A note on the hyper Cochrane sum
Let \(q\) and \(h\) be integers with \(q\geq 3\) and \((h,q)=1\). Let \(\mathbf{m}=\left(m_1,\cdots,m_k,m_{k+1}\right)\in \mathbb{Z}^{k+1}\). The hyper Cochrane sum is defined as following: \[ C\left(h,q;\mathbf{m},k\right)=\mathop{\sum_{a_1=1}^{q}}_{(a_1,q)=1}\cdots \mathop{\sum_{a_k=1}^{q}}_{(a_k,q)=1}\overline{B}_{m_1}\left(\frac{a_1}{q}\right)\cdots \overline{B}_{m_k}\left(\frac{a_k}{q}\right)\overline{B}_{m_{k+1}}\left(\frac{h\overline{a_1\cdots a_k}}{q}\right), \] where \(\overline{B}_{n}(x)\) is the \(n\)-th Bernoulli periodic function, \(\overline{y}\) is the multiplicative inverse of \(y\) modulo \(q\). This paper provides some upper bounds for \(C\left(h,q;\mathbf{m},k\right)\).
scientific article; zbMATH DE number 6197287