Distribution of alternative power sums and Euler polynomials modulo a prime

Abstract

For a fixed integer s >= 2, we estimate exponential sums with alternative power sums As(n) = Sigma(n)(i=0)(-1)(i)i(s) individually and on average, where A(s)(n) is computed modulo p. Our estimates imply that, for any epsilon > 0, the sets {A(s)(n) : n < p(1/2+epsilon)} and {(-1)E-n(s)(n) : n < p(1/2+epsilon)} are uniformly distributed modulo a sufficient large p, where E-s(x) are Euler polynomials. Comparing with the results in Garaev et al. (2006) [M. Z. Garaev, F. Luca and I. E. Shparlinski, Distribution of harmonic sums and Bernoulli polynomials modulo a prime, Math. Z., 253 (2006), 855-865], we see that the uniform distribution properties for the alternative power sums and Euler polynomials modulo a prime are better than those for the harmonic sums and Bernoulli polynomials. (C) 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved.