Counting Function Of Integers Research Articles

Some examples of well-behaved Beurling number systems

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called [ α , β ] [\alpha ,\beta ] -systems, where α \alpha and β \beta are connected to the optimal power saving in the prime and integer-counting functions. It is known that every [ α , β ] [\alpha ,\beta ] -system satisfies max { α , β } ≥ 1 / 2 \max \{\alpha ,\beta \}\ge 1/2 . In this paper we show there are [ α , β ] [\alpha ,\beta ] -systems for each α ∈ [ 0 , 1 ) \alpha \in [0,1) and β ∈ [ 1 / 2 , 1 ) \beta \in [1/2, 1) . Assuming the Riemann hypothesis, we also construct certain families of [ α , β ] [\alpha ,\beta ] -systems with β > 1 / 2 \beta >1/2 .

Beurling integers with RH and large oscillation

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)=Li(x)+O(x), while its integer counting function satisfies the oscillation estimate N(x)=ρx+Ω±(xexp⁡(−clog⁡xlog⁡log⁡x)) 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.

Arithmetic Properties of Integers in Chains and Reflections of g-ary Expansions

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base g ⩾ 2 and then for every infinite sequenceof g-ary digits we consider the counting function of integers n ⩽ N for which ∑n − 1i = 0digi is prime. We construct sequences for which grows fast enough, and show that for some constant ϑg < g there are at most O(ϑNg) initial elements (d0, …, dN − 1) of for which . We also discuss joint arithmetic properties of integers and mirror reflections of their g-ary expansions.

Rough values of Piatetski-Shapiro sequences

An integer is called y-rough if it is composed solely of primes \(> y\). Let \(\lfloor {.}\rfloor \) be the floor function. In this paper, we exhibit an asymptotic formula for the counting function of integers \(n \leqslant x\) such that \(\lfloor {n^c}\rfloor \) is y-rough uniformly for a range of y that depends on \(1< c < 2229/1949\).

On PNT equivalences for Beurling numbers

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.

Generalised prime systems with periodic integer counting function

We study generalised prime systems (both discrete and continuous) for which the `integer counting function' N(x) has the property that N(x) i cx is periodic for some c > 0. We show that this is extremely rare. In particular, we show that the only such system for which N is continuous is the trivial system with N(x) i cx constant, while if N has finitely many discontinuities per bounded interval, then N must be the counting function of the g-prime system containing the usual primes except for finitely many. Keywords and phrases: Generalised prime systems. I

Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function N P ( x ) is asymptotically well behaved, in the sense that N P ( x ) = ρ x + O ( x β ) , where ρ is a positive constant and β < 1 2 . For such systems, the associated zeta function ζ P ( s ) is holomorphic for σ = R s > β ( s ≠ 1 ) . We prove that for β < σ < 1 2 , ∫ 0 T | ζ P ( σ + i t ) | 2 d t = Ω ( T 2 − 2 σ − ε ) for any ε > 0 , and also for ε = 0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in N k P ( x ) − Res s = 1 ( ζ P ( s ) k x s / s ) , which is O ( x θ ) for some θ < 1 . Letting α k denote the infimum of such θ, we show that α k ⩾ 1 2 − 1 2 k .

Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems ${\user1{{\wp }}}\left( {1 < p_{1} \leqslant p_{2} \leqslant \cdot \cdot \cdot } \right.$ , with $p_{j} \in \mathbb{R}$ tending to infinity) and the associated Beurling zeta function $\zeta _{{\user1{{\wp }}}} {\left( s \right)} = {\prod\nolimits_{j = 1}^\infty {{\left( {1 - p^{{ - s}}_{j} } \right)}^{{ - 1}} } }$ . Under appropriate assumptions, we establish various analytic properties of $\zeta _{{\user1{{\wp }}}} {\left( s \right)}$ , 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 $\zeta _{{\user1{{\wp }}}} {\left( s \right)}$ . Further we study ‘well-behaved’ g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on $\mathbb{N}^{2} $ . Some of the above results are relevant to the second author’s theory of ‘fractal membranes’, whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.

A lower bound for the Lindelöf function associated to generalized integers

In this paper we study generalized prime systems for which the integer counting function N P ( x ) is asymptotically well-behaved, in the sense that N P ( x ) = ρ x + O ( x β ) , where ρ is a positive constant and β < 1 2 . For such systems, the associated zeta function ζ P ( s ) has finite order for σ = R s > β , and the Lindelöf function μ P ( σ ) may be defined. We prove that for all such systems, μ P ( σ ) ⩾ μ 0 ( σ ) for σ > β , where μ 0 ( σ ) = { 1 2 − σ if σ < 1 2 , 0 if σ ⩾ 1 2 .

Beurling primes with large oscillation

A Beurling generalized number system is constructed having integer counting function N B (x) = κ x +O(x θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π B (x) =li(x) + Ω(xexp(-c )), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability.

A property of the counting function of integers with no large prime factors

For logy log log x → ∞ as x → ∞, ψ( Cx, y) ≈ Cψ( x, y) uniformly for C in compact subsets of (0, 1]. The condition logy log log x → ∞ is necessary. For fixed sufficiently large K, x → ∞, and y = exp( K log log x), we have for 1 2 ≤ C ≤ 1 that Ψ(Cx,y)≈C 1−1/KΨ(x,y) .