Taiwanese Journal of Mathematics

ON ANALYTIC PROPERTIES AND CHARACTER ANALOGS OF HARDY SUMS

Yilmaz Simsek

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Abstract

The aim of this paper is to define $Y(h,k)$ sum which is related to the Hardy's sums $s_{5}(h,k)$. On the semi-group $G$, matrix operation of this sum is defined. Substituting mediants of Farey fractions into the matrix operation, $Y(h,k)$ sum is generalized. By using contour integration, the reciprocity theorem of the $Y(h,k)$ sum is proved. Moreover, by using $% L(1,\chi )$ function and Gauss sums, generalized character analogs of the Hardy sums are found.

Article information

Source
Taiwanese J. Math., Volume 13, Number 1 (2009), 253-268.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405282

Digital Object Identifier
doi:10.11650/twjm/1500405282

Mathematical Reviews number (MathSciNet)
MR2489317

Zentralblatt MATH identifier
1195.11057

Subjects
Primary: 11F20: Dedekind eta function, Dedekind sums
Secondary: 11B68: Bernoulli and Euler numbers and polynomials 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27] 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Keywords
$((x))$-function Dedekind sum Hardy sums Farey fraction Dirichlet character and $L$-function generalized Bernoulli numbers

Citation

Simsek, Yilmaz. ON ANALYTIC PROPERTIES AND CHARACTER ANALOGS OF HARDY SUMS. Taiwanese J. Math. 13 (2009), no. 1, 253--268. doi:10.11650/twjm/1500405282. https://projecteuclid.org/euclid.twjm/1500405282


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