some reformulations of the Riemann Hypothesis
The two classic reformulations of the Riemann Hypothesis are
"M(x) = o(x^{1/2 + a}) for all a
> 0" (WWN notes) and
"psi(x)  x = o(x^{1/2 + a})
for all a > 0" (WWN notes).
Here M is a step function satisfying M(0) = 0, and constant except at positive integers, having a jump of $\mu(n)$
at each n. Recall that the Möbius function $\mu(n)$
is defined to be zero if n is divisible by a square, and is
otherwise equal to (1)^{k} where k is the number
of distinct prime factors in n.
The function psi(x) is Chebyshev's
primecounting function, which the Prime Number Theorem
tells us is asymptotic to x.
Recall that the Landau notation f(x) = o(g(x))
means that lim_{x > 0} f(x)/g(x) = 0.
The following are all notes from WWN's workinprogress
"Zetafunctions and associated Riemann Hypotheses":
Equivalences to Riemann Hypothesis
The above contains links to abstracts of seven articles which are only accessible to
subscribers to MathSciNet. They can be alternatively accessed here:
V.V. Volchov, "On
an equality equivalent to the Riemann Hypothesis, Ukr. Math. J. 47 No.3 (1995) 491493
F. Amoroso, "On
the heights of a product of cyclotomic polynomials, Rend. Semin. Mat. Torino
53 No.3 (1995) 183191
A. Verjovsky, "Discrete
measures and the Riemann hypothesis, Kodai Math. J. 17 No.3 (1994) 596608
J. AlcántaraBode, "An
integral equation formulation of the Riemann hypothesis, Integral Equations Oper. Theory 17 No.2 (1993) 151168
A. Verjovsky, "Arithmetic,
geometry and dynamics in the unit tangent bundle of the modular orbifold", Dyamical Systems. Proceedings of
the 3rd international school of dynamical systems, Santiago de Chile, 1990 (R. Bamon, et.al., eds.) Longman Scientific and Tehcnical Pitman Res. Notes Math. Ser 285 (1993)
253298
W. Barrett, et.al., "On the spectral radius of a (0,1) matrix related to Mertens' function", Linear Algebra
Appl. 107 (1988) 151159
V.M. Popov, "On stability properties which are equivalent to Riemann hypothesis", Libertas Math.
5 (1985) 5561
RH equivalence to statement involving error
term in Prime Number Theorem
RH equivalence to statement involving order
of pi(x)li(x)
M. Riesz's equivalence from the article
"Sur l'hypothèse de Riemann", Acta Math. 40 (1916) 185190
Bombieri's refinement of Weil's positivity
criterion
XianJin Li's criterion
G. Robin's equivalence involving the 'sum of
divisors' function
Nicolas' equivalence involving Euler's totient
function
Massias, Nicolas and Robin's equivalence
involving the maximum order of an element in the symmetric group
The Riemann Hypothesis can also be reformulated in terms of a problem
involving Farey sequences. This is dealt with in the following:
J. Franel, "Les suites de Farey et les problemes des nombres
premiers." Gottinger Nachrichten, 198201 (1924)
E. Landau, "Bemerkungen zu der vorstehenden Abhandlung
von Herrn Franel." Gottinger Nachrichten, 202206 (1924)
A. Fujii, "A remark on the Riemann hypothesis."
Comment. Math. Univ. St. Pauli 29 (1980), 195201
A. Fujii, "Some explicit formulae in the theory
of numbers. A remark on the Riemann Hypothesis." Proc. Japan Acad.,
Ser. A 57(1981), 326330
S. Kanemitsu, and M. Yoshimoto, "Farey series and
the Riemann hypothesis." Acta Arith. 75 (1996), no.
4, 351374
S. Kanemitsu and M. Yoshimoto, "Farey series and
the Riemann hypothesis. III." Ramanujan J. 1 (1997),
no. 4, 363378
J. Kopriva, "Contribution to the relation of the
Farey series to the Riemann hypothesis on the zeros of the zeta function
(Czech), Casopis Pest. Mat. {\bf 78} (1953), 4955
J. Kopriva, "Contribution to the relation of the
Farey series to the Riemann hypothesis" (Czech), Casopis Pest.
