### Abstract

Levinson investigated the number of real zeros of the real or imaginary part of π^{-σ/2-it/2}Γ σ/2 + it/2 ζ(σ+it), where σ>0 and ζ(s) is the Riemann zeta function. By the functional equation, π^{-s/2} Γ s/2 ζ(s)=π^{-1-s/2} Γ1-s/2 ζ(1-s), we may assume σ 1/2. In this paper, we consider π^{-s+λ/2}Γ s+λ/2 ζ(s+λ) ±π^{-s-λ/2}Γ s-λ/2 ζ (s-λ) for any complex number s and any λ>0, as general forms of the real or imaginary part of the above function, and then we further study the zeros of the functions.

Original language | English |
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Pages (from-to) | 287-297 |

Number of pages | 11 |

Journal | Journal of Number Theory |

Volume | 107 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2004 Aug 1 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

Ki, H. (2004). On a theorem of Levinson.

*Journal of Number Theory*,*107*(2), 287-297. https://doi.org/10.1016/j.jnt.2004.04.003