View source for Covariance of the number of solutions

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A concept in the [[Dispersion method|dispersion method]] introduced for the comparison of the number of solutions of the equations

$$ \tag{1 }
n  =  \phi + D ^ { \prime } \nu
$$

and

$$ \tag{2 }
n  =  \psi + D ^ { \prime } \nu ,
$$

where  $  \phi $
and  $  \psi $
belong to certain sequences of positive integers,  $  D ^ { \prime } $
runs through some given system of intervals on the real axis

$$ 
( D)  =  [ D _ {1} , D _ {1} + D _ {2} ] ,
$$

and  $  \nu $
runs through a system of intervals on the real axis

$$ 
( \nu )  =  [ \nu _ {0} , \nu _ {0} + \nu _ {0}  ^  \prime  ] .
$$

Let

$$ 
U _ {1} ( m)  = \ 
\sum _ {\phi = m } 1 ,\ \ 
U _ {2} ( m)  = \ 
\sum _ {\psi = m } 1 ,
$$

then the dispersion of the difference of the solutions of (1) and (2) is

$$ 
V ^ { \prime }  = \ 
\sum _ {D ^ { \prime } \in ( D) }
\left ( {\sum _ {1} } \prime - {\sum _ {2} } \prime \right )  ^ {2} ,
$$

where

$$ 
{\sum _ {1} } \prime  = \ 
\sum _ {\nu \in ( \nu ) }
U _ {1} ( n - D ^ { \prime } \nu ) ,\ \ 
{\sum _ {2} } \prime  = \ 
\sum _ {\nu \in ( \nu ) }
U _ {2} ( n - D ^ { \prime } \nu ) .
$$

Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over  $  D ^ { \prime } $
to all of  $  D $
in  $  ( D) $. 
This can only increase the dispersion; thus

$$ 
V ^ { \prime }  \leq   V  = \ 
V _ {1} - 2 V _ {2} + V _ {3} ,
$$

where

$$ 
V _ {1}  = \ 
\sum _ {D \in ( D) }
( \sum _ {1} )  ^ {2} ,
$$

$$ 
V _ {3}  =  \sum _ {D \in ( D) } ( \sum _ {2} )  ^ {2} ,
$$

$$ 
V _ {2}  =  \sum _ {D \in ( D) } ( \sum _ {1} \sum _ {2} ) ;
$$

here

$$ 
\sum _ {1}  = \ 
\sum _ {\nu \in ( \nu ) }
U _ {1} ( n - D \nu ) ,
$$

$$ 
\sum _ {2}  =  \sum _ {\nu \in ( \nu ) } U _ {2} ( n - D \nu ) .
$$

In analogy with probability-theoretic concepts,  $  V _ {2} $
is called the covariance of the number of solutions of (1) and (2). An asymptotic estimate of  $  V _ {1} $, 
$  V _ {3} $
and the covariance  $  V _ {2} $
shows that the dispersion  $  V ^ { \prime } $
is relatively small, and this is essential in considering additive problems that lead to equations (1) and (2).

====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></table>

====Comments====
See also [[Circle method|Circle method]].