On some identities involving certain Hardy sums and a Kloosterman sum
type: Item
instance of: scholarly article
title: On some identities involving certain Hardy sums and a Kloosterman sum
zbMATH Open document ID: 1476.11108
author: M. Cihat Dağlı
published in: Ukrainian Mathematical Journal
publication date: 2021-06-21T00:00:00Z
review text: Several arithmetic sums similar to Dedekind sums \(s(h,p)\) are encountered in the theory of logarithms of the classical theta functions. They were studied by Hardy and Berndt and, therefore, they are now called Hardy or Hardy-Berndt sums. There are six sums of this kind, two of which for any positive integer \(p>1\) and integer \(n\) defined as follows: \[ \begin{gathered} s_{4}(h, p)=\sum_{a=1}^{p-1}(-1)^{[h a / p]},\quad s_{5}(h, p)=\sum_{a=1}^{p-1}(-1)^{a+[h a / p]}\left(\left(\frac{a}{p}\right)\right). \end{gathered} \] For any positive integer \(p>1\) and integer \(n\) a Kloosterman sum \(K(n, p)\) is defined by \[ K(n, p)=\mathop{\sum'}_{a=1}^{p} e\left(\frac{n a+\bar{a}}{p}\right) \] where \(\bar{a}\) denotes the solution of the congruence \(x a \equiv 1(\bmod p)\) and the prime denotes the operation of summation over all \(1 \leq a \leq p\) such that \( (a, p)=1\) and \(e(x)=e^{2 \pi i x}.\) The main results of the paper are three following theorems. Theorem 1.2. Let \(h\) and \(p\) be odd primes. Then \[ s_{5}(2 \bar{p}, h)+s_{5}(2 \bar{h}, p)=\frac{1}{2}-\frac{h^{2}+p^{2}}{4 h p} \] where \(\bar{p}\) and \(\bar{h}\) satisfy the congruence \(p \bar{p} \equiv 1\pmod h\) and \(h \bar{h} \equiv 1 \pmod p\). Theorem 1.3. Let \(p\) be an odd prime. Then \[ \sum_{m=1}^{p-1} \sum_{n=1}^{p-1} K(m, p) K(n, p) s_{4}(m \bar{n}, p)=p^{2}(p-1) \] Theorem 1.4. Let \(p\) be an odd prime. Then \[ \begin{aligned} &\sum_{m=1}^{p-1} \sum_{n=1}^{p-1}|K(m, p)|^{2}|K(n, p)|^{2} s_{4}(m \bar{n}, p) & \\ &= \begin{cases}p^{3}(p-1), & \text { if } \quad p \equiv 1(\bmod 4) \\ p^{3}(p-1)-36 p^{2} h_{p}^{2}, & \text { if } \quad p \equiv 3(\bmod 8) \\ p^{3}(p-1)-4 p^{2} h_{p}^{2}, & \text { if } \quad p \equiv 7(\bmod 8)\end{cases} \end{aligned} \] where \(h_{p}\) denotes the class number of the quadratic field \(\mathbb{Q}(\sqrt{-p})\).
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11F20
zbMATH DE Number: 7361383
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: Dedekind sums
zbMATH Keywords: Hardy sums
reviewed by: Alexey Ustinov
MaRDI profile type: Publication
full work available at URL: s11253-021-01883-5
OpenAlex ID: W3107806566
cites work: Q4101884
cites work: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan.
cites work: Q5539082
cites work: Mean values of Dedekind sums
cites work: On the Dedekind sums and its new reciprocity formula
cites work: Q3284684
cites work: Some identities involving certain Hardy sum and Kloosterman sum
cites work: Q5663963
cites work: Dedekind and Hardy sums
cites work: An exact formula for an average of L-series
cites work: A hybrid mean value related to the Dedekind sums and Kloosterman sums
cites work: On the identity involving certain Hardy sums and Kloosterman sums
cites work: On the mean values of Dedekind sums
DOI: 10.1007/S11253-021-01883-5
description: scientific article; zbMATH DE number 7361383