Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

Atul Dixit; Nicolas Robles; Arindam Roy; Alexandru Zaharescu

doi:10.1016/j.jnt.2014.10.004

Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

Atul Dixit
, Nicolas Robles
, Arindam Roy
, Alexandru Zaharescu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G.H. Hardy and W.L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform.

Original language	English (US)
Pages (from-to)	404-434
Number of pages	31
Journal	Journal of Number Theory
Volume	149
DOIs	https://doi.org/10.1016/j.jnt.2014.10.004
State	Published - Apr 1 2015

Keywords

Bounded vertical shifts
Critical line
Ramanujan's Lost Notebook
Riemann zeta-function
Self-reciprocal functions
Theta transformation formula
Zeros

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1016/j.jnt.2014.10.004

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abstract = "In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G.H. Hardy and W.L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform.",

keywords = "Bounded vertical shifts, Critical line, Ramanujan's Lost Notebook, Riemann zeta-function, Self-reciprocal functions, Theta transformation formula, Zeros",

author = "Atul Dixit and Nicolas Robles and Arindam Roy and Alexandru Zaharescu",

note = "The first author is funded in part by the grant NSF {\textendash} DMS 1112656 of Professor Victor H. Moll of Tulane University and sincerely thanks him for the support. The second author acknowledges partial support of SNF grant 200020 149150\textbackslash{}1 . The authors would like to express their sincere gratitude to the referee for his valuable comments on the earlier version of this paper.",

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AU - Robles, Nicolas

AU - Roy, Arindam

AU - Zaharescu, Alexandru

N1 - The first author is funded in part by the grant NSF – DMS 1112656 of Professor Victor H. Moll of Tulane University and sincerely thanks him for the support. The second author acknowledges partial support of SNF grant 200020 149150\1 . The authors would like to express their sincere gratitude to the referee for his valuable comments on the earlier version of this paper.

PY - 2015/4/1

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N2 - In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G.H. Hardy and W.L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform.

AB - In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G.H. Hardy and W.L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform.

KW - Bounded vertical shifts

KW - Critical line

KW - Ramanujan's Lost Notebook

KW - Riemann zeta-function

KW - Self-reciprocal functions

KW - Theta transformation formula

KW - Zeros

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JO - Journal of Number Theory

JF - Journal of Number Theory

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Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

Abstract

Keywords

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