Beurling's generalised prime construction
chronological bibliography and notes
E. Landau, "Neuer beweis des primzahlsatzes und beweis des primidealsatzes",
Math. Ann. 56 (1903) 645670.
A. Beurling, "Analyse de la loi asymptotique de la
distribution des nombres premiers généralisés", Acta Math. 68
(1937) 255291.
B. Nyman, "A general prime number theorem",
Acta Math. 81, (1949) 299307.
P. Malliavin, "Sur le reste de la loi asymptotique de répartition des nombres
premiers généralisés de Beurling" Acta Math. 106 (1961) 281298.
A. Beurling, "Construction and analysis of some convolution
algebras", Annales de l'Institut Fourier, Grenoble 14 (1964) 132.
R.S. Hall, "Theorems about
Beurling's generalised primes and the associated zeta function", Ph.D. Thesis, University of IllinoisUrbana, 1967
[abstract:] "Beurling defined a set, $P$, of generalized prime numbers as any nondecreasing unbounded
sequence of real numbers which are greater than one. The multiplicative semigroup, $N$, generated by the
elements of $P$ is called the associated set of generalized integers. The functions $Pi(x)$ and $N(x)$ are
defined, respectively, as the number of generalized prime and integers less than or equal to $x$. This paper
is concerned with deducing the behaviour of $Pi(x)$ from that of $N(x)$."
P.T. Bateman and H.G. Diamond, "Asymptotic distribution of Beurling's generalized prime numbers", Studies
in Number Theory, Mathematical Association of America Studies vol. 6
(W.J. Leveque, ed.), 1969, 152210.
H.G. Diamond, "The prime number theorem for
Beurling's generalized numbers", Journal of Number Theory 1
(1969) 200207.
H.G. Diamond, "Asymptotic distribution of
Beurling's generalized integers", Illinois Journal of Mathematics
14 (1970) 1228.
H.G. Diamond, "A set of generalized numbers
showing Beurling's theorem to be sharp", Illinois Journal of
Mathematics 14 (1970) 2934.
R.S. Hall, "The prime number theorem for generalized primes", Journal
of Number Theory 4 (1972) 313320.
H.G. Diamond, "Chebyshev estimates for Beurling generalized prime
numbers", Proceedings of the American Mathematical Society 39
(1973) 503508.
R.S. Hall, "Beurling generalized prime number systems in which the
Chebyshev inequalities fail", Proceedings of the American Mathematical
Society 40 (1973) 7982.
C. Ryavec, "The analytic continuation of Euler products with
applications to asymptotic formulae", Illinois Journal of Mathematics
17 (1973) 608618.
S. Segal, "Prime number
theorem analogues without primes", J. riene u. angew. Math. 265
(1974) 122.
H.G. Diamond, "Chebyshev type estimates in prime number theory", in
Seminaire de theorie des nombres, Universite de Bordeaux I, 19741975,
expose no. 24.
H.G. Diamond, "When do Beurling generalised integers have a density?", J. Reine
Angew. Math. 295 (1977) 2239
H. Müller, "Über die asymptotische Verteilung von Beurlingschen Zahlen", J. Reine Angew. Math.
289 (1977) 181187.
J.P. Borel, "Quelques resultats d'equipartion lies aux nombres premiers
generalises de Beurling", Acta Arith. 38 (1980) 255272
J.P. Borel, "Sur le prolongement des fonctions zeta associees a un systeme
de nombres premiers generalises de Beurling", Acta Arith. 43 (1984) 273282.
E. Stankus, "Analytic continuation of modified Lfunction",
Lithuanian Math. Journal 24 (1984) 176181.
W.B. Zhang, "Asymptotic distribution of Beurling's generalized prime numbers and integers",
Ph.D. Thesis, Univ. of Illinois, Urbana, 1986
WenBin Zhang, "Chebyshev estimates for Beurling generalized prime
numbers", Proceedings of the American Mathematical Society
101 (1987) 205212.
WenBin Zhang, "A generalization of Halász's theorem to Beurling's generalised
integers and its application", Illinois J. Math. 31 (1987) 645664
WenBin Zhang, "Density and Odensity of Beurling generalized integers",
Journal of Number Theory 30 (1988) 120139.
