TY - JOUR

T1 - Zeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shifts

AU - Dixit, Atul

AU - Kumar, Rahul

AU - Maji, Bibekananda

AU - Zaharescu, Alexandru

N1 - Funding Information:
The authors would like to sincerely thank the referee for his/her valuable comments and suggestions which improved the quality of the paper. The first author's research is supported by the SERB-DST grant ECR/2015/000070 whereas the third author is a SERB National Post Doctoral Fellow (NPDF) supported by the fellowship PDF/2017/000370 . Both sincerely thank SERB-DST for the support.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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 Ξ-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 Ξ-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.

AB - 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 Ξ-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 Ξ-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.

KW - Bounded vertical shifts

KW - Confluent hypergeometric function

KW - Riemann zeta function

KW - Theta transformation

KW - Zeros

UR - http://www.scopus.com/inward/record.url?scp=85047973625&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047973625&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2018.05.072

DO - 10.1016/j.jmaa.2018.05.072

M3 - Article

AN - SCOPUS:85047973625

VL - 466

SP - 307

EP - 323

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -