### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 404-434 |

Number of pages | 31 |

Journal | Journal of Number Theory |

Volume | 149 |

DOIs | |

State | Published - Apr 1 2015 |

### Keywords

- Bounded vertical shifts
- Critical line
- Ramanujan's Lost Notebook
- Riemann zeta-function
- Self-reciprocal functions
- Theta transformation formula
- Zeros

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Dixit, A., Robles, N., Roy, A., & Zaharescu, A. (2015). Zeros of combinations of the Riemann ξ-function on bounded vertical shifts.

*Journal of Number Theory*,*149*, 404-434. https://doi.org/10.1016/j.jnt.2014.10.004