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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
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References:

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[2] D. A. Burgess, On character sums and \(L\)-series, II. Proc. London Math. Soc. (3) 13 (1963), 524-536. · Zbl 0123.04404
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[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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