distribution of primes. Here the binary sequence
constructed by second difference of primes is used as samples. We find this sequence can't reach all the 'random standard'
of FIPS 1401/2, but still show obvious random feature. The interesting selfsimilarity is also observed in this sequence.
These results add the evidence that prime numbers is a chaos system."
M.Katori, M. Izumi and N. Kobayashi, "Two Bessel bridges conditioned never to collide, double
Dirichlet series, and Jacobi theta function" (preprint 11/07)
[abstract:] "It is known that the moments of the maximum value of a onedimensional conditional
Brownian motion, the threedimensional Bessel bridge with duration 1 started from the origin, are
expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which
noncolliding condition is imposed. We show that the moments of the maximum value is then expressed
using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions
and its derivatives. Since the present system will be provided as a diffusion scaling limit of a
version of vicious walker model, the ensemble of 2watermelons with a wall, the dominant terms in
longtime asymptotics of moments of height of 2watermelons are completely determined, for which only
the first moment, i.e. the average height, was recently studied by Fulmek by a method of enumerative
combinatorics."
L. Lacasa, B. Luque, O. Miramontes, "Phase transition and computational
complexity in a stochastic prime number generator" (preprint 12/2007)
[abstract:] "We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a
continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase
where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition,
both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model
as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The
computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easyhardeasy
pattern. We finally relate the nature of the phase transition to an averagecase classification of the problem."
Austrian National Research: Network
Analytic Combinatorics and Probabilistic Number Theory (20062008)
E. Kowalski, The
Large Sieve and its Applications, Arithmetic Geometry, Random Walks and Discrete Groups, Cambridge
Tracts in Mathematics 175 (CUP, 2008)
N.Kobayashi, M. Izumi, M. Katori, "Maximum distributions of noncolliding Bessel bridges" (preprint 08/2008)
[abstract:] "The onedimensional Brownian motion starting from the origin at time $t=0$, conditioned to return to the origin at time $t=1$ and to stay positive during time
interval $0 < t < 1$, is called the Bessel bridge with duration 1. We consider the $N$particle system of such Bessel bridges conditioned never to collide with each other in
$0 < t < 1$, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximumvalues of paths attained in the time
interval $t \in (0,1)$ are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution
function of maximum value is exactly determined for general $N$. We show that the present $N$path system of noncolliding Bessel bridges is realized as the
positiveeigenvalue process of the $2N \times 2N$ matrixvalued Brownian bridge in the symmetry class C. Using this fact computer simulations are performed and numerical
results on the $N$dependence of the maximumvalue distributions of the inner paths are reported. The present work demonstrates that the extremevalue problems of
noncolliding paths are related with the random matrix theory, representation theory of symmetry, and the number theory."
S. Torquato, A. Scardicchio, C.E. Zachary, "Point processes in arbitrary dimension from
fermionic gases, random matrix theory, and number theory" (preprint 09/2008)
[abstract:] "It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process
on the real line. Here we analytically provide exact generalizations of such a point process in $d$dimensional Euclidean space for any $d$, which are special cases of
determinantal processes. In particular, we obtain the $n$particle correlation functions for any n, which completely specify the point processes. We also demonstrate that
spinpolarized fermionic systems have these same $n$particle correlation functions in each dimension. The point processes for any $d$ are shown to be hyperuniform. The
latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law $r^{(d+1)}$. We graphically
display one and twodimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be
characterized by an effective "hardcore" diameter that grows like the square root of $d$. The nearestneighbor distribution functions for these point processes are also
evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius $r$ in dimension $d$ behaves like
a Poisson point process but in dimension $d+1$ for large r and finite $d$. We also show that as $d$ increases, the point process behaves effectively like a sphere packing
with a coverage fraction of space that is no denser than $1/2^d$."
N. Laskin, "Some Applications
of the Fractional Poisson Probability Distribution" (preprint 12/2008)
[abstract:] "New physical and mathematical applications of recently invented
fractional Poisson probability distribution have been presented. As a physical
application, a new family of quantum coherent states have been introduced and studied.
