On a conjecture of Montgomery and Soundararajan

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 $Ω$-results for the classical prime counting function.

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