论文标题
算术进展中素数的时刻
Moments of moments of primes in arithmetic progressions
论文作者
论文摘要
我们为在算术进展中的所有加权时刻建立了无条件的$ω$ - 分子。我们还研究了这些时刻的时刻，并在GRH下建立了下限。最后，在GRH和LI下，我们证明了相关限制分布的所有时刻的渐近分析，这反过来表明我们的无条件和GRH结果基本上是最好的。利用我们的概率结果，我们在具有精确相关的有效性范围的矩上制定了一个猜想，我们认为这也是最好的。最后的猜想意味着蒙哥马利 - 苏纳拉杰恩（Montgomery-Soundararajan）的$ q $ - 构图，猜想在短时间间隔刺激了素数的高斯分布。我们的证明中的想法包括在明确公式中的积极性的新颖应用，以及通过某些人固定的字符阵列组合。
We establish unconditional $Ω$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an asymptotic for all moments of the associated limiting distribution, which in turn indicates that our unconditional and GRH results are essentially best possible. Using our probabilistic results, we formulate a conjecture on the moments with a precise associated range of validity, which we believe is also best possible. This last conjecture implies a $q$-analogue of the Montgomery-Soundararajan conjecture on the Gaussian distribution of primes in short intervals. The ideas in our proofs include a novel application of positivity in the explicit formula and the combinatorics of arrays of characters which are fixed by certain involutions.
