TY - JOUR

T1 - Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

AU - Dixit, Atul

AU - Robles, Nicolas

AU - Roy, Arindam

AU - Zaharescu, Alexandru

N1 - Funding Information:
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

Y1 - 2015/4/1

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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U2 - 10.1016/j.jnt.2014.10.004

DO - 10.1016/j.jnt.2014.10.004

M3 - Article

AN - SCOPUS:84920265759

VL - 149

SP - 404

EP - 434

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -