View source for Hardy-Littlewood problem

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The problem of finding an asymptotic formula for the number  $  Q ( n) $
of solutions of the equation

$$ \tag{1 }
p + x  ^ {2} + y  ^ {2}  =  n,
$$

where  $  p $
is a prime number,  $  x $
and  $  y $
are integers, and  $  n $
is a natural number  $  ( n \rightarrow \infty ) $. 
An analogue of this problem is that of finding the asymptotic behaviour for the number of solutions of the equation

$$ \tag{2 }
p - x  ^ {2} - y  ^ {2}  =  l,
$$

where  $  l \neq 0 $
is a fixed integer, and  $  p \leq  n $(
$  n \rightarrow \infty $).

The problem was raised by G.H. Hardy and J.E. Littlewood in 1923 and treated by them on the basis of heuristic and hypothetical arguments.

The [[Dispersion method|dispersion method]] worked out by Yu.V. Linnik enabled him to find an asymptotic expansion for (1):

$$ 
Q ( n)  = \ 
\pi A _ {0} 
\frac{n}{ \mathop{\rm ln}  n }

\prod _ {p\mid  n }

\frac{( p - 1) ( p - \chi _ {4} ( p)) }{p  ^ {2} - p + \chi _ {4} ( p) }
 + R ( n),
$$

where

$$ 
A _ {0}  = \ 
\prod _ {p\mid  n }
\left ( 1 +

\frac{\chi _ {4} ( p) }{p ( p - 1) }

\right ) ,\ \ 
R ( n)  =  O
\left ( 
\frac{n}{(  \mathop{\rm ln}  n)  ^ {1.042} }

\right ) .
$$

From a similar formula for (2) it follows that the set of prime numbers of the form  $  p = x  ^ {2} + y  ^ {2} + l $
is infinite. By means of the dispersion method an asymptotic expansion has also been found for the number of solutions of the generalized Hardy–Littlewood equation  $  p + \phi ( x, y) = l $, 
where  $  p $
is a prime number and  $  \phi ( x, y) $
is a given primitive positive-definite quadratic form.

The discussion of the similar equation  $  p - \phi ( x, y) = l $
leads to a proof that the set of prime numbers of the form  $  p = \phi ( x, y) + l $
is infinite.

The Vinogradov–Bombieri theorem on the average [[Distribution of prime numbers|distribution of prime numbers]] in arithmetic progressions also gives a solution of the Hardy–Littlewood problem, by replacing the extended Riemann hypothesis by theorems of the type of the [[Large sieve|large sieve]].

====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,   Yu.V. Linnik,   "Asymptotic behaviour and ergodic properties of solutions of the generalized Hardy–Littlewood equation"  ''Mat. Sb.'' , '''71''' :  2  (1966)  pp. 145–161  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Bredikhin,   "The dispersion method and definite binary additive problems"  ''Russian Math. Surveys'' , '''20''' :  2  (1965)  pp. 85–125  ''Uspekhi Mat. Nauk'' , '''20''' :  2  (1965)  pp. 89–130</TD></TR></table>

====Comments====
Of course, this is just one of the many problems raised by Hardy and Littlewood.

[[Category:Number theory]]