Mat. 79 (1954), 7782
M. Mikolas, "Sur l'hypothese de Riemann." C.
R. Acad. Sci. Paris 228 (1949), 633636
M. Mikolas, "Farey series and their connection with
the prime number problem. I." Acta Univ. Szeged. Sect. Sci. Math.13
(1949), 93117
M. Mikolas, "Farey series and their connection with
the prime number problem. II." Acta Univ. Szeged. Sect. Sci. Math.14
(1951), 521
M. Mikolas, "On the asymptotic behaviour of Franel's
sum and the Riemann hypothesis." Results Math 21(1992)
no. 34, 368378
M. Yoshimoto, "Farey series and the Riemann hypothesis.
II." Acta Math.
Hungar. 78 (1998), no. 4, 287304
WWN notes on RH equivalences involving Farey series
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."
M.L. Lapidus and H. Maier, "Hypothese de Riemann, cordes fractales vibrantes
et conjecture de WeylBerry modifiee", C. R. Acad. Sci Paris Ser. I Math.
313 (1991) 1924.
(Abstract) "Jointly with C. Pomerance, the first author has recently proved in
dimension one the "modified WeylBerry conjecture" formulated in his
earlier work on the vibrations of fractal drums. Here, we show, in
particular, that (still in dimension one) the converse of this
conjecture is not true in the "midfractal" case and that it is
true everywhere else if and only if the Riemann hypothesis is
true. We thus obtain a new characterization of the Riemann hypothesis
by means of a inverse spectral problem."
The following contain interrelated reformulations of the RH in terms of scattering
theory:
P.D. Lax and R.S. Phillips, Scattering theory (Academic Press, 1967)
P.D. Lax and R.S. Phillips, "Scattering theory for automorphic functions", Bulletin
of the American Mathematical Society 2 (2) (1980) 261295.
L.D. Faddeev and B.S. Pavlov, "Scattering theory and automorphic functions", Proc.
Steklov Inst. Math. 27 (1972) 161193.
The follow purportedly reduces the Riemann Hypothesis to an inverse
(quantum) scattering problem, and (despite the humility of the title),
claims to contain a proof of the RH.
C. Castro, A. Granik, and J. Mahecha,
"On SUSYQM, fractal strings
and steps towards a proof of the Riemann hypothesis" (preprint 07/01)
The following appears to be related to the work of Castro, et.
al., insofar as it involves inverse scattering:
N.N. Khuri, "Inverse
scattering, the coupling constant spectrum, and the Riemann Hypothesis"
"We use inverse scattering methods, generalized for a specific
class of complex potentials, to construct a one parameter family of
complex potentials V(s,r) which have the property that the zero
energy swave Jost function, as a function of s alone, is identical
to Riemann's xi function whose zeros are the nontrivial zeros of
the zeta function. These potentials have an asymptotic expansion in
inverse powers of s(s1) with real coefficients
V_{n}(r) which are
explicitly calculated. We show that the validity of the Riemann
hypothesis depends essentially on simple integrability properties of
the first order coefficient, V_{1}(r). In the case studied in this
paper, this coefficient does not satisfy these conditions, but proof
of that fact does indicate several possibilities for proceeding
further."
W. Smith, "Cruel and unusual behavior of the
Riemann zeta function"
"We exhibit a sequence c_{n} such that the convergence
of c_{1}z + c_{2}z^{2} +
c_{3}z^{3} + ... for z < 1 is equivalent to
the Riemann Hypothesis. Numerical investigation of the c_{n}
revealed some astonishingly deceptive behavior."
J. Lagarias, "An elementary problem equivalent
to the Riemann Hypothesis"
J. Sondow, "The Riemann hypothesis, simple zeros and the asymptotic
convergence degree of improper Riemann sums", Proceedings of the
American Mathematical Society 126 (1998) 13111314.
[Abstract:] "We characterize the nonreal zeros of the Riemann zeta
function and their multiplicities, using the "asymptotic convergence
degree" of "improper Riemann sums" for elementary improper integrals.
