Asymptotic estimates for Stieltjes constants: a probabilistic approach

2010 ◽  
Vol 467 (2128) ◽  
pp. 954-963 ◽  
Author(s):  
José A. Adell
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Differential Calculus ◽  
Probabilistic Approach ◽  
Upper Bounds ◽  
Asymptotic Estimates ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

Let ( γ n ) n ≥ 0 be the sequence of Stieltjes constants appearing in the Laurent expansion of the Riemann zeta function. We obtain explicit upper bounds for | γ n |, whose order of magnitude is as n tends to infinity. To do this, we use a probabilistic approach based on a differential calculus for the gamma process.

scholarly journals A Probabilistic Proof for Representations of the Riemann Zeta Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 369
Author(s):  
Jiamei Liu ◽  
Yuxia Huang ◽  
Chuancun Yin
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Approach ◽  
Recurrence Formula ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Probabilistic Proof ◽  
Riemann Zeta ◽  
Positive Integers

In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.

New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

2008 ◽  
Vol 464 (2091) ◽  
pp. 711-731 ◽  
Author(s):  
Mark W Coffey
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Logarithmic Derivative ◽  
Bernoulli Numbers ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Power Series Expansions ◽  
Riemann Zeta ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of λ k is developed in terms of the Stieltjes constants γ j and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants η j entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s =1. We also demonstrate that the η j coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of η j and σ j , where are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function.

scholarly journals Low Pseudomoments of the Riemann Zeta Function and Its Powers

2020 ◽  
Author(s):  
Maxim Gerspach
Keyword(s):  
Zeta Function ◽  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Probabilistic Methods ◽  
Upper And Lower Bounds ◽  
Partial Sum ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

Abstract The $2 q$-th pseudomoment $\Psi _{2q,\alpha }(x)$ of the $\alpha $-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta ^\alpha $ on the critical line. Using probabilistic methods of Harper, we prove upper and lower bounds for these pseudomoments when $q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko et al., these bounds determine the size of all pseudomoments with $q> 0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of the partial sum, and it turns out that there are three different ranges with different growth behaviours. In particular, the results give the order of magnitude of $\Psi _{2 q, 1}(x)$ for all $q> 0$.

scholarly journals SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 70 (4) ◽  
pp. 1387-1396 ◽  
Author(s):  
Winston Heap ◽  
Maksym Radziwiłł ◽  
K Soundararajan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Upper Bounds ◽  
The Riemann Zeta Function ◽  
Fractional Moments ◽  
Riemann Zeta

Abstract We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin–Chandee–Radziwiłł.

scholarly journals Low Pseudomoments of Euler Products

2021 ◽  
Author(s):  
Maxim Gerspach ◽  
Youness Lamzouri
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Euler Products ◽  
Riemann Zeta

Abstract In this paper, we determine the order of magnitude of the 2 q-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0 \lt q\le 1/2$ and $0 \lt \alpha \lt 1$, completing the results of Bondarenko, Heap and Seip, and Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

The Riemann zeta-function and its derivatives

1995 ◽  
Vol 450 (1940) ◽  
pp. 477-499 ◽  
Keyword(s):  
Zeta Function ◽  
Error Bounds ◽  
Riemann Zeta Function ◽  
Summation Methods ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Higher Derivatives ◽  
Riemann Zeta ◽  
Significant Figures ◽  
Derivatives Of

Formulas for higher derivatives of the Riemann zeta-function are developed from Ramanujan’s theory of the ‘constant’ of series. By using the Euler-Maclaurin summation methods, formulas for ζ( n )( s ), ζ( n )(1 – s ) and ζ( n )(0) are obtained. Additional formulas involving the Stieltjes constants are also derived. Analytical expression for error bounds is given in each case. The formulas permit accurate derivative evaluation and the error bounds are shown to be realistic. A table of ζ'( s ) is presented to 20 significant figures for s = –20(0.1)20. For rational arguments, ζ(1/ k ), ζ'(1/ k ) are given for k = –10(1)10. The first ten zeros of ζ'( s ) are also tabulated. Because the Stieltjes constants appear in many formulas, the constants were evaluated freshly for this work. Formulas for the γ n are derived with new error bounds, and a tabulation of the constants is given from n = 0 to 100.

scholarly journals Divergent Integrals, The Riemann Zeta Function, and The Vacuum

2020 ◽  
Author(s):  
Siamak Tafazoli
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Vacuum Energy ◽  
Opposite Sign ◽  
Vacuum Energy Density ◽  
Power Functions ◽  
Vacuum States ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

This paper presents a new estimate for the vacuum energy density by summing the contributions of all quantum fields' vacuum states which turns out to be in the same order of magnitude (but with opposite sign) as the predictions of current cosmological models and all observational data to date. The basis for this estimate is the recent results on the analytical solution to improper integral of divergent power functions using the Riemann Zeta function.

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
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