# Titchmarsh problem

The problem of finding an asymptotic expression for

$$\tag{1 } Q ( n) = \ \sum _ {p \leq n } \tau ( p - l),$$

where $\tau ( m)$ is the number of divisors of $m$( cf. Divisor problems), $l$ is a fixed non-zero number and $p$ runs through all prime numbers. Analogous to this problem is the problem of finding an asymptotic expression for

$$\tag{2 } S ( n) = \ \sum _ {p \leq n - 1 } \tau ( n - p).$$

This problem was posed by E. Titchmarsh (1930) and was solved by him  under the assumption that the Riemann hypothesis is true (cf. Riemann hypotheses).

The dispersion method, developed by Yu.V. Linnik, allows one to find asymptotics for (1) and (2):

$$Q ( n) = \ \frac{315 \zeta ( 3) }{2 \pi ^ {4} } \prod _ {p \mid l } \frac{( p - 1) ^ {2} }{p ^ {2} - p + 1 } n + O ( n ( \mathop{\rm ln} n) ^ {- 1 + \epsilon } );$$

the formula for $S ( n)$ is analogous.

The Vinogradov–Bombieri theorem on the average distribution of prime numbers in arithmetic progressions also leads to a solution of the Titchmarsh problem. Here the assumption of the truth of the Riemann hypothesis is actually replaced by theorems of the large sieve type.

How to Cite This Entry:
Titchmarsh problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Titchmarsh_problem&oldid=48979
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article