On the Piatetski-Shapiro-Vinogradov theorem
Let \(\gamma\in (1,{53 \over 50})\) and let \({\mathcal P} \equiv {\mathcal P} (\gamma)= \{p:p= [n^\gamma]\), \(n\in\mathbb{N}\}\) be the set of Piatetski-Shapiro primes. Define \(T(N) \equiv T(\gamma_1, \gamma_2, \gamma_3;N) =|\{P_i\in {\mathcal P} (\gamma_i): P_1+ P_2+ P_3= N\}|\). The author proves that if \(73(1-\gamma_3) <9\), \(73(1-\gamma_2) +43(1-\gamma_3) <9\) and \(73(1-\gamma_1) +43(2-\gamma_2 -\gamma_3) <9\), then \[ T(N)= \bigl(1 +o(1)\bigr) {\Gamma (\gamma_1+1) \Gamma(\gamma_2+1) \Gamma(\gamma_3+1) \over \Gamma (\gamma_1+ \gamma_2+ \gamma_3)} \sigma(N) N^{\gamma_1 +\gamma_2+ \gamma_3-1}/ \log^3N, \] where \[ \sigma(N) =\prod_{p\mid N} \left(1- {1\over (p-1)^2} \right) \prod_{p \nmid N} \left(1+ {1\over (p-1)^3} \right) \] is the singular series from Vinogradov's result on the ternary Goldbach problem.
scientific article; zbMATH DE number 1132884