The Riemann Hypothesis and the conjecture that all the zeros are
simple then have elementary formulations."
L. de Branges, "A conjecture which implies the Riemann hypothesis",
Journal of Functional Analysis 121 (1994) 117184.
M.V. Berry has reformulated the Riemann
Hypothesis in terms of a search for a dynamical system with a very
particular set of properties.
A. Beurling, Proceedings of the National Academy of Sciences
41 (1955) 312.
L. Baez, "On Beurling's real variable reformulation of the Riemann
hypothesis", Advances in Mathematics 101 No.1 (1993) 1030.
E. Saias and M. Balazard, "The NymanBeurling equivalent form for
the Riemann hypothesis", Expositiones Mathematicae 18
(2) (2000)
J.F. Burnol,
"A lower
bound in an approximation problem involving the zeros of the Riemann
zeta function"
[abstract:] We slightly improve the lower bound of BaezDuarte,
Balazard, Landreau and Saias in the NymanBeurling formulation of the
Riemann Hypothesis as an approximation problem. We construct
Hilbert space vectors which could prove useful in the context of the
the so called 'HilbertPólya idea'.
J.F. Burnol, "On an analytic
estimate in the theory of the Riemann Zeta function and a Theorem of BaezDuarte"
[abstract:] "We establish a uniform upper estimate for the values of zeta(s)/zeta(s+A),
0<= A, on the critical line (conditionally on the Riemann Hypothesis). We use this to give
a variant, purely complex analytic, to BaezDuarte's proof of a strengthened
NymanBeurling criterion for the validity of the Riemann Hypothesis."
L. BaezDuarte, "A strengthening
of the NymanBeurling criterion for the Riemann Hypothesis" (preprint 02/02)
[abstract:] "Let $\rho(x)=x[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the
subspace $\B$ generated by $\{\rho_a  a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$.
By the NymanBeurling criterion the Riemann hypothesis is equivalent to the statement
$\chi\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that
the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$
where $\Bnat$ is the much smaller subspace generated by $\{\rho_a  a\in\Nat\}$."
L. BaezDuarte, "Möbiusconvolutions
and the Riemann hypothesis" (preprint 04/05)
[abstract:] "The wellknown necessary and sufficient criteria for the Riemann hypothesis of M. Riesz
and HardyLittlewood, based on the order of growth at infinity along the positive real axis of certain entire
functions, are here imbedded in a general theorem for a class of entire functions, which in turn is seen to be
a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved
sharpen what is hitherto known for the Riesz function."
J.F. Burnol, "An adelic causality problem related to abelian Lfunctions",
Journal of Number Theory 87 no.2 (2001) 253269.
In this paper, Burnol uses a LaxPhillips scattering framework to reveal "a natural
formulation of the Riemann Hypothesis, simultaneously for all Lfunctions, as a
property of causality."
M. Krishna, "xizeta relation", Proceedings
of the Indian Academy of Sciences 109 (4) (1999) 379383
[abstract:] "In this note we prove a relation between the Riemann zeta function
and the xi function (Krein spectral shift) associated with the Harmonic Oscillator in one
dimension. This gives a new integral representation of the zeta function and also a
reformulation of the Riemann hypothesis as a question in L^{1}(R)."
A. Connes, "Formule de trace en geometrie non commutative et hypothese
de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996)
12311235.;
(Abstract) "We reduce the Riemann hypothesis for Lfunctions on a
global field k to the validity (not rigorously justified) of a trace
formula for the action of the idele class group on the noncommutative
space quotient of the adeles of k by the multiplicative group of
k."
Berry and Keating refer to this article in their "H = xp
and the Riemann zeros", and explain that Connes has devised a Hermitian
operator whose eigenvalues are the Riemann zeros on the critical line.
This is almost the operator Berry seeks
in order to prove the Riemann Hypothesis, but unfortunately the possibility of zeros off the
critical line cannot be ruled out in Connes' approach.
His operator is
the transfer (PerronFrobenius) operator of a classical transformation.
