Distribution of harmonic sums and Bernoulli polynomials modulo a prime

Abstract

For a fixed integer s≥1, we estimate exponential sums with harmonic sums individually and on average, where H _s (n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H _s (n). For example, our estimates imply that for any ɛ>0, the set {H _s (n):n<p^1/2+ɛ} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as with max{n _ν |ν=1,. . . , 7}≤p^11/12+ɛ, and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values B _{p − r} (n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche.