### 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 |
---|---|

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

}

*Journal of Number Theory*, vol. 128, no. 9, pp. 2704-2755. https://doi.org/10.1016/j.jnt.2008.02.009

**On the zeros of sums of the Riemann zeta function.** / Ki, Haseo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the zeros of sums of the Riemann zeta function

AU - Ki, Haseo

PY - 2008/9/1

Y1 - 2008/9/1

N2 - In this article, we study the zeros of ζ (σ0 + s) ± ζ (σ0 - s) for a fixed σ0 ∈ R. We give a complete description where the zeros of the function are, except for frac(1, 2) ≤ σ0 ≤ frac(3, 4). It turns out that the behavior of zeros of the function with σ0 < frac(1, 2) is very different from that of the function with σ0 > frac(3, 4). Roughly speaking, zeros of the function for σ0 < 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 σ0 > frac(3, 4) are arbitrarily close to Re (s) = ± (σ0 - 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 (σ0 + s) + a ζ (σ0 - s) (σ0 > frac(3, 4) and | a | = 1; σ0 > frac(1, 2) and | a | ≠ 0, 1).

AB - In this article, we study the zeros of ζ (σ0 + s) ± ζ (σ0 - s) for a fixed σ0 ∈ R. We give a complete description where the zeros of the function are, except for frac(1, 2) ≤ σ0 ≤ frac(3, 4). It turns out that the behavior of zeros of the function with σ0 < frac(1, 2) is very different from that of the function with σ0 > frac(3, 4). Roughly speaking, zeros of the function for σ0 < 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 σ0 > frac(3, 4) are arbitrarily close to Re (s) = ± (σ0 - 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 (σ0 + s) + a ζ (σ0 - s) (σ0 > frac(3, 4) and | a | = 1; σ0 > frac(1, 2) and | a | ≠ 0, 1).

UR - http://www.scopus.com/inward/record.url?scp=48349112627&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=48349112627&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2008.02.009

DO - 10.1016/j.jnt.2008.02.009

M3 - Article

AN - SCOPUS:48349112627

VL - 128

SP - 2704

EP - 2755

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 9

ER -