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''in number theory''
 
''in number theory''
  
 
A method for solving certain binary equations (binary [[Additive problems|additive problems]]) of the form
 
A method for solving certain binary equations (binary [[Additive problems|additive problems]]) of the form
  
<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/d/d033/d033370/d0333701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\begin{equation}\label{eq1}
 +
\alpha+\beta = n,
 +
\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333703.png" /> belong to sequences of natural numbers which are sufficiently dense and sufficiently well distributed in the sequence of natural numbers.
+
where $\alpha$ and $\beta$ belong to sequences of natural numbers which are sufficiently dense and sufficiently well distributed in the sequence of natural numbers.
  
 
The method, which was developed by Yu.V. Linnik in 1958–1961 and is therefore also known as Linnik's dispersion method, combines elementary concepts of probability theory (in particular, the concept of dispersion and Chebyshev-type inequalities) with the analytic and algebraic ideas of I.M. Vinogradov and A. Weil. It may be described as follows (see also [[Additive number theory|Additive number theory]]).
 
The method, which was developed by Yu.V. Linnik in 1958–1961 and is therefore also known as Linnik's dispersion method, combines elementary concepts of probability theory (in particular, the concept of dispersion and Chebyshev-type inequalities) with the analytic and algebraic ideas of I.M. Vinogradov and A. Weil. It may be described as follows (see also [[Additive number theory|Additive number theory]]).
  
Equation (1) may be reduced to equations of the form
+
Equation \eqref{eq1} may be reduced to equations of the form
  
<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/d/d033/d033370/d0333704.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\begin{equation}\label{eq2}\nu D'+\beta=n;\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333706.png" /> run, independently of each other, through certain values from a rectangular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333708.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d0333709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337010.png" /> are certain intervals; moreover, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337011.png" /> are prime, while various additional conditions may be imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337013.png" /> be the number of solutions of this equation.
+
where $\nu$ and $D'$ run, independently of each other, through certain values from a rectangular domain $\nu\in(\nu)$, $D'\in (D)$, where $(\nu)$ and $(D)$ are certain intervals; moreover, the numbers $\nu$ are prime, while various additional conditions may be imposed on $D'$. Let $F$ be the number of solutions of this equation.
  
 
Now consider the equation
 
Now consider 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/d/d033/d033370/d03337014.png" /></td> </tr></table>
+
\[ \nu D+\beta=n \]
  
for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337015.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337016.png" /> be the number of its solutions found by some heuristic means. The (hypothetical) number of expected solutions of equation (2) is then written in the form
+
for an arbitrary $D\in(D)$, and let $A(n,D)$ be the number of its solutions found by some heuristic means. The (hypothetical) number of expected solutions of equation \eqref{eq2} is then written in the form
  
<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/d/d033/d033370/d03337017.png" /></td> </tr></table>
+
\[ S=\sum_{D'\in (D)}A(n,D').\]
  
An estimate of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337018.png" /> has the form
+
An estimate of the difference $F-S=V$ has the form
  
<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/d/d033/d033370/d03337019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\begin{equation}\label{eq3}V=\sum_{D'\in (D)}\left(\sum_{\nu D'+\beta=n} 1-A(n,D')\right).\end{equation}
  
 
An application of the Cauchy inequality yields
 
An application of the Cauchy inequality yields
  
<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/d/d033/d033370/d03337020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\begin{equation}\label{eq4} V^2\leq D_0V',\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337021.png" /> is the length of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337022.png" /> and
+
where $D_0$ is the length of the interval $(D)$ 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/d/d033/d033370/d03337023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\begin{equation}\label{eq5} V'=\sum_{D'\in(D)}\left(\sum_{\nu D'+\beta=n}1-A(n,D')\right)^2\end{equation}
  
is the dispersion of the number of solutions of equation (2).
+
is the dispersion of the number of solutions of equation \eqref{eq2}.
  
If the summation in (5) is extended over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337024.png" />, all additional conditions imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337025.png" /> in (2) are dropped. At the same time the value of the dispersion can only increase; accordingly,
+
If the summation in \eqref{eq5} is extended over all $D\in(D)$, all additional conditions imposed on $D'$ in \eqref{eq2} are dropped. At the same time the value of the dispersion can only increase; accordingly,
  
<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/d/d033/d033370/d03337026.png" /></td> </tr></table>
+
\[ V'\leq \sum_{D\in (D)} \left(\sum_{\nu D+\beta=n}1-A(n,D)\right)^2=\Sigma_1-2\Sigma_2+\Sigma_3.\]
  
The sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337029.png" /> can be computed asymptotically in certain cases. The computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337030.png" />, which is the basic sum in the dispersion method, is the most difficult. The asymptotic computation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337031.png" /> is effected using the [[Vinogradov method|Vinogradov method]] by computing, for certain functions, the number of fractional parts comprised in the given segment, and also by using the most recent estimates for trigonometric sums obtained by methods of algebraic geometry. Asymptotics for the sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337033.png" /> are found by elementary summations. If, as a result, the dispersion is found to be not too large, (3) and (4) yield an asymptotic expansion for the number of solutions of equation (2).
+
The sums $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ can be computed asymptotically in certain cases. The computation of $\Sigma_1$, which is the basic sum in the dispersion method, is the most difficult. The asymptotic computation of $\Sigma_1$ is effected using the [[Vinogradov method|Vinogradov method]] by computing, for certain functions, the number of fractional parts comprised in the given segment, and also by using the most recent estimates for trigonometric sums obtained by methods of algebraic geometry. Asymptotics for the sums $\Sigma_2$ and $\Sigma_3$ are found by elementary summations. If, as a result, the dispersion is found to be not too large, \eqref{eq3} and \eqref{eq4} yield an asymptotic expansion for the number of solutions of equation \eqref{eq2}.
  
