On the Wintner-Ingham-Segal summability method

S Kanemitsu (1) , T Kuzumaki (2) , Y Tanigawa (3)

1 Kyutech - Kyushu Institute of Technology
2 Faculty of Engineering, Gifu University
3 Graduate School of Mathematics [Nagoya]

The aim of this note is to establish a subclass of $\mathcal{F}$ considered by Segal if functions for which the Ingham-Wintner summability implies $\mathcal{F}$-summability as wide as possible. The subclass is subject to the estimate for the error term of the prime number theorem. We shall make good use of Stieltjes integration which elucidates previous results obtained by Segal.

Mots clés: (Ingham) summability; zero-free region; Stieltjes integral; weak Riemann hypothesis 2010 Mathematics Subject Classification 40D99; 11M99

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]

S Kanemitsu, T Kuzumaki, Y Tanigawa. On the Wintner-Ingham-Segal summability method. Hardy-Ramanujan Journal, 2019, Atelier Digit_Hum, pp.40 - 47. ⟨10.46298/hrj.2019.5104⟩. ⟨hal-01985935⟩