Such classical operators formally resemble quantum Hamiltonians, but
usually have complicated nondiscrete spectra and singular eigenfunctions.
Connes gets a discrete spectrum by making the operator act on an
abstract space where the primes appearing in the Euler product for the
Riemann zeta function are built in; the space is constructed from
collections of padic numbers (adeles) and the associated units
(ideles). The proof of the Riemann Hypothesis is thus reduced to
the proof of a certain classical trace formula.
relevant articles by Connes and videotaped
1998 lecture series
D. Goldfeld, "Explicit Formulae as Trace Formulae", from Number Theory, Trace Formulas
and Discrete Groups (K.E. Aubert, E. Bombieri and D. Goldfeld, eds.) (Academic, 1989)
281288
"In his epochmaking paper [2], Selberg developed a general trace
formula for discrete subgroups of GL(2,R). The analogies with the
explicit formulae of Weil [3] (relating very general
sums over primes with corresponding sums over the critical zeroes of the zetafunction)
are quite striking and have been the subject of much speculation over the years.
It is the object of this note to show that Weil's explicit formula can in fact be
interpreted as a trace formula on a suitable space. The simplest space we have been
able to construct for this purpose, at
present, is the semidirect product of the ideles of norm one with the
adeles, factored by the discrete subgroup
Q* X Q, the semideirect product of the multiplicative group of
rational numbers with the additive group of rational numbers. We will show that for
a suitable kernel function on this space, the conjugacy class side of the
Selberg trace formula, is precisely the sum over the
primes occuring in Weil's explicit formula.
This implies that the sum of the eigenvalues of the selfadjoint integral operator
associated to the aforementioned kernel function is precisely the sum over the critical
zeroes of the Riemann zetafunction occurring on the other side of Weil's formula. The
relation between the eigenvalues of this integral operator and the zeroes of the
zetafunction appears quite mysterious at present. What is lacking is a suitable
generalization of the Selberg transform in this situation.
Finally, we should point out that our approach leads to various new equivalences
to the Riemann Hypothesis, such as certain positivity hypotheses for the integral
operators. Although we have worked over Q, for simplicity of exposition, it
is not hard to generalize our results to Lfunctions of arbitrary number fields."
D. Mayer, "The thermodynamic
formalism approach to Selberg's zeta function for PSL(2,Z)", Bulletin
of the AMS 25(1991) 5560.
"The thermodynamic formalism...leads to a rather explicit representation
of the SmaleRuelle function and hence also of the Selberg function for
PSL(2,Z) in terms of Fredholm determinants of transfer operators
of the map T_{G}. Finally, combining our results with classical
ones for the Selberg function derived from
the trace formula suggests also
a seemingly new formulation of Riemann's hypothesis on his zeta function
in terms of the transfer operators of T_{G}."
According to A.
Strombergsson, the following reformulate the RH in terms of
problems involving the distribution of closed horocycles:
D. Zagier, "Eisenstein series and the Riemann zeta function", in
Automorphic Forms, Representation Theory and Arithmetic,
Tata Institute, Bombay (SpringerVerlag, 1981) 275301.
A. Verjovsky, "Arithmetic geometry and dynamics in the unit
tangent bundle of the modular orbifold", Dynamical Systems,
Santiago, 1990 (Longman, 1993) 263298.
Horocycles are most easily understood as horizonal lines in the
upperhalfplane model of the hyperbolic plane.
Strombergsson has also pointed out another paper by Verjovsky which
reformulates the RH in terms of a 'comparitively elementary equivalence,
not involving horocycles':
A. Verjovsky, "Discrete measures and the Riemann hypothesis", Kodai
Mathematics Journal 17 no. 3 (1994) 596608 (Workshop on
Geometry and Topology, Hanoi, 1993).
L D Pustyl'nikov, "On
a property of the classical zetafunction associated with the Riemann
hypothesis", Russ. Math. Surv. 54 (1) (1999), 262263.
This is an incredibly elegant reformulation of the RH: all even derivatives of the xi function at 1/2 are
positive. This is related to some other RH equivalences due to Li,
Lagarias and Bombieri.
