scholarly journals Dedekind sums take each value infinitely many times

2018 ◽  
Vol 14 (04) ◽  
pp. 1009-1012 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Dedekind Sums ◽  
Dedekind Sum

For [Formula: see text] and [Formula: see text], [Formula: see text], let [Formula: see text] denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs [Formula: see text], [Formula: see text], with [Formula: see text] tending to infinity as [Formula: see text] grows, such that [Formula: see text] for all [Formula: see text].

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 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.

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 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].

scholarly journals Dedekind sums s(a, b) and inversions modulo b

2015 ◽  
Vol 11 (08) ◽  
pp. 2325-2339
Author(s):  
Yiwang Chen ◽  
Nicholas Dunn ◽  
Campbell Hewett ◽  
Shashwat Silas
Keyword(s):  
Kloosterman Sums ◽  
Dedekind Sums ◽  
Roots Of Unity ◽  
Sufficient Condition ◽  
Dedekind Sum

We introduce the inversion polynomial for Dedekind sums fb(x) = ∑ x inv (a, b) to study the number of s(a, b) which have the same value for a given b. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new sufficient condition for the inequality of two Dedekind sums based on the reciprocity formula.

An integer-valued expression of Dedekind sums

2017 ◽  
Vol 13 (05) ◽  
pp. 1253-1259
Author(s):  
Simon Macourt
Keyword(s):  
General Result ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
The One

We prove a conjecture of Myerson and Phillips on when an expression involving Dedekind sums is an integer. We also provide a more general result and use this to extend the work of Myerson and Phillips studying whether the points of the graph of the one-variable Dedekind sum that fall on the line are dense.

scholarly journals On the fractional parts of Dedekind sums

2014 ◽  
Vol 11 (01) ◽  
pp. 29-38 ◽  
Author(s):  
Kurt Girstmair
Keyword(s):  
Rational Number ◽  
Fractional Part ◽  
Dedekind Sums ◽  
Dedekind Sum

We show that each rational number r, 0 ≤ r < 1, occurs as the fractional part of a Dedekind sum S(m, n). Further, we determine the number of integers x, 1 ≤ x ≤ n, (x, n) = 1, such that S(m, n) and S(x, n) have the same fractional parts.

DENOMINATORS OF DEDEKIND SUMS IN FUNCTION FIELDS

2013 ◽  
Vol 09 (06) ◽  
pp. 1423-1430 ◽  
Author(s):  
YOSHINORI HAMAHATA
Keyword(s):  
Rational Function ◽  
Upper Bound ◽  
Function Fields ◽  
Dedekind Sums ◽  
Dedekind Sum

We consider the Dedekind sum s(a, c) in rational function fields. It is very similar to the classical Dedekind sum D(a, c). The objective of this study is to give a good upper bound of the degree of the denominator of s(a, c). As an application of our result, we present a condition on the elements a1, a2, and c of 𝔽q[T] such that s(a1, c) = s(a2, c).

scholarly journals On a recent reciprocity formula for Dedekind sums

2019 ◽  
Vol 15 (07) ◽  
pp. 1469-1472
Author(s):  
Kurt Girstmair
Keyword(s):  
Main Tool ◽  
Dedekind Sums ◽  
Natural Numbers ◽  
Dedekind Sum ◽  
Term Relation ◽  
Special Case

Let [Formula: see text] denote the classical Dedekind sum and [Formula: see text]. Recently, Du and Zhang proved the following reciprocity formula. If [Formula: see text] and [Formula: see text] are odd natural numbers, [Formula: see text], then [Formula: see text] where [Formula: see text] and [Formula: see text]. In this paper, we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of [Formula: see text]-series, our main tool is the three-term relation for Dedekind sums.

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