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

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding

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.

2018 ◽  
Vol 61 (3) ◽  
pp. 622-627
Author(s):  
Helmut Maier ◽  
Michael Th. Rassias

AbstractA crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.


2011 ◽  
Vol 16 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Joern Steuding

We study the sequence of nontrivial zeros of the Riemann zeta-function with respect to sequences of zeros of other related functions, namely, the Hurwitz zeta-function and the derivative of Riemann's zeta-function. Finally, we investigate connections of the nontrivial zeros with the periodic zeta-function. On the basis of computation we derive several classifications of the nontrivial zeros of the Riemann zeta-function and stateproblems which mightbe ofinterestfor abetter understanding of the distribution of those zeros.


2018 ◽  
Vol 68 (4) ◽  
pp. 741-748 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Antanas Laurinčikas

Abstract We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).


2012 ◽  
Vol 08 (03) ◽  
pp. 589-597 ◽  
Author(s):  
XIAN-JIN LI

In [Complements to Li's criterion for the Riemann hypothesis, J. Number Theory77 (1999) 274–287] Bombieri and Lagarias observed the remarkable identity [1 - (1 - 1/s)n] + [1 - (1 - 1/(1 - s))n] = [1 - (1 - 1/s)n]⋅[1 - (1 - 1/(1 - s))n], and pointed out that the positivity in Li's criterion [The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory65 (1997) 325–333] has the same meaning as in Weil's criterion [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61]. Let λn = ∑ρ[1 - (1 - 1/ρ)n] for n = 1, 2, …, where ρ runs over the complex zeros of the Riemann zeta function ζ(s). In this note, a certain truncation of λn is expressed as Weil's explicit formula [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61] for each positive integer n. By using the Bombieri and Lagarias' identity, we prove that the positivity of these truncations implies the Riemann hypothesis. If these truncations have suitable upper bounds, we prove that all nontrivial zeros of the Riemann zeta function lie on the critical line.


2016 ◽  
Vol 162 (2) ◽  
pp. 293-317 ◽  
Author(s):  
XIANCHANG MENG

AbstractUnder the Riemann Hypothesis, we connect the distribution of k-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of ζ(s). Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-{x}/{\zeta(k)}$ and μk(n) is the characteristic function of k-free numbers. Finally, we make a conjecture about the maximum order of Mk(x) by heuristic analysis on the tail of the limiting distribution.


Author(s):  
Frederick Ira Moxley III

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.


Author(s):  
Frederick Ira Moxley

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Mu¨ller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta func- tion, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schro¨dinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.


2016 ◽  
Vol 8 (1) ◽  
pp. 16-20
Author(s):  
Yu.V. Basiuk ◽  
S.I. Tarasyuk

We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation. Also we obtain a new restatement of the Riemann Hypothesis.


Author(s):  
Andrés Chirre ◽  
Oscar E. Quesada-Herrera

Let [Formula: see text] be the argument of the Riemann zeta-function at the point [Formula: see text]. For [Formula: see text] and [Formula: see text] define its antiderivatives as [Formula: see text] where [Formula: see text] is a specific constant depending on [Formula: see text] and [Formula: see text]. In 1925, Littlewood proved, under the Riemann Hypothesis (RH), that [Formula: see text] for [Formula: see text]. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of [Formula: see text] and [Formula: see text]. This was extended by Fujii for [Formula: see text], when [Formula: see text]. Assuming the RH, we give the explicit asymptotic formula for the second moment of [Formula: see text] up to the second-order term, for [Formula: see text]. Our result conditionally refines Selberg’s and Fujii’s formulas and extends previous work by Goldston in [Formula: see text], where the case [Formula: see text] was considered.


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