View source for Weyl method

''in number theory''

A method for obtaining non-trivial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) 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/w/w097/w097720/w0977201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/w/w097/w097720/w0977202.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977203.png" /> are arbitrary real numbers. Developed by H. Weyl [[#References|[1]]] to establish criteria for [[Uniform distribution|uniform distribution]] (cf. [[Weyl criterion|Weyl criterion]]).

The method may be described as follows. The sum (1) is raised to the power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977204.png" /> by successive squaring operations in order to reduce the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977205.png" />. Thus, in the first stage

<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/w/w097/w097720/w0977206.png" /></td> </tr></table>

where the summations are performed over intervals of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977207.png" />; now

<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/w/w097/w097720/w0977208.png" /></td> </tr></table>

is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w0977209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772010.png" /> (the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772012.png" /> denote magnitudes of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772013.png" />). At the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772014.png" />-st step one obtains inner sum

<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/w/w097/w097720/w09772015.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/w/w097/w097720/w09772016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772018.png" />. Sums of the form (2) are estimated using the inequality

<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/w/w097/w097720/w09772019.png" /></td> </tr></table>

and the resulting estimate 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/w/w097/w097720/w09772020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<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/w/w097/w097720/w09772021.png" /></td> </tr></table>

The inequality (3) yields different estimates for the sum (1) in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772022.png" /> is small in comparison to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772023.png" />. These estimates depend on the accuracy with which the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772024.png" /> of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772025.png" /> can be approximated by rational fractions.

Example. 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/w/w097/w097720/w09772026.png" /></td> </tr></table>

Then

<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/w/w097/w097720/w09772027.png" /></td> </tr></table>

In particular, if

<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/w/w097/w097720/w09772028.png" /></td> </tr></table>

then

<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/w/w097/w097720/w09772029.png" /></td> </tr></table>

Weyl's method gives solutions, to a first approximation, of several important problems in number theory. The estimate (3) and its corollaries were used to study the [[Distribution modulo one|distribution modulo one]] of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772030.png" />. G.H. Hardy and J.E. Littlewood (1919) gave a solution of the [[Waring problem|Waring problem]] which was based on estimating the sums (1) by Weyl's method. They could thus estimate the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772031.png" /> for which 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/w/w097/w097720/w09772032.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772033.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772034.png" /> are integers, is solvable, and even gave an asymptotic formula for the number of solutions. A generalization of the estimate (3) to the case of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772035.png" /> which are not polynomials but are in a certain sense close to polynomials, resulted in the improvement of certain theorems in the theory of the [[Distribution of prime numbers|distribution of prime numbers]] (an estimate of the difference between two successive prime numbers and an estimate of the residual term in the asymptotic formula for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772036.png" /> of prime numbers not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772037.png" />).

The insufficient strength of the estimates obtained by Weyl's method is due to the high power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772038.png" /> to which the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772039.png" /> is raised. J. van der Corput proposed a somewhat improved method for estimating the sums (1). The [[Vinogradov method|Vinogradov method]] yields a very accurate upper bound for the integral

<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/w/w097/w097720/w09772040.png" /></td> </tr></table>

already for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772041.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772042.png" /> a constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772043.png" />). This estimate (cf. [[Vinogradov theorem about the average|Vinogradov theorem about the average]]) may be used to deduce essentially new estimates of Weyl sums (1) (with reduction factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772045.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097720/w09772046.png" /> a constant), which cannot be attained by Weyl's method.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,   "Ueber die Gleichverteilung von Zahlen mod. Eins"  ''Math. Ann.'' , '''77'''  (1916)  pp. 313–352</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR></table>