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

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

MathSciNet review:
678343

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Abstract | References | Similar Articles | Additional Information

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*.

**[1]**Harald Cramér,*Ein Mittelwertsatz in der Primzahltheorie*, Math. Z.**12**(1922), no. 1, 147–153 (German). MR**1544509**, https://doi.org/10.1007/BF01482072**[2]**Akio Fujii,*On the zeros of Dirichlet 𝐿-functions. I*, Trans. Amer. Math. Soc.**196**(1974), 225–235. MR**0349603**, https://doi.org/10.1090/S0002-9947-1974-0349603-2**[3]**P. X. Gallagher,*Pair correlation of zeros of the zeta function*(to appear).**[4]**P. X. Gallagher and Julia H. Mueller,*Primes and zeros in short intervals*, J. Reine Angew. Math.**303/304**(1978), 205–220. MR**514680****[5]**H. L. Montgomery,*Gaps between primes*(unpublished).**[6]**H. L. Montgomery,*The pair correlation of zeros of the zeta function*, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR**0337821****[7]**Atle Selberg,*Contributions to the theory of the Riemann zeta-function*, Arch. Math. Naturvid.**48**(1946), no. 5, 89–155. MR**0020594****[8]**-,*On the normal density of primes in short intervals and the difference between consecutive primes*, Arch. Math. Naturvid. B**47**(1943).

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Additional Information

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