Get Area, lattice points, and exponential sums PDF

By M. N. Huxley

ISBN-10: 0198534663

ISBN-13: 9780198534662

In analytic quantity thought many difficulties might be "reduced" to these regarding the estimation of exponential sums in a single or a number of variables. This e-book is an intensive therapy of the advancements bobbing up from the strategy for estimating the Riemann zeta functionality. Huxley and his coworkers have taken this technique and tremendously prolonged and more advantageous it. The strong strategies awarded right here pass significantly past older equipment for estimating exponential sums akin to van de Corput's approach. the opportunity of the tactic is much from being exhausted, and there's huge motivation for different researchers to attempt to grasp this topic. notwithstanding, somebody at the moment attempting to study all of this fabric has the bold activity of wading via various papers within the literature. This ebook simplifies that job via featuring all the suitable literature and an outstanding a part of the history in a single package deal. The booklet will locate its greatest readership between arithmetic graduate scholars and lecturers with a examine curiosity in analytic thought; particularly exponential sum equipment.

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Example text

When the B process is applied to y = TF(m/M), then we need F'(x) to be nonzero. Swinnerton-Dyer's method requires F" (x) not to vanish, and bounds for rounding error sums using exponential sum methods also require certain derivatives or determinants of derivatives to be non-zero. The derivatives of the function defining the curve in the final comparison problem will be rational functions in the derivatives F(T)( ) of the function F(x) defining the curve in the original problem, evaluated at one or more nearby points .

The use of n-tuples is based on Filaseta's use of second differences (1988), but it can also be regarded as an iteration of one of Swinnerton-Dyer's innovations (1974). , mR be a sequence of positive integers with

If L(nM) > 1, then the curve nMC enters each new square by cutting a lattice line. Since nMC is a convex curve, it meets lattice lines in two distinct points, or possibly in a closed interval. In the case of a closed interval, we take the endpoints of the interval to be the two points in which the lattice line is cut by the curve; they may coincide. Then L(nM), the number of squares cut, is at most the number of times that the curve nMC cuts a lattice line in this sense, and L(nM) < 4(nMA/2 + 1) = 2nMA + 4.

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Area, lattice points, and exponential sums by M. N. Huxley

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