Real zeros of Hurwitz zeta-functions and their asymptotic behavior in the interval (0,1)

• Published in: Journal of mathematical analysis and applications Vol. 473; no. 2; pp. 624 - 635
• Main Authors: Endo, Kenta, Suzuki, Yuta
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.05.2019
• Subjects: Hurwitz zeta-function; Real zeros; Real zeros; Hurwitz zeta-function
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2018.12.001

Let 0<a≤1,s∈C, and ζ(s,a) be the Hurwitz zeta-function. Recently, Nakamura showed that ζ(σ,a) does not vanish for any 0<σ<1 if and only if 1/2≤a≤1. In this paper, we show that ζ(σ,a) has precisely one zero in the interval (0,1) if 0<a<1/2. Moreover, we reveal the asymptotic behavior of this unique zero with respect to a.