WenBin Zhang, "Chebyshev
type estimates for Beurling generalized prime numbers. II"., Trans. Am. Math. Soc. 337 (1993)
651675
Sz. Revesz, "On Beurling's prime number theorem" Period. Math. Hungarica
28 (1994) 195210.
E. Stankus, "On some generalized integers", Lithuanian Math. Journal,
36 (1996) 115124.
J.P. Kahane,
"A Fourier formula for prime numbers" Dedicated to
the memory of Carl Herz, Canadian Mathematical Society Conference
Proceedings 21 (1997) 89102.
J.P. Kahane, "Sur les nombres generalises de Beurling. Preuve
d'une conjecture de Bateman et Diamanond", Journal de Theorie des
Nombres Bordeaux 9 (1997) 251266.
J.P. Kahane, "Le role de l'algebre H^{1} de Sobolev dans la theorie
des nombres premiers generalises de Beurling", C.R. Acad. Sci. Paris
324 (1997) 251266.
E. Stankus, "Modified zeta functions and the
number of gintegers", New Trends in Probability and
Statistics, Vol. 4 (Editors A. Laurincikas, et. al.) 247258.
E.P. Balanzario, "An example in Beurling's
theory of primes", Acta Arithmetica 87 (1998) 121139
J.P. Kahane, "Le role des algebres A de Wiener,
de Beurling et H^{1} de
Sobolev dans la theorie des nombres premiers generalises de
Beurling" Annales de l'Institut Fourier, Grenoble 48, 3
(1998) 611648.
J.P. Kahane, "Un theoreme de Littlewood pour les nombres premiers de
Beurling" Bulletin of the London Mathematical Society 31
(1999) 424430.
M. Balazard, "La version
de Diamond de la methode de l'hyperbole de Dirichlet", L'Enseignement Math.
45 (1999) 253270.
H. Montgomery, "Smooth
Beurling integers from irregular Beurling primes" (lecture presented at
University of Illinois, Urbana's Millennial Conference on Number Theory, May 2126, 2000)
H.G. Diamond, H.L. Montgomery and U.M.A. Vorhauer, "Beurling primes with large oscillation",
334 (2006) 136
[abstract:] "A Beurling generalized number system is constructed having integer counting function
$N_B(x)=\kappa x+O(x^\theta)$ with $\kappa>0$ and $1/2<\theta<1$, whose prime counting function satisfies the
oscillation estimate $\pi_B(x)==li(x)+\Omega(x\exp(c\sqrt{\log x}))$, and whose zeta function has infinitely
many zeros on the curve $\sigma=1a/\log t, t \geq 2$, and no zero to the right of this curve, where $a$ is
chosen so that $a>(4/e)(1\theta)$. The construction uses elements of classical analytic number theory and
probability."
A.S. Fainleib, "On the distribution of
Beurling integers", Journal of Number Theory 111 (2005) 227247
[abstract:] "Asymptotic behaviour of the counting function of Beurling integers is deduced from Chebyshev
upper bound and Mertens formula for Beurling primes. The proof based on some properties of corresponding
zetafunction on the right of its abscissa of convergence."
T.W. Hilberdink, "Wellbehaved
Beurling primes and integers", Journal of Number Theory 112 (2005) 332344
[abstract:] "In this paper, we study generalised prime systems for which both the prime and integer
counting functions are asymptotically wellbehaved, in the sense that they are approximately $li(x)$ and
$\rho x$, respectively (where $\rho$ is a positive constant), with error terms of order $O(x^{\theta_1})$ and
$O(x^{\theta_2})$ for some $\theta_1,\theta_2<1$. We show that it is impossible to have both $\theta_1$ and
$\theta_2$ less than 1/2."
T.W. Hilberdink, "A lower bound for
the Lindelöf function associated to generalized integers", Journal of Number Theory (in press)
Sz. Revesz, "Oscillation of the remainder term
in the Beurling prime number formula" (presentation given at Rencontres de théorie analytique et élémentaire des nombres,
Institut Henri Poincaré, 12/2006)
[abstract:] "Arne Beurling generalized the prime number theorem to the rather general situation when the role of
primes are taken over by some arbitrary reals, and integers are simply the reals of the freely generated
multiplicative subgroup of the primes given. If the "number of integers from $1$ to $x$" function takes the
form $N(x) = x + O(x^a)$, with $a < 1$, then the corresponding Beurling zeta function has a meromorphic
continuation to the halfplane to the right of $a$. Location of zeroes between the real part $= 1$ and real
part = a lines are then crucial to the oscillation of the prime number formula, as is wellknown in the
classical case. The lecture describes how these relations can be established even in the generality
of Beurling prime distribution."