Combining the numbers of solutions of all equations of the type (2) yields an asymptotic formula for the number of solutions of equation (1).
+
Combining the numbers of solutions of all equations of the type \eqref{eq2} yields an asymptotic formula for the number of solutions of equation \eqref{eq1}.
  
 
The above scheme of the dispersion method may be used also to solve equations of the type
 
The above scheme of the dispersion method may be used also to solve equations of the type
  
<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/d/d033/d033370/d03337034.png" /></td> </tr></table>
+
\[\alpha-\beta=l,\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337035.png" /> is a given non-zero integer.
+
where $l$ is a given non-zero integer.
  
Linnik, et al. (see [[#References|[3]]]) employed this method to solve a number of classical additive problems which, prior to the genesis of the dispersion method, could only be solved on the strength of heuristic or hypothetical conceptions. Such problems include the [[Additive divisor problem|additive divisor problem]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337038.png" />); the [[Titchmarsh problem|Titchmarsh problem]] on divisors (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337039.png" /> is a prime, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337040.png" />); and the [[Hardy–Littlewood problem|Hardy–Littlewood problem]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337041.png" /> is a prime, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337042.png" />).
+
Linnik, et al. (see {{Cite|Br}}) employed this method to solve a number of classical additive problems which, prior to the genesis of the dispersion method, could only be solved on the strength of heuristic or hypothetical conceptions. Such problems include the [[Additive divisor problem|additive divisor problem]] ($\alpha=x_1,\ldots,x_k$, $k=\textrm{const}$, $\beta=xy$); the [[Titchmarsh problem|Titchmarsh problem]] on divisors ($\alpha=p$ is a prime, $\beta=xy$); and the [[Hardy–Littlewood problem|Hardy–Littlewood problem]] ($\alpha=p$ is a prime, $\beta=x^2+y^2$).
  
 
The dispersion method also yielded solutions of certain analogues and generalizations of these problems; this includes, in particular, an asymptotic expansion for the number of solutions of the general Hardy–Littlewood equation
 
The dispersion method also yielded solutions of certain analogues and generalizations of these problems; this includes, in particular, an asymptotic expansion for the number of solutions of the general Hardy–Littlewood 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/d/d033/d033370/d03337043.png" /></td> </tr></table>
+
\[ p+\phi(\xi,\eta)=n,\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337044.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337045.png" /> is a given primitive positive-definite quadratic form. The existence of an infinite set of prime numbers of the type
+
where $p$ is a prime number and $\phi(\xi,\eta)$ is a given primitive positive-definite quadratic form. The existence of an infinite set of prime numbers of the type
  
<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/d/d033/d033370/d03337046.png" /></td> </tr></table>
+
\[ p=\phi(\xi,\eta)+l,\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033370/d03337047.png" /> is an arbitrary fixed integer, was proved.
+
where $l$ is an arbitrary fixed integer, was proved.
  
 
The domain of application of the dispersion method intersects with that of Linnik's [[Large sieve|large sieve]] method.
 
The domain of application of the dispersion method intersects with that of Linnik's [[Large sieve|large sieve]] method.
 +
  
 
====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) {{MR|0168543}} {{ZBL|0112.27402}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> K. Prachar, "Primzahlverteilung" , Springer (1957) {{MR|0087685}} {{ZBL|0080.25901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.M. Bredekhin, "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><TR><TD valign="top">[4]</TD> <TD valign="top"> B.M. Bredekhin, Yu.V. Linnik, "A new method in analytic number theory" , ''Current problems in analytic number theory'' , Minsk (1974) pp. 5–22 (In Russian) {{MR|337854}} {{ZBL|}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Br}}||valign="top"| B.M. Bredekhin, "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
 +
|-
 +
|valign="top"|{{Ref|BrLi}}||valign="top"| B.M. Bredekhin, Yu.V. Linnik, "A new method in analytic number theory" , ''Current problems in analytic number theory'' , Minsk (1974) pp. 5–22 (In Russian) {{MR|337854}} {{ZBL|}}  
 +
|-
 +
|valign="top"|{{Ref|Li}}||valign="top"| Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian) {{MR|0168543}} {{ZBL|0112.27402}}
 +
|-
 +
|valign="top"|{{Ref|Pr}}||valign="top"| K. Prachar, "Primzahlverteilung" , Springer (1957) {{MR|0087685}} {{ZBL|0080.25901}}
 +
|-
 +
|}

Latest revision as of 07:03, 11 June 2013


in number theory

A method for solving certain binary equations (binary additive problems) of the form

\begin{equation}\label{eq1} \alpha+\beta = n, \end{equation}

where $\alpha$ and $\beta$ belong to sequences of natural numbers which are sufficiently dense and sufficiently well distributed in the sequence of natural numbers.

