Arithmetic equivalent of essential simplicity of zeta zeros

Abstract:

Let $R(x)$ and $S(t)$ be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is $\psi (x) = x + R(x)$ and $N(t) = (1/2\pi )\int _0^t {\log (\tau /2\pi ) d\tau + S(t) + 7/8 + O(1/t)}$. We are interested in the following integrals: $J(T,\beta ) = \int _1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}}$ and $I(T,\alpha ) = \int _1^T {{{(S(t + \alpha /L) - S(t))}^2}dt}$, where $L = {(2\pi )^{ - 1}}\log T$. Furthermore, denote by $N(T,\alpha )(N^{\ast }(T))$ the number of pairs of zeros $\frac {1} {2} + i\Upsilon ,\frac {1} {2} + i\Upsilon ’$ with $0 < \Upsilon \leqslant T$ and $0 < (\Upsilon ’ - \Upsilon )L \leqslant \alpha ((\Upsilon ’ - \Upsilon )L = 0)$—i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and $({{\text {C}}_1},{{\text {C}}_2})$ are equivalent: for $\beta \to \infty$ and $\alpha \to 0$ as $T \to \infty$ we have (A) $J(T,\beta ) \sim \beta {T^{ - 1}}{\log ^2}T$, (B) $I(T,\alpha ) \sim \alpha T$ and $({{\text {C}}_1})\;N^{\ast }(T) \sim TL,({{\text {C}}_2})N(T,\alpha ) = o(TL)$. Hypothesis $({{\text {C}}_1},{{\text {C}}_2})$ is called the essential simplicity hypothesis.