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On the distribution of residues of products of integers. (English) Zbl 0629.10004

Es seien p eine Primzahl, M, N, S, T, \(a_ 1,...,a_ M\), \(b_ 1,...,b_ N\) ganze Zahlen mit \(1\leq M\), \(N\leq p-1\), \(1\leq T\leq p\), sowie (1) \(a_ i\not\equiv a_ j (mod p)\) für \(1\leq i<j\leq M\) und (2) \(b_ i\not\equiv b_ j (mod p)\) für \(1\leq i<j\leq N\). Wenn f(n) die Anzahl der Lösungen von \(a_ xb_ y\equiv n (mod p)\), \(1\leq x\leq M\), \(1\leq y\leq N\) bezeichnet, so wird die Abschätzung \[ | \sum^{S+T}_{n=S+1}f(n)-\frac{MNT}{p}| \quad <^ 2(pMN)^{1/2} \log p \] bewiesen. Speziell mit \(S=0\), \(T=[2p^{3/2} M^{-1/2} N^{- 1/2} \log p]+1\) kann daraus \(\sum^{T}_{n=1}f(n)>0\) hergeleitet werden. Andererseits werden für \(p>900\), \(M=N=T=(p-1)/2\) ganze Zahlen \(a_ 1,...,a_ M\), \(b_ 1,...,b_ N\), S konstruiert, so daß (1) und (2) erfüllt sind, aber \[ | \sum^{S+T}_{n=S+1}f(n)- \frac{MNT}{p}| \quad >\quad \frac{1}{28}(pMN)^{1/2} \] ist. Die Methoden sind elementar.
Reviewer: D.Leitmann

MSC:

11A07 Congruences; primitive roots; residue systems
11L40 Estimates on character sums
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References:

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