素数作为概率测度谱的条件算术障碍
A conditional arithmetic obstruction to the prime numbers as a spectrum of a probability measure
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中文总结 AI 辅助
该研究证明，若每个足够大的正偶数都可表示为两素数之差（即Polignac猜想成立），则素数集无法成为实数域上任何Borel概率测度的指数标准正交基的频率谱。
中文摘要 AI 辅助
设$\Pp=\{2,3,5,7,\ldots\}$表示素数集。我们证明：若每个足够大的正偶数都可表示为两个素数之差，则不存在实数$\bR$上的Borel概率测度$\mu$，使得$\left\{e^{2\pi i p x}:p\in\Pp\right\}$是$L^2(\mu)$的一组标准正交基。特别地，在Polignac猜想成立的前提下，素数集$\Pp$不可能是$\bR$上任意概率测度的谱（即指数标准正交基的频率集合）。
英文摘要
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 $μ$ on $\R$ for which \(\left\{e^{2πi p x}:p\in\Pp\right\}\) is an orthonormal basis of $L^2(μ)$. 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 $\R$.