F. Roesler, "Riemann's
hypothesis as an eigenvalue problem. III", Linear Algebra and
its Applications 141 (1990) 146
[abstract:] "We give
conditional induction proofs for the existence of a small zerofree
strip inside the critical strip of Riemann's zeta function $\zeta(s)$.
The starting point is some formulas for the eigenvalues $\lambda$ of
certain matrices A_{N} over the integers, whose
determinants are connected with Riemann's hypothesis by the equation
$det_{A}N = N!\Sum_{1 \leq n \leq N} \mu(n)/n$, where $\mu$ denotes
the MÃ¶bius function. The conditions of the proofs refer to properties
of the characteristic polynomials X_{N}(x) of the
matrices A_{N} near x = 0 and/or the existence
of small eigenvalues. A typical example: If for every
N > N_{0} at least one of the
polynomials X_{M}(x),
$N \leq M \leq N + N^{1 + \epsilon} has a zero $\lambda$ such that
0.09 < $\lambda$ < 1.04, then $\zeta(s) \neq 0$ if
$\Re s > 1  \epsilon$."
D. Merlini, "The Riemann magneton of
the primes" (preprint 04/04)
[abstract:] "We present a calculation involving a function related
to the Riemann Zeta function and suggested by two recent works
concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and
the other by Volchkov. We define an integral m(r)
involving the Zeta function in the complex variable s = r
+ it and find a particularly interesting expression for
m(r) which is rigorous at least in some range of r.
In such a range we find that there are two discontinuities of the
derivative m'(r) at r = 1 and r = 0, which
we calculate exactly. The jump at r = 1 is given by 4*Pi. The
validity of the expression for m(r) up to r =
1/2 is equivalent to the truth of the Riemann Hypothesis (RH).
Assuming RH the expression for m (r) gives m = 0
at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) =
4.95 at r = 1/2 (where gamma = 0.577215... is the Euler
constant). As a consequence, if the expression for m(r)
can be continued up to r = 1/2, then if we interpret
m(r) as a magnetization in the presence of a magnetic
field h = r  1/2 (or as a "free energy" at inverse
temperature beta proportional to r), there is a first order
phase transition at r = 1/2 (h = 0) with a jump of
m'(r) given by 2*Pi times the first Lin coefficient
lambda_1 = [1+gamma/2(1/2)ln(4*Pi)] = 0.0230957. Independently of the
RH, by looking at the behavior of the convergent Taylor expansion of
m(r) at r = 1, m(r = 1/2+) as well
as the first Lin coefficient may be evaluated using the Euler product
formula, in terms of the primes. This gives further evidence for the
possible truth of the Riemann Hypothesis."
S. Beltraminelli and D. Merlini, "The criteria of
Riesz, HardyLittlewood et. al. for the Riemann Hypothesis revisited using similar functions" (preprint 01/06)
[abstract:] "The original criteria of Riesz and of HardyLittlewood concerning the truth of the Riemann Hypothesis
(RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of BaezDuarte
concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the
emergence of Poissonlike distributions and we present some numerical experiments which indicate that the RH may barely
be true."
M. McGuigan, "Riemann
Hypothesis and short distance fermionic Green's functions" (preprint 04/05)
[abstract:] "We show that the Green's function of a two dimensional fermion with a
modified dispersion relation and short distance parameter a is given by the Lerch
zeta function. The Green's function is defined on a cylinder of radius R and we show
that the condition R = a yields the Riemann zeta function as a quantum
transition amplitude for the fermion. We formulate the Riemann hypothesis physically as a
nonzero condition on the transition amplitude between two special states associated with
the point of origin and a point half way around the cylinder each of which are fixed points
of a Z_{2} transformation. By studying partial sums we show that that the transition
amplitude formulation is analogous to neutrino mixing in a low dimensional context. We also
derive the thermal partition function of the fermionic theory and the thermal divergence at
temperature 1/a. In an alternative harmonic oscillator formalism we discuss the
relation to the fermionic description of two dimensional string theory and matrix models.
Finally we derive various representations of the Green's function using energy momentum
integrals, point particle path integrals, and string propagators.
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