A note on a generalization of Euler's \(\varphi\)-function (Q919372)

For relatively prime integers \(s\) and \(d\) let \(\varphi(s,d,n)\) denote the number of elements in the arithmetic progression \(s,s+d,\ldots,s+(n-1)d\) which are relatively prime to \(n\), see \textit{P. G. Garcia} and \textit{S. Ligh} [Fibonacci Q. 21, 26--28 (1983; Zbl 0503.10004)]. Previously \(\varphi(s,d,n)\) has been studied as a function of the multiplicative variable \(n\). The present author investigates \(\varphi(s,d,n)\) as a function of the non-multiplicative variable \(d\). He notes that \(\varphi(s,\cdot,n)\) is an even function \(\pmod n\) and derives an asymptotic formula for the summatory function \(\sum_{d\leq x} \varphi(s,d,n)\).