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Exponential sums of sum-of-digit functions

1986; Duke University Press; Volume: 30; Issue: 4 Linguagem: Inglês

10.1215/ijm/1256064238

1945-6581

Alan H. Stein,

Analytic Number Theory Research

Let q(x) E,<,,qt("), where q > 0 and fl(n) represents the sum of the digits of n written to a given base b.If q b 2, then q,(x) represents the number of odd integers in the first x rows of Pascal's triangle.This case has been studied by Harborth [4] and Stolarsky [9], while the author has previously extended many of their results for arbitrary q > 0 [7].We shall show that many of the same properties hold for arbitrary base b. Define(1.1)B q(b), 0 log B/log b, p(x) (x)/x .We shall establish that q(x) is on the order of x , develop an exact formula for q,(x) and extend q(x) to a continuous function on R +.In the course of doing so, we will also examine interesting properties of the function 1 qt log 1 q h(t) logt 2. We begin by developing several formulas for q(x), each of which also yields a formula for q (x).Formula 1. q(bx) Bq(x).Proof Write ck(bx) .n<x'm<bqfl(bn+m).Since fl(bn + m) [3(n) + fl(m) if m < b, we can factor out qO(") to get(2.1) ck ( bx ) n<x m<b n<x and Formula 1 follows immediately.Corresponding to Formula 1 we have: Formula 1'./(bx) (x).(Note.We frequently take advantage of B b.)