TY - JOUR

T1 - Distribution of harmonic sums and Bernoulli polynomials modulo a prime

AU - Garaev, Moubariz Z.

AU - Luca, Florian

AU - Shparlinski, Igor E.

PY - 2006/8

Y1 - 2006/8

N2 - For a fixed integer s≥.1, we estimate exponential sums with harmonic sums [InlineMediaObject not available: see fulltext.] 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 >0, the set {H s (n):n1/2+ε} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as [InlineMediaObject not available: see fulltext.] with max{n ν |ν=1,. . . , 7}≥p 1/2+ε , 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.

AB - For a fixed integer s≥.1, we estimate exponential sums with harmonic sums [InlineMediaObject not available: see fulltext.] 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 >0, the set {H s (n):n1/2+ε} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as [InlineMediaObject not available: see fulltext.] with max{n ν |ν=1,. . . , 7}≥p 1/2+ε , 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.

UR - http://www.scopus.com/inward/record.url?scp=33744753110&partnerID=8YFLogxK

U2 - 10.1007/s00209-006-0939-5

DO - 10.1007/s00209-006-0939-5

M3 - Article

AN - SCOPUS:33744753110

SN - 0025-5874

VL - 253

SP - 855

EP - 865

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 4

ER -