Zeros of Derivatives Of the Riemann Zeta-Function Near the Critical Line

1990 ◽  
pp. 95-110 ◽  
Author(s):  
J. B. Conrey ◽  
A. Ghosh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of ◽  
Zeros Of Derivatives

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.

scholarly journals Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

2018 ◽  
Vol 72 (3) ◽  
pp. 500-535 ◽  
Author(s):  
Louis-Pierre Arguin ◽  
David Belius ◽  
Paul Bourgade ◽  
Maksym Radziwiłł ◽  
Kannan Soundararajan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Short Interval ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Small values of the Riemann zeta function on the critical line

2015 ◽  
Vol 169 (3) ◽  
pp. 201-220 ◽  
Author(s):  
Justas Kalpokas ◽  
Paulius Šarka
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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.

Null trajectories for the symmetrized Hurwitz zeta function

2006 ◽  
Vol 463 (2077) ◽  
pp. 303-319 ◽  
Author(s):  
Ross C McPhedran ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Hurwitz Zeta Function ◽  
Trigonometric Functions ◽  
Asymptotic Results ◽  
End Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

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