# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Arithmetic equivalent of essential simplicity of zeta zerosHTML articles powered by AMS MathViewer

by Julia Mueller
Trans. Amer. Math. Soc. 275 (1983), 175-183 Request permission

## 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.
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