Commentary on: [L 112] More than One Third of Zeros of Riemann’s Zeta-Function are on σ = 1/2 [L 113] Zeros of Derivative of Riemann’s ξ-Function and Corrigendum [L 116] At least One-Third of Zeros of Riemann’s Zeta-Function are on σ = 1/2 [L 117] Generalization of Recent Method Giving Lower Bound for N0(T) of Riemann’s Zeta-Function [L 118] (with H. L. Montgomery) Zeros of the Derivatives of the Riemann Zeta-Function [L 120] A Simplification of the Proof that N 0(T) > (1 /3)N(T) for Riemann’s Zeta-Function [L 121] Deduction of Semi-Optimal Mollifier for Obtaining Lower Bound for N 0(T) of Riemann’s Zeta-Function

Author(s):  
Brian Conrey
10.37236/1195 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
David W. Farmer

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.


2018 ◽  
Vol 14 (02) ◽  
pp. 371-382
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu

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.


2011 ◽  
Vol 16 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Joern Steuding

We study the sequence of nontrivial zeros of the Riemann zeta-function with respect to sequences of zeros of other related functions, namely, the Hurwitz zeta-function and the derivative of Riemann's zeta-function. Finally, we investigate connections of the nontrivial zeros with the periodic zeta-function. On the basis of computation we derive several classifications of the nontrivial zeros of the Riemann zeta-function and stateproblems which mightbe ofinterestfor abetter understanding of the distribution of those zeros.


Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding

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.


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