scholarly journals A lower bound for the zeros of riemann’s zeta function on the critical line

2006 ◽  
pp. 176-182
Author(s):  
Enrico Bombieri
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Critical Line ◽  
Riemann’S 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.

The Riemann-Siegel expansion for the zeta function: high orders and remainders

1995 ◽  
Vol 450 (1939) ◽  
pp. 439-462 ◽  
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Critical Line ◽  
High Accuracy ◽  
Stokes Line ◽  
Power Dependence ◽  
Multiplier Function ◽  
Riemann’S Zeta Function

On the critical line s ═ ½ + i t ( t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar 'factorial' divided by power' dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r ═ r * ≈ 2π t and then increase. The form of the remainder when the expansion is truncated near r * is determined; it is of order exp(-π t ), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t .

scholarly journals On the zeros of Riemann's zeta-function on the critical line

2016 ◽  
Vol 165 ◽  
pp. 203-269
Author(s):  
Siegfred Alan C. Baluyot
Keyword(s):  
Zeta Function ◽  
Critical Line ◽  
Riemann’S Zeta Function

scholarly journals More than five-twelfths of the zeros of $$\zeta $$ are on the critical line

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Kyle Pratt ◽  
Nicolas Robles ◽  
Alexandru Zaharescu ◽  
Dirk Zeindler
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Dirichlet Polynomial ◽  
Real Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Combinatorial Process

AbstractThe second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $$(\mu \star \Lambda _1^{\star k_1} \star \Lambda _2^{\star k_2} \star \cdots \star \Lambda _d^{\star k_d})$$(μ⋆Λ1⋆k1⋆Λ2⋆k2⋆⋯⋆Λd⋆kd) is computed unconditionally by means of the autocorrelation of ratios of $$\zeta $$ζ techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of $$\begin{aligned} \zeta (s) + \lambda _1 \frac{\zeta '(s)}{\log T} + \lambda _2 \frac{\zeta ''(s)}{\log ^2 T} + \cdots + \lambda _d \frac{\zeta ^{(d)}(s)}{\log ^d T}, \end{aligned}$$ζ(s)+λ1ζ′(s)logT+λ2ζ′′(s)log2T+⋯+λdζ(d)(s)logdT,where $$\zeta ^{(k)}$$ζ(k) stands for the kth derivative of the Riemann zeta-function and $$\{\lambda _k\}_{k=1}^d$${λk}k=1d are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

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