# On a theorem of Levinson

Research output: Contribution to journalArticle

7 Citations (Scopus)

### Abstract

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.

Original language English 287-297 11 Journal of Number Theory 107 2 https://doi.org/10.1016/j.jnt.2004.04.003 Published - 2004 Aug 1

### Fingerprint

Number of Real Zeros
Complex number
Theorem
Riemann zeta function
Functional equation
Zero
Form

### All Science Journal Classification (ASJC) codes

• Algebra and Number Theory

### Cite this

Ki, Haseo. / On a theorem of Levinson. In: Journal of Number Theory. 2004 ; Vol. 107, No. 2. pp. 287-297.
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In: Journal of Number Theory, Vol. 107, No. 2, 01.08.2004, p. 287-297.

Research output: Contribution to journalArticle

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AU - Ki, Haseo

PY - 2004/8/1

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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.

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