## Character sums and products of factorials modulo $$p$$.(English)Zbl 1101.11027

Let $$p$$ be prime and let $$u$$, $$v$$, $$S$$, $$T$$ be integers. Suppose $$C_ 1,\ldots, C_ u$$ and $$D_ 1, \ldots, D_ v$$ are integers such that no two distinct $$C_ i$$ and no two distinct $$D_ j$$ are congruent mod $$p$$, where $$1 \leq u,v \leq p-1$$. Let $$f(n)$$ denote the number of solutions of $$C_ iD_ j\equiv n, \bmod\,p$$. Suppose $$1 \leq T \leq p$$, and denote $$\Sigma_ {S,T}= \sum_ {S<n \leq S+T}f(n)$$.
A. Sarközy [Acta Math. Hung. 49, 397–401 (1987; Zbl 0629.10004)] showed, by an elementary method, that $$\bigl| \Sigma_ {S,T}- uvT/p\bigr| <2\sqrt{puv}\log p$$. Adjusting the “main term” slightly, the authors establish an inequality $$\bigl| \Sigma_ {S,T}- uvT/(p-1)\bigr| \ll u+v+\sqrt{uv}\,B(T,p)\log p$$, in which a factor $$B(T,p) = T \smash{p^ {1/4r^ 2}(p^ {1/4}/T)^ {1/r}}$$ appears following an appeal to a character-sum estimate of D. Burgess. The expression in terms of Dirichlet characters of the quantity to be estimated is obtained following a method of A. A. Karatsuba [Sov. Math., Dokl. 11, 707–711 (1970); translation from Dokl. Akad. Nauk SSSR 192, 724–727 (1970; Zbl 0224.10043)].
When $$p$$ is large and $$T$$ is close to $$p^ {1/4}$$ a choice of $$r$$ so that $$1/4r^ 2<\varepsilon$$ yields an estimate $$\ll u+v+p^ {1/4+\varepsilon}\sqrt{uv}$$, improving upon Sarközy’s elementary estimate.
The authors deduce a similar improvement from $$1\over2$$ to $$1\over4$$ in the exponent of a prime $$p$$ in a result of F. Luca and P. Stǎnicǎ [Colloq. Math. 96, No. 2, 191–205 (2003; Zbl 1042.11002)] about products of factorials mod $$p$$.

### MSC:

 11L40 Estimates on character sums 11A07 Congruences; primitive roots; residue systems 11B65 Binomial coefficients; factorials; $$q$$-identities

### Keywords:

congruences; number of solutions

### Citations:

Zbl 0629.10004; Zbl 0224.10043; Zbl 1042.11002
Full Text:

### References:

 [1] W. Banks, F. Luca, I. E. Shparlinski, H. Stichtenoth, On the value set of $$n!$$ modulo a prime. Turkish J. Math. 29 no. 2 (2005), 169-174. · Zbl 1161.11386 [2] D. A. Burgess, On character sums and $$L$$-series, II. Proc. London Math. Soc. (3) 13 (1963), 524-536. · Zbl 0123.04404 [3] M. Z. Garaev, F. Luca, I. E. Shparlinski, Character sums and congruences with $$n!$$. Trans. Amer. Math. Soc. 356 (2004), 5089-5102. · Zbl 1060.11046 [4] M. Z. Garaev, F. Luca, I. E. Shparlinski, Exponential sums and congruences with factorials. J. reine angew. Math., to appear. · Zbl 1071.11051 [5] M. Z. Garaev, F. Luca, I. E. Shparlinski, Waring problem with factorials mod $$p$$. Bull. Australian Math. Soc. 71 (2005), 259-264. · Zbl 1076.11013 [6] A. A. Karatsuba, The distribution of products of shifted prime numbers in arithmetic progressions. Dokl. Akad. Nauk SSSR 192 (1970), 724-727 (in Russian). · Zbl 0224.10043 [7] F. Luca, P. Stănică, Products of factorials modulo $$p$$. Colloq. Math. 96 no. 2 (2003), 191-205. · Zbl 1042.11002 [8] A. Sárkőzy, On the distribution of residues of products of integers. Acta Math. Hung. 49 (3-4) (1987), 397-401. · Zbl 0629.10004
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