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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|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 [[#References|[1]]] under the assumption that the Riemann hypothesis is true (cf. [[Riemann hypotheses|Riemann hypotheses]]).

The [[Dispersion method|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|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|large sieve]] type.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,   "The dispersion method in binary additive problems" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR></table>