scholarly journals THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS

2010 ◽  
Vol 06 (08) ◽  
pp. 1933-1944 ◽  
Author(s):  
SANDRO BETTIN
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Second Moment ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

2018 ◽  
Vol 14 (02) ◽  
pp. 371-382
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
Riemann Zeta ◽  
Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

scholarly journals The second moment of Sn(t) on the Riemann hypothesis

2021 ◽  
pp. 1-24
Author(s):  
Andrés Chirre ◽  
Oscar E. Quesada-Herrera
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Order Term ◽  
Asymptotic Formulas ◽  
Second Moment ◽  
Second Moments ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Let [Formula: see text] be the argument of the Riemann zeta-function at the point [Formula: see text]. For [Formula: see text] and [Formula: see text] define its antiderivatives as [Formula: see text] where [Formula: see text] is a specific constant depending on [Formula: see text] and [Formula: see text]. In 1925, Littlewood proved, under the Riemann Hypothesis (RH), that [Formula: see text] for [Formula: see text]. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of [Formula: see text] and [Formula: see text]. This was extended by Fujii for [Formula: see text], when [Formula: see text]. Assuming the RH, we give the explicit asymptotic formula for the second moment of [Formula: see text] up to the second-order term, for [Formula: see text]. Our result conditionally refines Selberg’s and Fujii’s formulas and extends previous work by Goldston in [Formula: see text], where the case [Formula: see text] was considered.

On the zeros of linear combinations of derivatives of the Riemann zeta function

2016 ◽  
Vol 12 (06) ◽  
pp. 1703-1723 ◽  
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

In this paper, the zeros of linear combinations of the Riemann zeta function and its derivatives are studied. We establish an asymptotic formula for the number of zeros in a rectangle of height [Formula: see text]. We also find a sharp asymptotic formula for the supremum of the real parts of zeros of such combinations in a certain family.

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.

On the Riemann–Siegel formula

2007 ◽  
Vol 463 (2086) ◽  
pp. 2557-2568 ◽  
Author(s):  
A Kuznetsov
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Critical Point ◽  
Riemann Zeta Function ◽  
Error Term ◽  
Error Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this article, we derive a generalization of the Riemann–Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point, we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann–Siegel formula.

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