scholarly journals An explicit formula for the fourth power mean of the Riemann zeta-function

1993 ◽  
Vol 170 (2) ◽  
pp. 181-220 ◽  
Author(s):  
Yoichi Motohashi
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Fourth Power ◽  
Power Mean ◽  
Fourth Power Mean ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals A new derivation of $\zeta(-r)=-\frac{B_{r+1}}{r+1}$

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Master Theorem

In this note, we give a new derivation for the fact that $\zeta(-r)=-\frac{B_{r+1}}{r+1}$ where $\zeta(s)$ represents the Riemann zeta function, and $B_{r}$ represents the Bernoulli numbers. Our proof uses the well-known explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, and the Ramanujan's master theorem to obtain an integral representation for the Riemann zeta function.

Power mean values of the Riemann zeta‐function

Mathematika ◽  
1982 ◽  
Vol 29 (2) ◽  
pp. 202-212 ◽  
Author(s):  
J.-M. Deshouillers ◽  
H. Iwaniec
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Power Mean ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

ON THE EXPLICIT FORMULA IN THE THEORY OF PRIME NUMBERS

2012 ◽  
Vol 08 (03) ◽  
pp. 589-597 ◽  
Author(s):  
XIAN-JIN LI
Keyword(s):  
New York ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Li's Criterion

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.

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