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.
Publisher Copyright:
© 2018 Elsevier Inc.
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
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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 -