Distribution of alternative power sums and Euler polynomials modulo a prime
- Li, Yan
- Kim, Min-Soo
- Hu, Su
Publication date
March 2012
DOI Publisher
Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. ISSN
0019-3577Abstract
AbstractFor a fixed integer s≥2, we estimate exponential sums with alternative power sums As(n)=∑i=0n(−1)iis individually and on average, where As(n) is computed modulo p. Our estimates imply that, for any ϵ>0, the sets {As(n):n<p1/2+ϵ}and{(−1)nEs(n):n<p1/2+ϵ} are uniformly distributed modulo a sufficient large p, where Es(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
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