### Abstract

In this article, we study the zeros of ζ (σ_{0} + s) ± ζ (σ_{0} - s) for a fixed σ_{0} ∈ R. We give a complete description where the zeros of the function are, except for frac(1, 2) ≤ σ_{0} ≤ frac(3, 4). It turns out that the behavior of zeros of the function with σ_{0} < frac(1, 2) is very different from that of the function with σ_{0} > frac(3, 4). Roughly speaking, zeros of the function for σ_{0} < frac(1, 2) tend to be located on the imaginary axis or the real axis. On the other hand, almost all zeros of the functions for σ_{0} > frac(3, 4) are arbitrarily close to Re (s) = ± (σ_{0} - frac(1, 2)) and there are fewer zeros in any strip which does not contain these axes. We have the analogues for the function ζ (σ_{0} + s) + a ζ (σ_{0} - s) (σ_{0} > frac(3, 4) and | a | = 1; σ_{0} > frac(1, 2) and | a | ≠ 0, 1).

Original language | English |
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Pages (from-to) | 2704-2755 |

Number of pages | 52 |

Journal | Journal of Number Theory |

Volume | 128 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2008 Sep 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*128*(9), 2704-2755. https://doi.org/10.1016/j.jnt.2008.02.009