Karatsuba's divisor problem and related questions

Abstract

We prove that $$\sum _{p\le x}\frac{1}{\tau (p-1)}\asymp \frac{x}{(\log x{)}^{3/2}},\sum _{n\le x}\frac{1}{\tau (n^2+1)}\asymp \frac{x}{(\log x{)}^{1/2}},$$ where $\tau (n)={\sum }_{d|n}1$ is the number of divisors of $n$, and the summation in the first sum is over primes.