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On an equation with prime numbers. (English) Zbl 1023.11054

The authors consider a hybrid of the Goldbach and Piatetski-Shapiro problems. Let \(c>1\) be a real, non-integral number. Let \(R(N)\) be defined as \(\sum^* (\log p_1)(\log p_2)(\log p_3)\), where \(\sum^*\) denotes that the sum is over triples of primes \((p_1,p_2,p_3)\) such that \([p_1^c]+ [p_2^c]+ [p_3^c]= N\). The authors prove an asymptotic formula for \(R(N)\) valid in the range \(1< c< \frac{12}{11}\). This improves a result of M. B. S. Laporta and D. I. Tolev [Mat. Zametki 57, 926-929 (1995; Zbl 0858.11054)], who had \(\frac{17}{16}\) in place of \(\frac{12}{11}\). The proof uses Heath-Brown’s identity for trigonometric sums over primes, van der Corput’s method, and an estimate of a double exponential sum due to Kolesnik.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11L20 Sums over primes

Citations:

Zbl 0858.11054
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