2021/07/06 by Atul Dixit, Dixit, Atul, Rajat Gupta +3
Engineering · Mathematics · #33E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Primary 30D05 #Probability and Statistical Research #Secondary 30E15 #Sports Dynamics and Biomechanics
paper · pdf · doi:10.48550/arxiv.2107.02607
openalex publication_date 2021/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F(x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F(x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study \mathscrFk,N(x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by \mathscrFk,N(x). Radchenko and Zagier gave a beautiful relation between the integral ∫01(log(1+tx))/(1+t) dt and F(x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between \mathscrFk,N(x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of \mathscrFk, N(x) and some generalized Lambert series are also obtained along with other supplementary results.