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) Zbl 0080.25901