scholarly journals On the location of the zero-free half-plane of a random Epstein zeta function

2017 ◽  
Vol 371 (3-4) ◽  
pp. 1191-1227 ◽  
Author(s):  
Andreas Strömbergsson ◽  
Anders Södergren
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Epstein Zeta Function

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

scholarly journals A Minimum Problem for the Epstein Zeta-Function

1953 ◽  
Vol 1 (4) ◽  
pp. 149-158 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Hexagonal Lattice ◽  
Mean Value ◽  
Minimum Problem ◽  
Constant Multiple ◽  
Packing And Covering ◽  
Closest Packing ◽  
Epstein Zeta Function ◽  
The Mean

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s=. Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z() would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s.

scholarly journals On Manin's conjecture for singular del Pezzo surfaces of degree four, II

2007 ◽  
Vol 143 (3) ◽  
pp. 579-605 ◽  
Author(s):  
R. DE LA BRETÈCHE ◽  
T. D. BROWNING
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Open Subset ◽  
Height Function ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Height Zeta Function

AbstractThis paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four$X \subset \bfP^4$. In fact, ifU⊂Xis the open subset formed by deleting the lines fromX, andHis the usual projective height function on$\bfP^4(\Q)$, then the height zeta function$ \sum_{x \in U(\Q)}{H(x)^{-s}} $is analytically continued to the half-plane ℜe(s) > 17/20.

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Euler Products ◽  
Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

scholarly journals SELF-APPROXIMATION FOR THE RIEMANN ZETA FUNCTION

2012 ◽  
Vol 87 (3) ◽  
pp. 452-461 ◽  
Author(s):  
TAKASHI NAKAMURA ◽  
ŁUKASZ PAŃKOWSKI
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Critical Strip ◽  
Approximation Of Functions ◽  
Joint Universality ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Right

AbstractIn the paper we deal with self-approximation of the Riemann zeta function in the half plane $\operatorname {Re} s\gt 1$ and in the right half of the critical strip. We also prove some results concerning joint universality and joint value approximation of functions $\zeta (s+\lambda +id\tau )$ and $\zeta (s+i\tau )$.

scholarly journals A lemma about the Epstein zeta-function

1964 ◽  
Vol 6 (4) ◽  
pp. 198-201 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Positive Definite ◽  
Simple Pole ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shapewith y > 0. We write (for s > 1)The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

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