## Arithmetic equivalent of essential simplicity of zeta zeros

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- by Julia Mueller PDF
- 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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*Pair correlation of zeros of the zeta function*(to appear).

*Gaps between primes*(unpublished).

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## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 175-183 - MSC: Primary 10H05; Secondary 10H15
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678343-0
- MathSciNet review: 678343