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.
Publisher Copyright:
© 2014 Elsevier Inc.
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
SN - 0022-314X
VL - 149
SP - 404
EP - 434
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -