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Picturesque Exponential Sums, I

D. H. Lehmer and Emma Lehmer
The American Mathematical Monthly
Vol. 86, No. 9 (Nov., 1979), pp. 725-733
DOI: 10.2307/2322021
Stable URL: http://www.jstor.org/stable/2322021
Page Count: 9
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Picturesque Exponential Sums, I
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Abstract

Let k be a fixed positive integer ⩾ 2 and let ξ = exp(2π i/k). For any integer n ⩾ 0 we define b(n) to be the sum of the digits of n when written to the base k. In this paper we consider the exponential sum Sj (m) = ∑m n=0 ξb(n)+jn. The properties of Sj(m) are exhibited by plotting their graphs Gj(k) for k ⩽ 9. These properties are proved for general k. The paper is the forerunner of a more complicated paper [2] in which b(n) is replaced by the sum of products of all pairs of two consecutive digits of n.

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