Zeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shifts

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:466

Article

Dixit, Atul¹  Kumar, Rahul¹  Maji, Bibekananda¹  Zaharescu, Alexandru^2,3 

[1] Indian Inst Technol Gandhinagar, Discipline Math, Gandhinagar 382355, Gujarat, India

[2] Univ Illinois, Dept Math, 1409 West Green St, Urbana, IL 61801 USA

[3] Romanian Acad, Simion Stoilow Inst Math, POB 1-764, RO-014700 Bucharest, Romania

关键词: Riemann zeta function;    Theta transformation;    Confluent hypergeometric function;    Bounded vertical shifts;    Zeros;   

DOI  :  10.1016/j.jmaa.2018.05.072

来源: Elsevier

【 摘 要 】

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann Xi-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Xi-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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