On Eisenstein series in M2k(Γ0(N)) and their applications

Let k, N ∈ N with N square-free and k > 1. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any f(z) ∈ M2k(Γ0(N)) in terms of sum of divisors function. In particular, if f(z) ∈ E2k(Γ0(N)), then the computation will to yield to an expressio...

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Main Author: Aygin, Zafer Selcuk
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2020
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Online Access:https://hdl.handle.net/10356/143236
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1432362023-02-28T19:47:02Z On Eisenstein series in M2k(Γ0(N)) and their applications Aygin, Zafer Selcuk School of Physical and Mathematical Sciences Science::Mathematics Sum of Divisors Function Convolution Sums Let k, N ∈ N with N square-free and k > 1. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any f(z) ∈ M2k(Γ0(N)) in terms of sum of divisors function. In particular, if f(z) ∈ E2k(Γ0(N)), then the computation will to yield to an expression for the Fourier coefficients of f(z). Then we apply our main theorem to give formulas for convolution sums of the divisor function to extend the result by Ramanujan, and to eta quotients which yields to formulas for number of representations of integers by certain families of quadratic forms. At last we give essential results to derive similar results for modular forms in a more general setting. Ministry of Education (MOE) Accepted version I would like to thank Song Heng Chan for his helpful comments throughout the course of this research, and I am grateful to Heng Huat Chan, whose feedback helped to give a more elegant proof of Lemma 3.1. I am also indebted to Kenneth S. Williams for his encouraging remarks and corrections on an earlier version of this work. The author was supported by the Singapore Ministry of Education Academic Research Fund, Tier 2, project number MOE2014-T2-1-051, ARC40/14. 2020-08-14T04:58:05Z 2020-08-14T04:58:05Z 2018 Journal Article Aygin, Z. S. (2019). On Eisenstein series in M2k(Γ0(N)) and their applications. Journal of Number Theory, 195, 358-375. doi:10.1016/j.jnt.2018.06.010 0022-314X https://hdl.handle.net/10356/143236 10.1016/j.jnt.2018.06.010 2-s2.0-85050163291 195 358 375 en Journal of Number Theory © 2018 Elsevier Inc. All rights reserved. This paper was published in Journal of Number Theory and is made available with permission of Elsevier Inc. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Sum of Divisors Function
Convolution Sums
spellingShingle Science::Mathematics
Sum of Divisors Function
Convolution Sums
Aygin, Zafer Selcuk
On Eisenstein series in M2k(Γ0(N)) and their applications
description Let k, N ∈ N with N square-free and k > 1. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any f(z) ∈ M2k(Γ0(N)) in terms of sum of divisors function. In particular, if f(z) ∈ E2k(Γ0(N)), then the computation will to yield to an expression for the Fourier coefficients of f(z). Then we apply our main theorem to give formulas for convolution sums of the divisor function to extend the result by Ramanujan, and to eta quotients which yields to formulas for number of representations of integers by certain families of quadratic forms. At last we give essential results to derive similar results for modular forms in a more general setting.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Aygin, Zafer Selcuk
format Article
author Aygin, Zafer Selcuk
author_sort Aygin, Zafer Selcuk
title On Eisenstein series in M2k(Γ0(N)) and their applications
title_short On Eisenstein series in M2k(Γ0(N)) and their applications
title_full On Eisenstein series in M2k(Γ0(N)) and their applications
title_fullStr On Eisenstein series in M2k(Γ0(N)) and their applications
title_full_unstemmed On Eisenstein series in M2k(Γ0(N)) and their applications
title_sort on eisenstein series in m2k(γ0(n)) and their applications
publishDate 2020
url https://hdl.handle.net/10356/143236
_version_ 1759854006182608896