Geodesic
Karatsuba's divisor problem and related questions
Sbornik. Mathematics, Tome 214 (2023) no. 7, pp. 919-933 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}} \quad\text{and}\quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $\tau(n)=\sum_{d\mid n}1$ is the number of divisors of $n$, and the first sum is taken over prime numbers. Bibliography: 14 titles.
Keywords: divisor function, sums of values of functions, shifted primes and squares. 
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M. R. Gabdullin; S. V. Konyagin; V. V. Iudelevich. Karatsuba's divisor problem and related questions. Sbornik. Mathematics, Tome 214 (2023) no. 7, pp. 919-933. http://geodesic.mathdoc.fr/item/SM_2023_214_7_a1/

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