scholarly journals A Novel Integral Equation for the Riemann Zeta Function and Large t-Asymptotics

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 650
Author(s):  
Konstantinos Kalimeris ◽  
Athanassios S. Fokas
Keyword(s):  
Integral Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Substantial Improvement ◽  
New Approach ◽  
Absolute Value ◽  
The Riemann Zeta Function ◽  
Lindelöf Hypothesis ◽  
Riemann Zeta ◽  
The Absolute

Based on the new approach to Lindelöf hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equation. This approach suggests a substantial improvement of the current large t - asymptotics estimate for ζ 1 2 + i t .

scholarly journals The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios

2016 ◽  
Vol 207 (1) ◽  
pp. 23-113 ◽  
Author(s):  
Reda Chhaibi ◽  
Joseph Najnudel ◽  
Ashkan Nikeghbali
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
New Approach ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Unitary Ensemble

Universality of $L$-Dirichlet functions and nontrivial zeros of the Riemann zeta-function

2019 ◽  
Vol 210 (12) ◽  
pp. 1753-1773 ◽  
Author(s):  
A. Laurinčikas ◽  
J. Petuškinaitė
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals On The Tails of the Exponential Series

1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
Maclaurin Series ◽  
Exponential Series ◽  
Asymptotic Estimation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

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