Cubic Waring-Goldbach problem with Piatetski-Shapiro primes
Linji Long, Jinjiang Li, Min Zhang, Yankun Sui
TL;DR
This work advances the Waring-Goldbach problem by showing that nine cubes of primes restricted to Piatetski–Shapiro sets sum to every sufficiently large odd integer, provided the type parameters satisfy $\gamma \in (\tfrac{317}{320},1)$. The authors implement the Hardy–Littlewood circle method, translating representations into exponential sums and proving a key uniform bound for $F_i(\alpha)-G(\alpha)$ through a detailed analysis of Type I/II sums via Heath–Brown's identity and van der Corput-type estimates. The main result yields an asymptotic formula with a singular series factor, refining prior results of Akbal and Güloğlu (2018) by lowering the admissible range of $\gamma$ to $\tfrac{317}{320}$ and providing corollaries for equal or mixed $\gamma_i$. The approach combines delicate exponential-sum bounds, structured decompositions, and optimization over auxiliary parameters to control error terms and establish the desired representation count.
Abstract
In this paper, it is proved that, for $γ\in(\frac{317}{320},1)$, every sufficiently large odd integer can be written as the sum of nine cubes of primes, each of which is of the form $[n^{1/γ}]$. This result constitutes an improvement upon the previous result of Akbal and Güloğlu [1].