ATS Theorem - History of The Problem

History of The Problem

In some fields of mathematics and mathematical physics, sums of the form

$S = \sum_{a<k\le b} \varphi(k)e^{2\pi i f(k)} \ \ \ (1)$

are under study.

Here and are real valued functions of a real argument, and Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation.

The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson.

We shall define the length of the sum to be the number (for the integers and this is the number of the summands in ).

Under certain conditions on and the sum can be substituted with good accuracy by another sum

$S_1 = \sum_{\alpha<k\le \beta} \Phi(k)e^{2\pi i F(k)}, \ \ \ (2)$

where the length is far less than

First relations of the form

$S = S_1 + R, \ \ \ (3)$

where are the sums (1) and (2) respectively, is a remainder term, with concrete functions and were obtained by G. H. Hardy and J. E. Littlewood, when they deduced approximate functional equation for the Riemann zeta function $\zeta(s)$ and by I. M. Vinogradov, in the study of the amounts of integer points in the domains on plane. In general form the theorem was proved by J. Van der Corput, (on the recent results connected with the Van der Corput theorem one can read at ).

In every one of the above-mentioned works, some restrictions on the functions and were imposed. With convenient (for applications) restrictions on and the theorem was proved by A. A. Karatsuba in (see also,).

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