We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies $π(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = ρx + Ω_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ for some $c>0$, where $ρ>0$ is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and combines saddle-point analysis with the Diamond-Montgomery-Vorhauer probabilistic method via random prime number system approximations.
In 1997 H.G.Diamond gave a condition on Beurling's generalized prime numbers in order that the corresponding generalized integers have a density. We give a new proof of this condition (Theorem 1) and a proof that it is not necessary (Theorem 2 and Examples). However, it is very near to be necessary (Theorem 3). Both proofs of Theorems 1 and 2 rely on Fourier analysis, mainly the Wiener algebra, and partly on probability methods.
We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.
Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $ρ_0$ of the Beurling zeta function $ζ_P$ for a given number system generated by the primes $P$, the corresponding error term $Δ_P(x):=ψ_{P}(x)-x$, where $ψ_{P}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough…
We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x} μ(n_k)/n_k=o(1)$, with $μ$ the Beurling analog of the Moebius function.
We show that Halász's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generalized integers and a Chebyshev upper bound for the generalized primes.
In classical prime number theory several asymptotic relations are considered to be "equivalent" to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.
It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k} ρ(\frac{θ_{k}}{x})$, where $ρ(x)\adef\pfrac{x}$, and $a_{k}\in\cc$, $0<θ_{k}\leqslant 1$ satisfies $\sum_{k=1}^{n}a_{k} θ_{k}=0$. Parsevall Identity $\norma{f(x)+1}^{2}=\sum_{n\in\zz}\modulo{c(n)}^{2}$ is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients $c(n)$ of $f+1$, when $f$ is a function defined as a…
We study generalised prime systems $\mathcal{P}$ $(1<p_1\leq p_2\leq...,$ with $p_j\in\R$ tending to infinity) and the associated Beurling zeta function $ζ_{\mathcal{P}}(s) =\prod_{j=1}^{\infty} (1-p_j^{-s})^{-1}$. Under appropriate assumptions, we establish various analytic properties of $ζ_{\mathcal{P}}(s)$, including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of $ζ_{\math…
In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $ψ(x):=\sum_{g\in G, |g|\le x} Λ_{G}(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $Λ_{G}$ is the corresponding von Mangoldt function attaining $\log|p|$ for $g=p^k$ with a prime element $p\in G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $ζ_{G}$, belonging to $G$, to the number of zeroes of $ζ_G$ in various regions, in particular within the critical strip where the analytic continuation exists…
We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $Δ(x):=ψ(x)-x$, given any zero-free region $D(η):=\{s=σ+it\in\mathbb{C}~:~ σ:=\Re s \ge 1-η(t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding …
Given $β\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^β x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $π(x) = \mathrm{Li}(x) + Ω\bigl(x\exp(-c'(\log x)^{\fracβ{β+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.
Euler wrote a formula expressing that l(n)/n is a completely multiplicative function with sum 0 (a CMO function) , where l(n) is the completely multiplicative function equal to -1 on the prime numbers (the Liouville function). We extend this formula to Beurling generalized numbers, using a result of Diamond. As an application we show how to construct a CMO function carried by a set of integers whose counting function is of the form D x^a (1+o(1) ) for any a between 0 and 1.
We show that for Beurling generalized numbers the prime number theorem in remainder form $$π(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n \in \mathbb{N},$$ where $N$ and $π$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number th…
For completely multiplicative functions f(n) taking values 1 and -1, under certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log log x)sqrt(log x)) times as n runs through the integers <= x.
This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For $N$ bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic (N -> infinity) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum…
We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the known leading-order expression [M. Tran {\it et al.}, Ann.\ Phys.\ (N.Y.) {\bf 311}, 204 (2004)], which yield a faster convergence to the average values of the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean tr…
