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A concept in the [[Dispersion method|dispersion method]] introduced for the comparison of the number of solutions of the equations
 
A concept in the [[Dispersion method|dispersion method]] introduced for the comparison of the number of solutions of the equations
  
<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/c/c026/c026830/c0268301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \phi + D ^ { \prime } \nu
 +
$$
  
 
and
 
and
  
<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/c/c026/c026830/c0268302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= \psi + D ^ { \prime } \nu ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c0268303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c0268304.png" /> belong to certain sequences of positive integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c0268305.png" /> runs through some given system of intervals on the real axis
+
where $  \phi $
 +
and $  \psi $
 +
belong to certain sequences of positive integers, $  D ^ { \prime } $
 +
runs through some given system of intervals on the real axis
  
<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/c/c026/c026830/c0268306.png" /></td> </tr></table>
+
$$
 +
( D)  = [ D _ {1} , D _ {1} + D _ {2} ] ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c0268307.png" /> runs through a system of intervals on the real axis
+
and $  \nu $
 +
runs through a system of intervals on the real axis
  
<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/c/c026/c026830/c0268308.png" /></td> </tr></table>
+
$$
 +
( \nu )  = [ \nu _ {0} , \nu _ {0} + \nu _ {0}  ^  \prime  ] .
 +
$$
  
 
Let
 
Let
  
<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/c/c026/c026830/c0268309.png" /></td> </tr></table>
+
$$
 +
U _ {1} ( m)  = \
 +
\sum _ {\phi = m } 1 ,\ \
 +
U _ {2} ( m)  = \
 +
\sum _ {\psi = m } 1 ,
 +
$$
  
 
then the dispersion of the difference of the solutions of (1) and (2) is
 
then the dispersion of the difference of the solutions of (1) and (2) is
  
<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/c/c026/c026830/c02683010.png" /></td> </tr></table>
+
$$
 +
V ^ { \prime }  = \
 +
\sum _ {D ^ { \prime } \in ( D) }
 +
\left ( {\sum _ {1} } \prime - {\sum _ {2} } \prime \right )  ^ {2} ,
 +
$$
  
 
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/c/c026/c026830/c02683011.png" /></td> </tr></table>
+
$$
 +
{\sum _ {1} } \prime  = \
 +
\sum _ {\nu \in ( \nu ) }
 +
U _ {1} ( n - D ^ { \prime } \nu ) ,\ \
 +
{\sum _ {2} } \prime  = \
 +
\sum _ {\nu \in ( \nu ) }
 +
U _ {2} ( n - D ^ { \prime } \nu ) .
 +
$$
  
Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683012.png" /> to all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683014.png" />. This can only increase the dispersion; thus
+
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
  
<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/c/c026/c026830/c02683015.png" /></td> </tr></table>
+
$$
 +
V ^ { \prime }  \leq  V  = \
 +
V _ {1} - 2 V _ {2} + V _ {3} ,
 +
$$
  
 
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/c/c026/c026830/c02683016.png" /></td> </tr></table>
+
$$
 +
V _ {1}  = \
 +
\sum _ {D \in ( D) }
 +
( \sum _ {1} )  ^ {2} ,
 +
$$
  
<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/c/c026/c026830/c02683017.png" /></td> </tr></table>
+
$$
 +
V _ {3}  = \sum _ {D \in ( D) } ( \sum _ {2} )  ^ {2} ,
 +
$$
  
<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/c/c026/c026830/c02683018.png" /></td> </tr></table>
+
$$
 +
V _ {2}  = \sum _ {D \in ( D) } ( \sum _ {1} \sum _ {2} ) ;
 +
$$
  
 
here
 
here
  
<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/c/c026/c026830/c02683019.png" /></td> </tr></table>
+
$$
 +
\sum _ {1}  = \
 +
\sum _ {\nu \in ( \nu ) }
 +
U _ {1} ( n - D \nu ) ,
 +
$$
  
<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/c/c026/c026830/c02683020.png" /></td> </tr></table>
+
$$
 +
\sum _ {2}  = \sum _ {\nu \in ( \nu ) } U _ {2} ( n - D \nu ) .
 +
$$
  
In analogy with probability-theoretic concepts, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683021.png" /> is called the covariance of the number of solutions of (1) and (2). An asymptotic estimate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683023.png" /> and the covariance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683024.png" /> shows that the dispersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026830/c02683025.png" /> is relatively small, and this is essential in considering additive problems that lead to equations (1) and (2).
+
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====
 
====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>
 
<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====
 
====Comments====
 
See also [[Circle method|Circle method]].
 
See also [[Circle method|Circle method]].

Latest revision as of 17:31, 5 June 2020


A concept in the 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

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

Comments

See also Circle method.

How to Cite This Entry:
Covariance of the number of solutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariance_of_the_number_of_solutions&oldid=46541
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article