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 -