Distribution of harmonic sums and Bernoulli polynomials modulo a prime

Moubariz Z. Garaev*, Florian Luca, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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):n<p 1/2+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.

Original languageEnglish
Pages (from-to)855-865
Number of pages11
JournalMathematische Zeitschrift
Issue number4
Publication statusPublished - Aug 2006


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