View source for Hardy-Littlewood problem

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

<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>

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

<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>

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" />).

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):

<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>

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>

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.

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.

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.