Mathematical applications are related to the number theory. We have developed fractional
generalization of the Bell polynomials, the Bell numbers, and the Stirling numbers of the
second kind. The fractional Bell polynomials appearance is natural if one evaluates the
diagonal matrix element of the evolution operator in the basis of newly introduced
quantum coherent states. The fractional Stirling numbers of the second kind have been
introduced and applied to evaluate skewness and kurtosis of the fractional Poisson
probability distribution function. A new representation of the Bernoulli numbers in terms
of fractional Stirling numbers of the second kind has been found. In the limit case when
the fractional Poisson distribution becomes the wellknown Poisson probability
distribution all of the above listed new developments and implementations turn into the
wellknown results of the quantum optics and the number theory."
B. Holdom, "Correlations, scale invariance and the Riemann Hypothesis" (preprint 03/2009)
[abstract:] "Negative correlations in the distribution of prime numbers are found to display a scale invariance. There are similarities and differences when compared to the scale invariant correlations of fractional Brownian motion. We conjecture that a violation of the Riemann hypothesis is equivalent to a breakdown of the scale invariance."
G. GarciaPerez, M. Angeles Serrano and M. Boguna, "The complex architecture of primes and natural numbers" (preprint 02/2014)
[abstract:] "Natural numbers can be divided in two nonoverlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameterfree nonMarkovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cram\'er's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cram\'er's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help not only to conjecture but also to prove results about primes and the complex architecture of natural numbers."
B.M. Weon and Y. Kim, "Randomness in coin tosses and last digits of primes" (preprint 09/2014)
[abstract:] "Randomness is a central concept to statistics and physics. A new statistical analysis provides evidence that tossing coins and finding last digits of prime numbers are identical problems regarding equally likely outcomes. This analysis explains why randomness of equally likely outcomes is valid at large numbers."
G. França and A. LeClair, "On the validity of the Euler product inside the critical strip" (preprint 10/2014)
[abstract:] "The Euler Product Formula relates Riemann's zeta function $\zeta(s)$ to an infinite product over primes, and is known to be valid for $\Re(s) > 1$. We provide arguments that the formula is actually valid for $\Re(s) > 1/2$ and $\Im(s)\neq 0$ due to the conditional convergence of the infinite product in this regime. The argument relies on four ingredients: the prime number theorem, an Abel transform, a central limit theorem for the Random Walk of the Primes series $\sum_p\cos(t\log p)$, where $p$ is a prime number, and the Cauchy criterion for convergence. The significance of $\Re(s) > 1/2$ arises from the universality of the $N^{1/2}$ growth of fluctuations in various central limit theorems for independent and weakly dependent random processes, which are common in statistical physics for systems of size $N$. Numerical evidence of this surprising result is presented, and some of its consequences are discussed."
D. Ostrovsky, "On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral" (preprint 06/2015)
[abstract:] "Rescaled Mellintype transforms of the exponential functional of the Bourgade–Kuan–Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot–Bacry–Muzy. The conjecture implies that a nontrivial, loginfinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral."
A. Abdesselam, "Towards threedimensional conformal probability" (preprint 10/2015)
[abstract:] "In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higherdimensional conformal bootstrap which has developed at a breathtaking pace over the last five years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for a very general first and secondquantized Kolmogorov–Chentsov Theorem. Firstquantized refers to regularity of sample paths. Secondquantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. Finally, we present a summary of progress made on a $p$adic hierarchical model and point out possible connections to number theory."
A. LeClair, "Riemann Hypothesis and random walks: The zeta case" (preprint 01/2016)
[abstract:] "In previous work it was shown that if certain series based on sums over primes of nonprincipal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$function is valid to the right of the critical line $\Re (s) > 1/2$, and the Riemann Hypothesis for this class of $L$functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$functions. We use our results to argue that $ S_\delta (t) \equiv \lim_{\delta \to 0^+} \dfrac{1}{\pi} \arg\zeta(\tfrac{1}{2}+ \delta + it) =
O(1)$, and that it is nearly always on the principal branch. We conjecture that a 1point correlation function of the Riemann zeros has a normal distribution. This leads to the construction of a probabilistic model for the zeros. Based on these results we describe a new algorithm for computing very high Riemann zeros as a kind of stochastic process, and we calculate the $10^{100}$th zero to over 100 digits."
S.C.L. Srivastava, A. Lakshminarayan, S. Tomsovic and A. Bäcker, "Ordered level spacing probability densities" (preprint 10/2018)
[abstract:] "Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3\times 3$ matrix model. The closest neighbour density is generalized to the $k$th closest neighbour spacing density, which allows for investigating longrange correlations. For larger $k$ the probability density of $k$th closest neighbour spacings is well described by a Gaussian. Using these $k$th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found."
