A note on Titchmarsh divisor problem under the generalized Riemann hypothesis. (English) Zbl 1457.11134
Summary: Let \(\Lambda(n)\) be the von Mangoldt function and \(\tau(n)\) the divisor function. We focus on the summation \[T(x)=\sum_{1<n\leq x} \Lambda(n)\tau(n-1).\] Under the Riemann hypothesis related to Dirichlet \(L\)-functions, it is proved that \[T(x)=x P_1 (\log x)+O(x^{1-\theta})\] holds for \(\theta<\frac{1}{6} \times 10^{-4}\). Here \(P_1(t)\) is a polynomial of degree \(1\).
MSC:
| 11N37 | Asymptotic results on arithmetic functions |
| 11F30 | Fourier coefficients of automorphic forms |
| 11L20 | Sums over primes |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
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