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''in number theory''

A method for obtaining non-trivial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) of the form

$$ \tag{1 }
S( f  )  =  \sum _ {1 \leq  x \leq  P } e ^ {2 \pi i f( x) } ,
$$

where

$$ 
f( x)  =  \alpha _ {n} x  ^ {n} + \dots + \alpha _ {1} x
$$

and  $  \alpha _ {n} \dots \alpha _ {1} $
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  $  \rho _ {0} = 2  ^ {n-} 1 $
by successive squaring operations in order to reduce the degree of the polynomial  $  f( x) $. 
Thus, in the first stage

$$ 
S  ^ {2} ( f  )  =  \sum _ {\lambda _ {1} \neq 0 } \sum _ { x }
e ^ {2 \pi i ( f( x+ \lambda _ {1} )- f( x)) } + O( P),
$$

where the summations are performed over intervals of lengths  $  \ll  P $; 
now

$$ 
f( x+ \lambda _ {1} ) - f( x)  = \ 
n \alpha _ {n} \lambda _ {1} x  ^ {n-} 1 + \dots
$$

is a polynomial of degree  $  n - 1 $
in  $  x $(
the symbols  $  O( P) $, 
$  \ll  P $
denote magnitudes of order  $  P $). 
At the  $  ( n - 1) $-
st step one obtains inner sum

$$ \tag{2 }
S( \alpha )= \sum _ {x= a+ 1 } ^ { {a+ }  P _ {1} }
e ^ {2 \pi i \alpha x } ,
$$

where  $  P _ {1} \leq  P $, 
$  \alpha = n! \lambda _ {1} {} \dots \lambda _ {n-} 1 \alpha _ {n} $, 
$  \lambda _ {i} \neq 0 $. 
Sums of the form (2) are estimated using the inequality

$$ 
| S( \alpha ) |  \leq   \min
\left ( P _ {1} , 
\frac{1}{| \sin  \pi \alpha | }
 \right ) ,
$$

and the resulting estimate is:

$$ \tag{3 }
| S( f  ) | ^ {\rho _ {0} }  \leq   P ^ {\rho _ {0} - 1 } +
$$

$$ 
+ 
P ^ {\rho _ {0} {- n + \epsilon } } \sum _ {0 < y \leq  P  ^ {n-} 1 } \min \left
( P _ {1} , 
\frac{1}{| \sin  \pi y \alpha _ {n} | }
 \right ) .
$$

The inequality (3) yields different estimates for the sum (1) in case  $  1/ | \sin  \pi y \alpha _ {n} | $
is small in comparison to  $  P $. 
These estimates depend on the accuracy with which the coefficient  $  \alpha _ {n} $
of the polynomial  $  f( x) $
can be approximated by rational fractions.

Example. Let

$$ 
\left | \alpha _ {n} - 
\frac{p}{q}
 \right |  \leq   
\frac{1}{q  ^ {2} }
 ,
\  ( a, q) = 1.
$$

Then

$$ 
| S( f  ) |  \ll   P ^ {1 + \epsilon } q  ^  \epsilon 
\left ( 
\frac{1}{P}
 + 
\frac{1}{q}
 + 
\frac{q}{P  ^ {n} }
 \right ) ^ {\rho _ {0} } .
$$

In particular, if

$$ 
P  \leq   q  \leq   P  ^ {n-} 1 ,
$$

then

$$ 
| S( f) |  \ll   P ^ {1- \rho _ {0} + \epsilon } .
$$

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  $  f( x) $. 
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  $  r= r( k) $
for which the equation

$$ 
N = x _ {1}  ^ {k} + \dots + x _ {r}  ^ {k} ,
$$

where  $  N > 0 $
is an integer and  $  x _ {i} $
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  $  f( x) $
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  $  \pi ( N) $
of prime numbers not exceeding  $  N $).

The insufficient strength of the estimates obtained by Weyl's method is due to the high power  $  \rho _ {0} $
to which the sum  $  S( f  ) $
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

$$ 
\int\limits _ { 0 } ^ { 1 }  \dots \int\limits _ { 0 } ^ { 1 } 
| S( f  ) |  ^ {2b}  d \alpha _ {n} \dots d \alpha _ {1}  $$

already for  $  b \geq  cn  ^ {2} $(
$  c > 0 $
a constant,  $  n \geq  2 $). 
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  $  P ^ {- \rho } $, 
$  \rho = c _ {1} n  ^ {2}  \mathop{\rm ln}  n $; 
$  c _ {1} > 0 $
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>

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[[Category:Mathematics]]