G. Chavez and A. Allawala, "Prime zeta function statistics and Riemann zerodifference repulsion" (preprint 02/2021)
[abstract:] "We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ``repulsion'' between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the $1$line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zerodifference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related Dirichlet $L$functions."
G. Mussardo and A. LeClair, "Randomness of Mobius coefficents and brownian motion: Growth of the Mertens function and the Riemann Hypothesis" (preprint 01/2021)
[abstract:] "The validity of the Riemann Hypothesis (RH) on the location of the nontrivial zeros of the Riemann zetafunction is directly related to the growth of the Mertens function: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \simeq x^{1/2+\epsilon}$. We show that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function. To this aim, we focus the attention on the squarefree numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of squarefree numbers, the average number of prime divisors, the Erdos–Kac theorem for squarefree numbers, etc. These results lead us to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk, therefore with an asymptotic behaviour given by $x^{1/2+\epsilon}$. We also argue how the Riemann Hypothesis implies the Generalised Riemann Hypothesis for the Dirichlet $L$functions. Next we study the local properties of the Mertens function dictated by the Mobius coefficients restricted to the squarefree numbers. We perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity. The successful outputs of all these tests (with a level of confidence of 99% that all the subsequences analyzed are indeed random) can be seen as impressive experimental confirmations of the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however ridiculously improbable."
papers by A. Khrennikov, et al. on padic
probability, etc.
A. Khrennikov
, "padic probability and statistics", Dokl. Akad.
Nauk 322 (1992) 10751079.
D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors
of random dynamical systems over padic numbers and a model of noisy cognitive
processes", Physica D 130 (1999) 112
M. Endo and A. Khrennikov "Unboundedness of the padic Gaussian
distribution" Izvestia Akad. Nauk USSR, ser. Matem., 56, no. 5,
(1992) 11161103.
A. Khrennikov, "Axiomatics of the padic theory of probability",
Dokl. Akad. Nauk 26, no. 5 (1992) 10751079.
A. Khrennikov, "padic statistical models", Dokl. Akad. Nauk
330, no. 3 (1993) 300304.
A. Khrennikov, "padic probability theory and its applications. The
principle of statistical stabization of frequencies", Theoretical and
Mathematical Physics 97, no. 3 (1993) p. 348363.
A. Khrennikov, "Bernoulli probabilities with padic values",
Dokl. Akad. Nauk 338, No. 3 (1994) 313316.
A. Khrennikov, "An algorithmic approach to padic probability
theory", Dokl. Akad. Nauk 335, no. 1 (1994) 3538.
A. Khrennikov, "On probablity distributions on the field of padic
numbers", Theory of Probability and Applications 40 no.1 (1995)
189192.
A. Khrennikov, "An extension of the frequency approach of R. von Mises and
the axiomatic approach of N.A. Kolmogorov to the padic theory of
probability", Theory of Probability and Applications 40, no. 2
(1995) 458463.
A. Khrennikov, "The statistical simulation over fields of padic
numbers", Matem. Modelirovanie 7, no. 4 (1995) 8798.
A. Khrennikov, "A limit theorem for padic probabilities",
Izvestia Akad. Nauk., ser. Matem., 59, no. 3 (1995) 207223.
M. Endo and A. Khrennikov, "On the annihilators of the padic Gaussian
distributions", Comm. Math. Univ. Sancti Pauli 144, no. 1
(1995) 105108.
A. Khrennikov, "padic analogues of the law of large numbers
and the central limit theorem", Indag. Math. 8 (1) (1997) 6177.
A. Khrennikov, "padic asymptotic of Bernoulli probabilities",
Theory of Probability and its Applications 42, no. 4 (1997)
839845.
A. Khrennikov, "The Bernoulli theorem for probabilities that take
padic values", Dokl. Akad. Nauk 354, no. 4 (1998)
461464 [also Dokl. Math. 55 no. 3, 402405.]
A. Khrennikov, "padic behaviour of Bernoulli probabilities",
Statistics and Probability Letters 37, no. 4 (1998) 375380.
A. Khrennikov, S. Yamada, and A. van Rooij, "Measuretheoretical approach
to padic probability theory", Annals Math. Blaise Pascal
6, no. 1 (1999) 2132.
A. Khrennikov, "padic information spaces, infinitely small
probabilities and anomalous phenomena", Journal of Scientific
Exploration 4, no. 13 (1999) 665680.
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