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 and be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is and . We are interested in the following integrals: and , where . Furthermore, denote by the number of pairs of zeros with and --i.e., off-diagonal and diagonal pairs.

Theorem. *Assume the Riemann hypothesis. The following three hypotheses* (A), (B) *and* *are equivalent: for* *and* *as* *we have* (A) , (B) *and* . Hypothesis is called the *essential simplicity hypothesis*.

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0678343-0

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