Distribution of harmonic sums and Bernoulli polynomials modulo a prime

For a fixed integers >= 1, we estimate exponential sums with harmonic sums H-s(n)= Sigma(n)(i=1) 1/i(s) individually and on average, where Hs(n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H, (n). For example, our estimates imply that for any epsilon > 0, the set {H-s (n) : n < p(1/2+epsilon)} is uniformly distributed modulo a sufficiently large p. We also show that every residue class; can be represented as H-s(n(1))+ (. . .) + H-s(n(7)) equivalent to lambda (mod p) with max {n(v) vertical bar v = 1, . . . , 7} <= p(11/12+epsilon), and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values Bp-r(n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche.