Exponential Sums and the Riemann Zeta Function V
Exponential Sums and the Riemann Zeta Function V
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指数和和黎曼 Zeta 函数 V
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发表时间：
2005
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通讯作者：
M. Huxley
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作者：
M. Huxley
一个货车德尔科普特指数和是S = exp（2 π i f（m）），其中m的大小为M，函数f（x）的大小为T，α =（log M）/ log T < 1。在α的不同范围内，S有不同的界限。在α接近1/大于2的中间区域，S=O（MTθ+ θ）。这个θ限制了黎曼zeta函数在其临界线Re s = 1/over 2上的增长指数。货车德尔科普特使用了一种迭代法，在每一步都改变α。虽然仍然基于均方，但Bjerieri-Iwaniec方法引入了数论思想和问题。第二个间隔问题是当y = f '（x）的图的两条弧在整数格的自同构后大致重合时，计算和的短间隔之间的共振次数。在本系列的前一篇论文[Proc.伦敦数学学会会刊（3）66（1993）1-40]和专著《面积、格点和指数和》中，我们看到，重合意味着有一个整数点接近某个“共振曲线”，这是某个对偶空间中的一组曲线之一，现在在论文“Bjueri-Iwaniec方法中的共振曲线”中精确计算，它将出现在函数中。约置评.数学
A Van der Corput exponential sum is S = Σ exp (2 π i f(m)) where m has size M, the function f(x) has size T and α = (log M) / log T < 1. There are different bounds for S in different ranges for α. In the middle range where α is near 1/over 2, S=O(MTθ+ϵ) . This θ bounds the exponent of growth of the Riemann zeta function on its critical line Re s = 1/over 2. Van der Corput used an iteration which changed α at each step. The Bombieri–Iwaniec method, whilst still based on mean squares, introduces number‐theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of y = f′(x) coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1–40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ‘resonance curve’, one of a family of curves in some dual space, now calculated accurately in the paper ‘Resonance curves in the Bombieri–Iwaniec method’, which is to appear in Funct. Approx. Comment. Math.