scholarly journals Evaluating Fractional Derivatives of the Riemann Zeta Function

2020 ◽  
pp. 94-101
Author(s):  
Ricky E. Farr ◽  
Sebastian Pauli ◽  
Filip Saidak
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Fractional Derivatives ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

scholarly journals Counting distinct zeros of the Riemann zeta-function

10.37236/1195 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
David W. Farmer
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Simple Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

Bounds on the number of simple zeros of the derivatives of a function are used to give bounds on the number of distinct zeros of the function.

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 On some averages at the zeros of the derivatives of the Riemann zeta-function

2011 ◽  
Vol 131 (10) ◽  
pp. 1939-1961 ◽  
Author(s):  
Yunus Karabulut ◽  
Cem Yalçın Yıldırım
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

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 Zeros of the derivatives of the Riemann zeta-function

1974 ◽  
Vol 133 (0) ◽  
pp. 49-65 ◽  
Author(s):  
Norman Levinson ◽  
H. L. Montgomery
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of
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