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On a conjecture of Montgomery and Soundararajan

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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.

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  1. One can replace differentiability by a Lipschitz condition if for instance \(\eta \) is compactly supported in \({\mathbb {R}}\) and monotonic on \({\mathbb {R}}_{\geqslant 0}\).

  2. We can take for example \(\eta = \eta _0 \star \eta _0\) for some smooth and rapidly decaying \(\eta _0\).

  3. Instead of assuming that \(\eta \) is differentiable, one can assume that it is Lipschitz, compactly supported in \({\mathbb {R}}\) and monotonic on \({\mathbb {R}}_{\geqslant 0}\).

  4. The integrability of \(\xi h(\xi )\) implies that \({{\widehat{h}}}\) is differentiable (see [21, p. 430]).

  5. One can obtain a slightly weaker but unconditional lower bound by applying (2.10) at the end of the argument.


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Correspondence to Régis de la Bretèche.

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Communicated by Kannan Soundararajan.

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Bretèche, R.d.l., Fiorilli, D. On a conjecture of Montgomery and Soundararajan. Math. Ann. 381, 575–591 (2021).

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