AI 中文总结
研究整数作为两个素数平方和的表示数的第\(k\)个矩，证明\(k\geq4\)时其正确量级，此前\(k = 4\)上界已知部分情况，\(k\geq4\)下界有条件已知，还给出移位素因子函数矩下界更简单证明。
AI 中文摘要
我们证明了，对于每个固定整数\(k\geq4\)，计算整数作为两个素数平方和的表示数的函数的第\(k\)个矩的正确量级。\(k = 4\)时的上界之前已知到\(\log\log\log x\)，\(k\geq4\)时的下界仅在Sabuncu关于素数线性方程的Green-Tao定理的推测统一版本的条件下已知。作为我们方法的应用，我们给出了移位素因子函数矩的下界的更简单证明，从而恢复了Gabdullin关于Fan和Pomerance猜想的最新结果的下界部分。
英文摘要
We prove, for every fixed integer $k\ge 4$, the correct order of magnitude for the $k$th moments of the function that counts the number of representations of an integer as sums of two prime squares. The upper bound for $k=4$ was previously known up to $\log\log\log x$, and the lower bound for $k\ge 4$ was only known conditionally on a conjectural uniform version of the Green-Tao theorem on linear equations in primes by the work of Sabuncu \cite{Sabuncu2024}. As an application of our method, we give a simpler proof of the lower bounds for the moments of the shifted prime divisor function, thereby recovering the lower-bound part of Gabdullin's recent result on a conjecture of Fan and Pomerance.