W.B. Zhang, "Beurling primes with RH and Beurling primes with large
oscillation", Mathematische Annalen 337 (2007) 671704
[abstract:] "Two Beurling generalized number systems, both with
$N(x)=kx+O(x^{1/2}}\exp\{c(\log\,x)^{2/3}\})$ and $k > 0$, are constructed. The associated zeta
function of the first satisfies the RH and its prime counting function satisfies $\pi(x) = li(x) + O(x^{1/2})$.
The associated zeta function of the second has infinitely many zeros on the curve $s = 11/log t$ and no zeros
to the right of the curve and the Chebyshev function $\psi(x)$ of its primes satisfies $\limsup\,
(\psi(x)x)/(x\exp\{2\sqrt{\log\,x}\})=2$ and $\liminf\, (\psi(x)x)/(x\exp\{2\sqrt{\log\,x}\})=2$. A
sharpened form of the DiamondMontgomeryVorhauer random approximation and elements of analytic number theory
are used in the construction."
The only use of Beurling's generalisation by a physicist of which I am aware:
B. Julia, "Thermodynamic limit in
number theory: RiemannBeurling gases" Physica A 203 (1994)
425436.
"We study the grand canonical version of a solved statistical model,
the Riemann gas: a collection of bosonic oscillators with energies the
logarithms of the prime numbers. The introduction of a chemical potential
$\mu$ amounts to multiply each prime by $e^{\mu}$, the
corresponding gases could be called Beurling gases because they are
defined by the choice of appropriate generalized primes when considered
as canonical ensembles; one finds generalized Hagedorn singularities in the
temperature. The discrete spectrum can be treated as continuous in its high
energy region; this approximation allows us to study the high energy level
density and is applied to Beurling gases. It is expected to be accurate for
the high temperature behaviour. One model (the logarithmic gases) will be
studied in more detail, it corresponds to the choice of all the integers strictly
larger than one as Beurling primes; we give an explicit formula for its grand
canonical thermodynamic potential $F  \mu N$ in terms of a
hypergeometric function and check the approximation on the Hagedorn
phenomenon. Related physical situations include string theories and quark
deconfinement where one needs a better understanding of the nature of the
Hagedorn transitions."
A set of Beurling generalised integers has the Delone property if the
gaps between succesive members are bounded above and below. This was introduced
by the Russian crystallographer and number theorist B.N. Delone in 1937, motivated
by the question of whether the crystallike nature of Z^{+} is relevant
to the (presumed) truth of Riemann hypothesis, according to the following:
B.N. Delone, N.P. Dolbilin, M.I. Shtogrin, R.V. Galiulin, "A local criterion
for regularity of a system of points", Sov. Math. Dok. 17 (1976) 319322
This was brought to my attention by:
J. Lagarias, "Beurling generalized
integers with the Delone property", Forum Math. 11 (1999) 295312
T.W.
Hilberdink and M.L. Lapidus,
"Beurling zeta functions, generalised primes, and fractal membranes" (preprint 08/04)
Here is an excerpt from the introductory section, which refers to Lapidus's emerging
idea of a (noncommutative) flow of Beurling gprime systems.
H. Delille, who in 2004 began making
and retracting claims that he had a proof of the Riemann hypothesis,
provided this summary of his work, which
involves the analytic continuation of Beurling zeta functions (August 2004).
In discussion with H. Diamond:
After seeing the original "prime evolution"
notes, Martin Huxley contacted me
to point out that I had reinvented the
basic idea of Beurling's construction. I
then contacted Harold Diamond who
appeared to be one of the leading experts in this field. Diamond examined
the notes and made the following comments:
"Prof. Huxley is correct: this topic lies in the area of Beurling
generalized number theory. The essence of this theory is that if a
sequence of gintegers is generated by a sequence of gprimes,
and if one of the sequences is distributed sufficiently like its
classical counterpart, then so too is the other."
