A Tauberian theorem with remainder for the Laplace transform and its application to the riemann zeta-function

1991 ◽  
Vol 43 (10) ◽  
pp. 1269-1278
Author(s):  
V. I. Mel'nik
Keyword(s):  
Laplace Transform ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
The Riemann Zeta Function ◽  
The Laplace Transform ◽  
Riemann Zeta

scholarly journals The Proof for A Convergent Integral and Another Nonzero Integral--Respectively Using the Riemann Zeta Function and the Trigonometric Sums

2016 ◽  
Vol 8 (4) ◽  
pp. 74
Author(s):  
Hao-Cong Wu
Keyword(s):  
Laplace Transform ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mathematical Induction ◽  
Trigonometric Sums ◽  
The Riemann Zeta Function ◽  
The Laplace Transform ◽  
Riemann Zeta ◽  
Infinite Descent

In this paper, there are the applications of the main inequalities, and show how to use the analytic properties of the Zeta function and the Laplace transform to prove the convergence of the desired integral. In addition, show how to use the trigonometric sums and the mathematical induction with the method of infinite descent to prove the non-zero value of another integral. In this way, we can obtain the important proofs concerning the Riemann Zeta function and the sum of two primes.

scholarly journals Ingham Tauberian theorem with an estimate for the error term

2003 ◽  
Vol 2003 (64) ◽  
pp. 4025-4031
Author(s):  
E. P. Balanzario ◽  
E. Marmolejo-Olea
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
Error Term ◽  
Elementary Proof ◽  
Weak Form ◽  
Free Region ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We estimate the error term in the Ingham Tauberian theorem. This estimation of the error term is accomplished by considering an elementary proof of a weak form of Wiener's general Tauberian theorem and by using a zero-free region for the Riemann zeta function.

A Tauberian Theorem and Analogues of the Prime Number Theorem

1968 ◽  
Vol 20 ◽  
pp. 362-367 ◽  
Author(s):  
T. M. K. Davison
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
Prime Number Theorem ◽  
Negative Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and1then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).

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 .

Sign in / Sign up

Export Citation Format

Share Document