Integer points, exponential sums and the Riemann Zeta function

Huxley, M. N. 2024. Integer points, exponential sums and the Riemann Zeta function. Berndt, Bruce, ed. Number Theory for the Millennium II, London: A K Peters/CRC Press, pp. 275-290. (10.1201/9780138747060-13)

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Abstract

One of the problems at the origin of mathematics is to define and calculate the area of a plane region D. If parts of the boundary of D are curved, then this problem leads to the integral calculus. Archimedes’ construction used two polygons, one inside D, the other containing D. After Descartes’ coordinate system, it is natural to consider a zigzag polygon whose vertices are of the form x = mδ, y = nδ, where δ is the size of the smallest square of the printed grid on the graph paper. To find the area of the inner and outer polygons, one counts squares of side δ. This gives upper and lower bounds for the area. A better estimate is given by counting the square bounded by the four points x = mδ, (m + 1)δ, y = nδ, (n + 1)δ if and only if the point (mδ, nδ) lies in D, so the integer lattice point (m, n) lies in δ -1 D.

Item Type:	Book Section
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Additional Information:	First Published 2002 eBook Published 31 July 2024
Publisher:	A K Peters/CRC Press
ISBN:	9780138747060
Last Modified:	05 Dec 2024 13:45
URI:	https://orca.cardiff.ac.uk/id/eprint/172748

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