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arXiv · 2608.12400

A conditional arithmetic obstruction to the prime numbers as a spectrum of a probability measure

Zhi-Yi Wu·Qian Zhao
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Abstract

Let $\Pp=\{2,3,5,7,\ldots\}$ denote the set of prime numbers. We prove that if every sufficiently large positive even integer can be represented as a difference of two primes, then there is no Borel probability measure $\mu$ on $\R$ for which \(\left\{e^{2\pi i p x}:p\in\Pp\right\}\) is an orthonormal basis of $L^2(\mu)$. In particular, under the Polignac conjecture, the prime numbers $\Pp$ cannot be a spectrum (i.e., the set of frequencies of an exponential orthonormal basis) of any probability measure on $\R$.

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Zhi-Yi Wu, Qian Zhao. 2026-08-09. A conditional arithmetic obstruction to the prime numbers as a spectrum of a probability measure. https://arxiv.org/abs/2608.12400

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