Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).

The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

On the Distribution of the Nontrivial Zeros for the Dirichlet L-Functions

This paper addresses a variant of the product for the Dirichlet $L$--functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet $L$--functions, which states that the real part of the nontrivial zeros is $1/2$. The Wang and Hardy--Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach's conjecture.

Joint universality of periodic zeta-functions with multiplicative coefficients. II

In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurinčikas, M. Tekorė, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.

Zeros of GL2 𝐿-functions on the critical line

We use Levinson’s method and the work of Blomer and Harcos on the GL 2 \mathrm{GL}_{2} shifted convolution problem to prove that at least 6.96 % of the nontrivial zeros of the 𝐿-function of a GL 2 \mathrm{GL}_{2} automorphic form lie on the critical line.

A HARMONIC SUM OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

We consider the sum $\sum 1/\gamma $ , where $\gamma $ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$ , and examine its behaviour as $T \to \infty $ . We show that, after subtracting a smooth approximation $({1}/{4\pi }) \log ^2(T/2\pi ),$ the sum tends to a limit $H \approx -0.0171594$ , which can be expressed as an integral. We calculate H to high accuracy, using a method which has error $O((\log T)/T^2)$ . Our results improve on earlier results by Hassani [‘Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function’, Appl. Math. E-Notes16 (2016), 109–116] and other authors.