On some identities involving certain Hardy sums and a Kloosterman sum
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 \(\mathbb{Q}(\sqrt{-p})\).
scientific article; zbMATH DE number 7361383