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 [1] 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.

References

 [1] Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian) [2] B.M. Bredikhin, "The dispersion method and binary additive problems" Russian Math. Surveys , 20 : 2 (1965) pp. 85–125 Uspekhi Mat. Nauk , 20 : 2 (1965) pp. 89–130 [3] K. Prachar, "Primzahlverteilung" , Springer (1957)
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