scholarly journals Dedekind sums arising from newform Eisenstein series

2020 ◽  
Vol 16 (10) ◽  
pp. 2129-2139
Author(s):  
T. Stucker ◽  
A. Vennos ◽  
M. P. Young
Keyword(s):  
Eisenstein Series ◽  
Short Proof ◽  
Explicit Construction ◽  
Transformation Rule ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Dirichlet Characters

For primitive nontrivial Dirichlet characters [Formula: see text] and [Formula: see text], we study the weight zero newform Eisenstein series [Formula: see text] at [Formula: see text]. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of [Formula: see text]. We also give a short proof of the reciprocity formula for this Dedekind sum.

Non-random behavior in sums of modular symbols

2021 ◽  
pp. 1-25
Author(s):  
Alex Cowan
Keyword(s):  
Eisenstein Series ◽  
Spectral Decomposition ◽  
Fourier Coefficients ◽  
Dirichlet Character ◽  
Common Phenomenon ◽  
Modular Symbols ◽  
Dirichlet Characters ◽  
Random Behavior ◽  
The Common ◽  
Automorphic Functions

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

Dedekind sums and the transformation formula in function fields

2016 ◽  
Vol 12 (08) ◽  
pp. 2061-2072 ◽  
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Function Fields ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Sum Formula

Dedekind used the classical Dedekind sum [Formula: see text] to describe the transformation of [Formula: see text] under the substitution [Formula: see text]. In this paper, we use the Dedekind sum [Formula: see text] in function fields to describe the transformation of a certain series under the substitution [Formula: see text].

scholarly journals Franel integrals of order four

1996 ◽  
Vol 60 (2) ◽  
pp. 192-203 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Unit Interval ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Elementary Method ◽  
The Third ◽  
Reciprocity Law ◽  
Order Four

AbstractLet ((x)) =x−⌊x⌋−1/2 be the swatooth function. Ifa, b, cand e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

scholarly journals Approximation of rational numbers by Dedekind sums

2014 ◽  
Vol 10 (05) ◽  
pp. 1241-1244 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Dedekind Sums ◽  
Dedekind Sum

Given a rational number x and a bound ε, we exhibit m, n such that |x - s(m, n)| < ε. Here s(m, n) is the classical Dedekind sum and the parameters m and n are completely explicit in terms of x and ε.

scholarly journals The Saito–Kurokawa lift and Siegel’s theta series

2017 ◽  
Vol 13 (07) ◽  
pp. 1679-1693
Author(s):  
Roland Matthes
Keyword(s):  
Spectral Analysis ◽  
Eisenstein Series ◽  
Short Proof ◽  
Theta Series ◽  
Converse Theorem ◽  
Selberg Integral ◽  
Real Analytic

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.

scholarly journals The norm of a Ree group

2010 ◽  
Vol 199 ◽  
pp. 15-41
Author(s):  
Tom De Medts ◽  
Richard M. Weiss
Keyword(s):  
Short Proof ◽  
Explicit Construction ◽  
Basic Properties ◽  
Ree Group

AbstractWe give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

scholarly journals The norm of a Ree group

2010 ◽  
Vol 199 ◽  
pp. 15-41
Author(s):  
Tom De Medts ◽  
Richard M. Weiss
Keyword(s):  
Short Proof ◽  
Explicit Construction ◽  
Basic Properties ◽  
Ree Group

AbstractWe give an explicit construction of the Ree groups of typeG2as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

DIFFERENTIAL EQUATIONS FOR CUBIC THETA FUNCTIONS

2011 ◽  
Vol 07 (07) ◽  
pp. 1945-1957 ◽  
Author(s):  
TIM HUBER
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Theta Function ◽  
Modular Group ◽  
Short Proof ◽  
Nonlinear Differential Equations ◽  
Theta Functions ◽  
Coupled Systems ◽  
Function Identity ◽  
Theta Function Identity

We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujan's differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujan's original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.

scholarly journals The largest values of Dedekind sums

2016 ◽  
Vol 13 (06) ◽  
pp. 1579-1583 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Positive Integer ◽  
Dedekind Sums ◽  
Dedekind Sum

Let [Formula: see text] denote the classical Dedekind sum, where [Formula: see text] is a positive integer and [Formula: see text], [Formula: see text]. For a given positive integer [Formula: see text], we describe a set of at most [Formula: see text] numbers [Formula: see text] for which [Formula: see text] may be [Formula: see text], provided that [Formula: see text] is sufficiently large. For the numbers [Formula: see text] not in this set, [Formula: see text].

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