Qian Zhao

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

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...