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

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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

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

}

*Journal of Number Theory*, vol. 107, no. 2, pp. 287-297. https://doi.org/10.1016/j.jnt.2004.04.003

**On a theorem of Levinson.** / Ki, Haseo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On a theorem of Levinson

AU - Ki, Haseo

PY - 2004/8/1

Y1 - 2004/8/1

N2 - Levinson investigated the number of real zeros of the real or imaginary part of π-σ/2-it/2Γ 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/2Γ s+λ/2 ζ(s+λ) ±π-s-λ/2Γ 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.

AB - Levinson investigated the number of real zeros of the real or imaginary part of π-σ/2-it/2Γ 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/2Γ s+λ/2 ζ(s+λ) ±π-s-λ/2Γ 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.

UR - http://www.scopus.com/inward/record.url?scp=4344676749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344676749&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2004.04.003

DO - 10.1016/j.jnt.2004.04.003

M3 - Article

AN - SCOPUS:4344676749

VL - 107

SP - 287

EP - 297

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -