Generalizing the Titchmarsh divisor problem (Q2880326)

Let \(d(n)\) denote the number of positive divisors of \(n\) and \(p\) be a prime. In this paper the author proves the asymptotic formula NEWLINE\[NEWLINE\sum_{\substack{ p\leq x\\ p\equiv a\pmod k}} d\Biggl({p-a\over k}\Biggr)= c_k x+ O\Biggl({x\over\log x}\Biggr),NEWLINE\]NEWLINE where \(c_k\) is a constant depending on \(k\) and \(a\), and the implied constant depends on \(k\). The case \(k=1\) (with any \(a\)) corresponds to the Titchmarsh divisor problem and the analogous formula was established unconditionally by Linnik with an extra factor \(\log\log x\) in the error term, later removed by various authors. A problem related to Artin's conjecture on primitive roots is also considered in this paper. For \(p\nmid a\) let \(f_a(p)\) denote the order of \(a\pmod p\). Then the author shows that the average order of magnitude over \(a\leq y\) of \(S(x):=\sum_{p\leq x} {1\over f_a(p)}\) is approximately \(\log x\) provided \({x\over\log x}= o(y)\).NEWLINENEWLINE \textit{M. Ram Murty} and \textit{S. Srinivasan} proved in [Can. Math. Bull. 30, No. 1--3, 80--85 (1987; Zbl 0574.10005)] that if \(S(x)\ll x^{1/4}\) then Artin's conjecture is true. The proofs in the paper under review depend on results concerning primes in arithmetic progression and involve complicated manipulations of multiple sums.