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Transactions of the American Mathematical Society

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



Arithmetic equivalent of essential simplicity of zeta zeros

Author: Julia Mueller
Journal: Trans. Amer. Math. Soc. 275 (1983), 175-183
MSC: Primary 10H05; Secondary 10H15
MathSciNet review: 678343
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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^{\prime}$ with $ 0 < \Upsilon \leqslant T$ and $ 0 < (\Upsilon^{\prime} - \Upsilon)L \leqslant \alpha \,((\Upsilon^{\prime} - \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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Keywords: Prime number theorem, Riemann-von Mangoldt formula, remainder term, Riemann zeta function, zeros, simple zeros, pairs of zeros, essential simplicity of zeros
Article copyright: © Copyright 1983 American Mathematical Society

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