scholarly journals Resurgent expansion of Lambert series and iterated Eisenstein integrals

2021 ◽  
Vol 15 (1) ◽  
pp. 1-57
Author(s):  
Daniele Dorigoni ◽  
Axel Kleinschmidt
Keyword(s):  
Lambert Series

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

scholarly journals Linear independence of certain Lambert series

2014 ◽  
Vol 142 (10) ◽  
pp. 3411-3419 ◽  
Author(s):  
Florian Luca ◽  
Yohei Tachiya
Keyword(s):  
Linear Independence ◽  
Lambert Series

Generalized Lambert series

1996 ◽  
pp. 357-370 ◽  
Author(s):  
Ronald Evans
Keyword(s):  
Lambert Series

Lambert series associated to Hilbert modular form

2021 ◽  
pp. 1-15
Author(s):  
Rishabh Agnihotri
Keyword(s):  
Asymptotic Expansion ◽  
Modular Form ◽  
Cusp Form ◽  
Riemann Zeta Function ◽  
Hilbert Modular Form ◽  
Lambert Series ◽  
Nontrivial Zeros ◽  
Hilbert Modular ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Liouville identities with two functions

2016 ◽  
Vol 12 (05) ◽  
pp. 1345-1363
Author(s):  
Şaban Alaca ◽  
Greg Doyle
Keyword(s):  
Power Series ◽  
Lambert Series ◽  
Convolution Sums

We express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.

scholarly journals An identity involving Nörlund polynomials

1991 ◽  
Vol 43 (2) ◽  
pp. 307-315
Author(s):  
A.E. Özlük ◽  
C. Snyder
Keyword(s):  
Lattice Points ◽  
Transformation Formula ◽  
Lambert Series

We prove an identity involving Nörlund polynomials, the proof of which is elementary and involves the enumeration of lattice points. The identity is slightly stronger than an identity of Carlitz which he obtained by using Apostol's transformation formula for Lambert series.

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