[in reference to my remarks
on the equal spacing of the integers:]
"Equally spaced. I believe that the Riemann Hypothesis will not
be settled until methods have been developed which make effective use
of the equal spacing of integers. Note that the functional
equation of the Riemann zeta function does have this
information encoded into it."
I later asked him to elaborate on this intriguing prediction, and
received the following reply:
"The functional equation of the Riemann zeta function
does contain information about the equal spacing of the integers, but it is
very subtle. It shows itself in a few ways:
1) If one requires only an asymptotic law for the integer counting function of the
form N(x) = x + O(1), one could make examples in which
the associated generalized zeta function
had a
natural boundary on the imaginary axis, i.e. the zeta function could not be
continued further to the left, while the Riemann zeta
function is continuable to the entire complex plane (except for the pole
point s = 1).
2) If you extend the 1/2 plane of convergence of the Riemann zeta function by the
EulerMaclaurin summation formula, the good spacing of the integers allows
you to keep the process going ad infinitum.
3) If you extend the Riemann zeta function to the complex plane by use of
the functional
equation, the proofs of the functional equation all contain as an element the
good spacing of the integers.
The reason for the claim that equal spacing of integers is likely to be essential for a proof of
the Riemann hypothesis is this: there are quite simple Beurlingtheory
examples of nice counting functions N(x) for which such a good
approximation as N(x) = x + O(1) exists, but for which
the zeta function has zeros off the line Re s = 1/2. The examples
have proper multiplicative structure, the log of the zeta function is of a
suitable type, and there is good approximability of N(x) by
x, but they lack additive structure."
[in reference to my speculations
on the necessary conditions for gintegers to have asymptotic
density 1:]
"There is quite a bit known about gintegers having asymptotic
density 1. Roughly speaking, in addition to the gprimes having to
be distributed rather like the usual ones, an additional condition
is needed to cause the density be exactly 1.
A trivial remark: the set of generalized integers
consisting of all odd numbers (the prime 2 has been killed) also
has the property of being equally spaced. It says that the sequence
of standard primes is not the only one that generates equally
spaced integers. Note that in this example we have density 1/2.
Brief commments on
my questions about modified zeta functions.
First, this is very appropriate for studying such problems.
Questions of half planes of convergence of the zeta function and
behavior of the zeta function at the point 1 are easily answered
in terms of the rate of growth of the integer counting function.
Questions of an extension of the generalized zeta function to  {1} and its properties in the newly
gained territory appear to be
very difficult. It is not obvious how zeros of the zeta function
move around under perturbations of the primes. If you start with
the classical primes, and do your permutations one prime at a time,
you will not affect the classical zeta function zeros, but you will
induce a bunch of zeros and poles of the new zeta function on the
imaginary axis. If you are permuting everything at the same time,
I don't have a clue about the behavior of the zeros.
Relating to my
point about continuous prime distributions in
section 3.6, in some sense Dirac measure at the point 1 + Lebesgue
measure on represents another distinguished limiting
configuration that is related to the counting measure of positive
integers and is generated by a measure whose integral is close to
.
I have exploited this relation in several of my articles."
I asked Professor Diamond to explain what he meant by "distinguished
limiting configuration", and got the following reply:
"This is a bit queasy, but here goes.
Imagine that we set out to transmogrify the primes into a
continuous mass distribution by
successively cashing in classical primes and replacing them with a couple of nearby numbers,
each with reduced multiplicity. For
example, the factor (1  2^{s})^{1} in the product
form of the Riemann zeta function could be replaced by a couple of
factors of the form (1  a_{j}^{s})^{b_j}
for suitable numbers a_{j} around 2 and suitable
(small!) exponents b_{j}. The
notion of `suitable' is to keep the counting function N(x)
pretty close to x. The process is repeated ad infinitum, at
each stage causing each of the 'primes' to be replaced by a bunch of numbers
each with much lower multiplicity. The distribution that I
discussed, with counting measure Dirac measure at the point 1 + Lebesgue
measure on has integral that is exactly
x, at
least for x > 1 and moreover, the associated zeta function has a logarithm which is very
analogous to the log of the Riemann zeta function.
I have written about this stuff in some of the Beurling articles."
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