The method, which was developed by Yu.V. Linnik in 1958–1961 and is therefore also known as Linnik's dispersion method, combines elementary concepts of probability theory (in particular, the concept of dispersion and Chebyshev-type inequalities) with the analytic and algebraic ideas of I.M. Vinogradov and A. Weil. It may be described as follows (see also Additive number theory).

Equation \eqref{eq1} may be reduced to equations of the form

\begin{equation}\label{eq2}\nu D'+\beta=n;\end{equation}

where $\nu$ and $D'$ run, independently of each other, through certain values from a rectangular domain $\nu\in(\nu)$, $D'\in (D)$, where $(\nu)$ and $(D)$ are certain intervals; moreover, the numbers $\nu$ are prime, while various additional conditions may be imposed on $D'$. Let $F$ be the number of solutions of this equation.

Now consider the equation

\[ \nu D+\beta=n \]

for an arbitrary $D\in(D)$, and let $A(n,D)$ be the number of its solutions found by some heuristic means. The (hypothetical) number of expected solutions of equation \eqref{eq2} is then written in the form

\[ S=\sum_{D'\in (D)}A(n,D').\]

An estimate of the difference $F-S=V$ has the form

\begin{equation}\label{eq3}V=\sum_{D'\in (D)}\left(\sum_{\nu D'+\beta=n} 1-A(n,D')\right).\end{equation}

An application of the Cauchy inequality yields

\begin{equation}\label{eq4} V^2\leq D_0V',\end{equation}

where $D_0$ is the length of the interval $(D)$ and

\begin{equation}\label{eq5} V'=\sum_{D'\in(D)}\left(\sum_{\nu D'+\beta=n}1-A(n,D')\right)^2\end{equation}

is the dispersion of the number of solutions of equation \eqref{eq2}.

If the summation in \eqref{eq5} is extended over all $D\in(D)$, all additional conditions imposed on $D'$ in \eqref{eq2} are dropped. At the same time the value of the dispersion can only increase; accordingly,

\[ V'\leq \sum_{D\in (D)} \left(\sum_{\nu D+\beta=n}1-A(n,D)\right)^2=\Sigma_1-2\Sigma_2+\Sigma_3.\]

The sums $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ can be computed asymptotically in certain cases. The computation of $\Sigma_1$, which is the basic sum in the dispersion method, is the most difficult. The asymptotic computation of $\Sigma_1$ is effected using the Vinogradov method by computing, for certain functions, the number of fractional parts comprised in the given segment, and also by using the most recent estimates for trigonometric sums obtained by methods of algebraic geometry. Asymptotics for the sums $\Sigma_2$ and $\Sigma_3$ are found by elementary summations. If, as a result, the dispersion is found to be not too large, \eqref{eq3} and \eqref{eq4} yield an asymptotic expansion for the number of solutions of equation \eqref{eq2}.

Combining the numbers of solutions of all equations of the type \eqref{eq2} yields an asymptotic formula for the number of solutions of equation \eqref{eq1}.

The above scheme of the dispersion method may be used also to solve equations of the type

\[\alpha-\beta=l,\]

where $l$ is a given non-zero integer.

Linnik, et al. (see [Br]) employed this method to solve a number of classical additive problems which, prior to the genesis of the dispersion method, could only be solved on the strength of heuristic or hypothetical conceptions. Such problems include the additive divisor problem ($\alpha=x_1,\ldots,x_k$, $k=\textrm{const}$, $\beta=xy$); the Titchmarsh problem on divisors ($\alpha=p$ is a prime, $\beta=xy$); and the Hardy–Littlewood problem ($\alpha=p$ is a prime, $\beta=x^2+y^2$).

The dispersion method also yielded solutions of certain analogues and generalizations of these problems; this includes, in particular, an asymptotic expansion for the number of solutions of the general Hardy–Littlewood equation

\[ p+\phi(\xi,\eta)=n,\]

where $p$ is a prime number and $\phi(\xi,\eta)$ is a given primitive positive-definite quadratic form. The existence of an infinite set of prime numbers of the type

\[ p=\phi(\xi,\eta)+l,\]

where $l$ is an arbitrary fixed integer, was proved.

The domain of application of the dispersion method intersects with that of Linnik's large sieve method.


References

[Br] B.M. Bredekhin, "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
[BrLi] B.M. Bredekhin, Yu.V. Linnik, "A new method in analytic number theory" , Current problems in analytic number theory , Minsk (1974) pp. 5–22 (In Russian) MR337854
[Li] Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian) MR0168543 Zbl 0112.27402
[Pr] K. Prachar, "Primzahlverteilung" , Springer (1957) MR0087685 Zbl 0080.25901
How to Cite This Entry:
Dispersion method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dispersion_method&oldid=23813
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article