We establish lower bounds for all weighted even moments of primes up to X in intervals which are in agreement with a conjecture of Montgomery and Soundararajan. Our bounds hold unconditionally for an unbounded set of values of X, and hold for all X under the Riemann Hypothesis. We also deduce new unconditional \(\Omega \)-results for the classical prime counting function.

中文翻译：

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关于蒙哥马利和桑达拉然的猜想

我们为所有不超过X的素数加权加权偶数矩建立下界，这与蒙哥马利和Soundararajan的猜想是一致的。对于无限制的X值，我们的边界成立，并且在黎曼假设下对所有X成立。我们还推导了新的无条件\（\ Omega \）-经典素数计数函数的结果。