A mean value of Cochrane sum
Let \(q\) be a positive integer. For an integer \(a\) relatively prime to \(q\), let \(\overline a\) denote the multiplicative inverse of \(a\) modulo \(q\). Then one can define the so-called Cochrane sum \(C(h,q)\) by \[ C(h,q)=\sum_{{a=1}\atop{(a,q)=1}}^q\left(\left(\frac aq\right)\right)\left(\left(\frac{\overline{ a}h}q\right)\right) \] for arbitrary integers \(h\), where for a real number \(x\), we let \(((x))=x-\lfloor x\rfloor-1/2\) if \(x\) is not an integer and \(((x))=0\) if \(x\) is an integer. The purpose of the paper under review is to obtain an asymptotic formula for the sum \(\sum_{a<q/4}\sum_{b<q/4}C(ab,q)\).
scientific article; zbMATH DE number 5629495