scholarly journals On the logarithm of the Riemann zeta-function near the nontrivial zeros

2021 ◽  
pp. 1
Author(s):  
Fatma Cicek
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals QUESTIONS AROUND THE NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION. COMPUTATIONS AND CLASSIFICATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Joern Steuding
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Hurwitz Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

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.

scholarly journals Riemann zeta zeros and zero-point energy

2014 ◽  
Vol 29 (09) ◽  
pp. 1450051 ◽  
Author(s):  
J. G. Dueñas ◽  
N. F. Svaiter
Keyword(s):  
Scalar Field ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Vacuum Energy ◽  
Zero Point ◽  
Point Energy ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Adjoint Operators

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

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
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
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.

scholarly journals On a problem of Ivi\'c.

2000 ◽  
Vol Volume 23 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Upper Bound ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Implicit Constant ◽  
International Audience ◽  
Riemann Zeta

International audience Let $\gamma$ denote the imaginary parts of the nontrivial zeros of the Riemann zeta-function $\zeta(s)$. For sufficiently large $T$ and $\varepsilon>0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.

Sign in / Sign up

Export Citation Format

Share Document