On the mean values of Dedekind sum and certain Hardy–Berndt sums

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Publication Details

Journal
Mathematica Slovaca
Published
2026-07-14
DOI
https://doi.org/10.1515/ms-2026-0017
Primary Topic
Analytic Number Theory Research
Type
article
Field-Weighted Citation Impact
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article

On the mean values of Dedekind sum and certain Hardy–Berndt sums

Merve Kara, Mümün Can, Muhammet Cihat Dağlı
Mathematica Slovaca
Analytic Number Theory Research
article

On the mean values of Dedekind sum and certain Hardy–Berndt sums

Merve Kara, Mümün Can, Muhammet Cihat Dağlı
article en

Abstract

Abstract This paper investigates the mean value behavior of Apostol-type Dedekind and certain Hardy–Berndt sums when evaluated along sequences defined by second-order linear recurrences. By employing reciprocity formulas together with the structural properties of these sequences, we derive explicit and asymptotic formulas for a broad class of sums, including <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> ${s}_{p}\left(a,c\right)$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> ${S}_{p}\left(a,c\right)$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfenced> </m:mrow> </m:msubsup> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> ${S}_{p}^{\left(3\right)}\left(a,c\right)$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:mfenced> </m:mrow> </m:msubsup> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> ${S}_{p}^{\left(5\right)}\left(a,c\right)$ . In particular, we establish the unboundedness of these sums in a generalized setting and show that both their pointwise values and associated mean values exhibit a relative asymptotic growth behavior that is essentially independent of the parameter p , revealing a certain universality phenomenon.

Mathematica Slovaca
Akdeniz University (TR)
Openalex Percentile: Top 4%
Analytic Number Theory Research
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