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The problem of finding an asymptotic formula for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463601.png" /> of solutions of the equation
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<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/h/h046/h046360/h0463602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463603.png" /> is a prime number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463605.png" /> are integers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463606.png" /> is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463607.png" />. An analogue of this problem is that of finding the asymptotic behaviour for the number of solutions of the equation
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The problem of finding an asymptotic formula for the number $  Q ( n) $
 +
of solutions of the equation
  
<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/h/h046/h046360/h0463608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
 +
p + x  ^ {2} + y  ^ {2}  = n,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h0463609.png" /> is a fixed integer, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636011.png" />).
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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 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.
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The [[Dispersion method|dispersion method]] worked out by Yu.V. Linnik enabled him to find an asymptotic expansion for (1):
 
The [[Dispersion method|dispersion method]] worked out by Yu.V. Linnik enabled him to find an asymptotic expansion for (1):
  
<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/h/h046/h046360/h04636012.png" /></td> </tr></table>
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$$
 +
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
 
where
  
<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/h/h046/h046360/h04636013.png" /></td> </tr></table>
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$$
 +
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} }
  
From a similar formula for (2) it follows that the set of prime numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636016.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636017.png" /> is a given primitive positive-definite quadratic form.
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\right ) .
 +
$$
  
The discussion of the similar equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636018.png" /> leads to a proof that the set of prime numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046360/h04636019.png" /> is infinite.
+
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]].
 
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]].
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====References====
 
====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>
 
<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====
 
====Comments====
 
Of course, this is just one of the many problems raised by Hardy and Littlewood.
 
Of course, this is just one of the many problems raised by Hardy and Littlewood.
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 +
[[Category:Number theory]]

Latest revision as of 19:43, 5 June 2020


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 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 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.

References

[1] Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian)
[2] 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)
[3] 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

Comments

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

How to Cite This Entry:
Hardy-Littlewood problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy-Littlewood_problem&oldid=22549
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article