Weyl’s Sums
Weyl’s Sums
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韦尔求和
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
N. M. Korobov
中科院分区：
文献类型：
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作者：
N. M. Korobov
In Chapter I, the Weyl sums of the first degree were considered and it was shown that the estimate
$$ left| {sumlimits_{x = 1}^P {{e^{2pi ialpha x}}} }
ight| leqslant min left( {P,frac{1}{{2left| alpha
ight|}}}
ight) $$
holds for them. The basic idea of Weyl’s method consists in reducing the estimation of a sum of an arbitrary degree n ⩾ 2
$$ Sleft( P
ight) = sumlimits_{x = 1}^P {{e^{2pi ileft( {{alpha _1}x + ldots + {alpha _n}{x^n}}
ight)}}} $$
to the estimation of a sum of degree n − 1 and, ultimately, to the use of the estimate (140). We have already met the reduction of the degree of an exponential sum in proving the theorem on the modulus of the Gauss sum. In the Gauss theorem, the square of the modulus of the exponential sum of the second degree was transformed with the help of linear change of variable in summation into a double sum, in which one of summations was reduced to the evaluation of a sum of the first degree. Similar but technically more complicated considerations are used for the reduction of the degree of sums in the Weyl method as well.
In Chapter I, the Weyl sums of the first degree were considered and it was shown that the estimate
$$ left| {sumlimits_{x = 1}^P {{e^{2pi ialpha x}}} }
ight| leqslant min left( {P,frac{1}{{2left| alpha
ight|}}}
ight) $$
holds for them. The basic idea of Weyl’s method consists in reducing the estimation of a sum of an arbitrary degree n ⩾ 2
$$ Sleft( P
ight) = sumlimits_{x = 1}^P {{e^{2pi ileft( {{alpha _1}x + ldots + {alpha _n}{x^n}}
ight)}}} $$
to the estimation of a sum of degree n − 1 and, ultimately, to the use of the estimate (140). We have already met the reduction of the degree of an exponential sum in proving the theorem on the modulus of the Gauss sum. In the Gauss theorem, the square of the modulus of the exponential sum of the second degree was transformed with the help of linear change of variable in summation into a double sum, in which one of summations was reduced to the evaluation of a sum of the first degree. Similar but technically more complicated considerations are used for the reduction of the degree of sums in the Weyl method as well.