Arithmetic equivalent of essential simplicity of zeta zeros
Author:
Julia Mueller
Journal:
Trans. Amer. Math. Soc. 275 (1983), 175183
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 Riemannvon 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., offdiagonal 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, http://dx.doi.org/10.1007/BF01482072
 [2]
Akio
Fujii, On the zeros of Dirichlet
𝐿functions. I, Trans. Amer. Math.
Soc. 196 (1974),
225–235. MR 0349603
(50 #2096), http://dx.doi.org/10.1090/S00029947197403496032
 [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
(80b:10060)
 [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H. L. Montgomery, Gaps between primes (unpublished).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
(49 #2590)
 [7]
Atle
Selberg, Contributions to the theory of the Riemann
zetafunction, Arch. Math. Naturvid. 48 (1946),
no. 5, 89–155. MR 0020594
(8,567e)
 [8]
, On the normal density of primes in short intervals and the difference between consecutive primes, Arch. Math. Naturvid. B 47 (1943).
 [1]
 H. Cramér, Ein Mittelwertsatz in der Primzahltheorie, Math. Z. 12 (1922), 147153. MR 1544509
 [2]
 A. Fujii, On the zeros of Dirichlet functions. I. Trans. Amer. Math. Soc. 196 (1974), 225235. MR 0349603 (50:2096)
 [3]
 P. X. Gallagher, Pair correlation of zeros of the zeta function (to appear).
 [4]
 P. X. Gallagher and J. Mueller, Primes and zeros in short intervals, J. Reine Angew. Math. 303/304 (1977), 205220. MR 514680 (80b:10060)
 [5]
 H. L. Montgomery, Gaps between primes (unpublished).
 [6]
 , The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R. I., 1973, pp. 181193. MR 0337821 (49:2590)
 [7]
 A. Selberg, Contributions to the theory of the Riemann zetafunction, Arch. Math. Naturvid. 48 (1946), 89155. MR 0020594 (8:567e)
 [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:
http://dx.doi.org/10.1090/S00029947198306783430
PII:
S 00029947(1983)06783430
Keywords:
Prime number theorem,
Riemannvon Mangoldt formula,
remainder term,
Riemann zeta function,
zeros,
simple zeros,
pairs of zeros,
essential simplicity of zeros
Article copyright:
© Copyright 1983
American Mathematical Society
