A new arithmetical function and its mean value properties (Q825014)

For odd primes \(p\le q\) let \(d(n;p,q)\) be the number of factorizations \(n=ab\), where \(a\) is a primitive root mod \(p\) and \(b\) is a primitive root mod \(q\). The authors establish for \(x\ge q^2\) the equality \[ \sum_{n\le x}d(n;p,q)= \frac{\varphi(p-1)\varphi(q-1)}{pq}x\log x + c(p,q)x + O_{p,q}\left(x^{1/3}\right). \]