Covariance of the number of solutions
A concept in the dispersion method introduced for the comparison of the number of solutions of the equations
![]() | (1) |
and
![]() | (2) |
where and
belong to certain sequences of positive integers,
runs through some given system of intervals on the real axis
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and runs through a system of intervals on the real axis
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Let
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then the dispersion of the difference of the solutions of (1) and (2) is
![]() |
where
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Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over to all of
in
. This can only increase the dispersion; thus
![]() |
where
![]() |
![]() |
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here
![]() |
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In analogy with probability-theoretic concepts, is called the covariance of the number of solutions of (1) and (2). An asymptotic estimate of
,
and the covariance
shows that the dispersion
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.
Covariance of the number of solutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariance_of_the_number_of_solutions&oldid=19216