Picturesque Exponential Sums, I
Derrick Henry Lehmer, Emma T. Lehmer · American Mathematical Monthly · 1979
Let k be a fixed positive integer ⩾2 and let ζ = exp(2πi/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 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.