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The problem of finding an asymptotic expression for
 
The problem of finding an asymptotic expression for
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
Q ( n)  = \
 +
\sum _ {p \leq  n }
 +
\tau ( p - l),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928902.png" /> is the number of divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928903.png" /> (cf. [[Divisor problems|Divisor problems]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928904.png" /> is a fixed non-zero number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928905.png" /> runs through all prime numbers. Analogous to this problem is the problem of finding an asymptotic expression for
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
 +
S ( n)  = \
 +
\sum _ {p \leq  n - 1 }
 +
\tau ( n - p).
 +
$$
  
 
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]]).
 
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]]).
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The [[Dispersion method|dispersion method]], developed by Yu.V. Linnik, allows one to find asymptotics for (1) and (2):
 
The [[Dispersion method|dispersion method]], developed by Yu.V. Linnik, allows one to find asymptotics for (1) and (2):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928907.png" /></td> </tr></table>
+
$$
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092890/t0928908.png" /> is analogous.
+
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.
 
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.

Revision as of 08:25, 6 June 2020


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=18419
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article