Kumchev, A.; Nedeva, T. On an equation with prime numbers. (English) Zbl 1023.11054 Acta Arith. 83, No. 2, 117-126 (1998). 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. Reviewer: S.W.Graham (Mount Pleasant) Cited in 3 ReviewsCited in 27 Documents MSC: 11P32 Goldbach-type theorems; other additive questions involving primes 11L20 Sums over primes Citations:Zbl 0858.11054 PDFBibTeX XMLCite \textit{A. Kumchev} and \textit{T. Nedeva}, Acta Arith. 83, No. 2, 117--126 (1998; Zbl 1023.11054) Full Text: DOI EuDML