On some Lambert-like series

P Agarwal , S Kanemitsu (1) , T Kuzumaki

1 Kyutech - Kyushu Institute of Technology

In this note, we study radial limits of power and Laurent series which are related to the Lerch zeta-function or polylogarithm function. As has been pointed out in [CKK18], there have appeared many instances in which the imaginary part of the Lerch zeta-function was considered by eliminating the real part by considering the odd part only. Mordell studied the properties of the power series resembling Lambert series, and in particular considered whether the limit function is a rational function or not. Our main result is the elucidation of the threshold case of b_n = 1/n studied by Mordell [Mor63], revealing that his result is the odd part of Theorem 1.1 in view of the identities (1.9), (1.5). We also refer to Lambert series considered by Titchmarsh [Tit38] in connection with Estermann's zeta-functions.

Mots clés: Lambert-like series; Lambert series; Estermann zeta function; trigonometric series; Dirichlet-Abel theorem; Riemann's posthumous fragment II

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

P Agarwal, S Kanemitsu, T Kuzumaki. On some Lambert-like series. Hardy-Ramanujan Journal, 2020, Volume 42 - Special Commemorative volume in honour of Alan Baker - 2019, pp.100 - 110. ⟨10.46298/hrj.2020.6461⟩. ⟨hal-02554284⟩