A pair correlation hypothesis and the exceptional set in Goldbach’s problem. (English) Zbl 0884.11042

The authors consider the following generalization of H. L. Montgomery’s pair correlation hypothesis (MC) [Analytic number theory, Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 181-193 (1973; Zbl 0268.10023)] to the zeros of Dirichlet \(L\)-functions. Assume the Generalized Riemann Hypothesis (GRH). Put \[ F(X, T, q,a)= \sum_{\chi_1,\chi_2\text{ mod }q} \chi_1(a)\overline\chi_2(a) \tau(\overline\chi_1) \tau(\chi_2) \sum_{|\gamma_1|, |\gamma_2|< T} X^{i(\gamma_1- \gamma_2)} w(\gamma_1- \gamma_2) \] (\((a,q)= 1\), \(\rho_j={1\over 2}+ i\gamma_j\) denotes non-trivial zeros of \(L(s, \chi_j)\), \(w(u)= 4/(4+ u^2)\)). Denote by \(\text{GMC}(\vartheta)\) \((0<\vartheta\leq{1\over 2})\) the following hypothesis. For every \(\varepsilon>0\), \(V= X^{1-\vartheta}/q\), \(V\leq T\leq X\), \(q\leq X^\vartheta\) the estimate \(F(X, T, q,a)\ll_\varepsilon q^2TX^\varepsilon\) holds. This would improve the trivial bound by a factor \(q\).
The authors study the influence of \(\text{GMC}(\vartheta)\) on the set of exceptions in the binary Goldbach problem. One of their corollaries is: (GRH) and (\(\text{GMC}(\vartheta)\)) imply \[ \#\{n\leq X,\;n\equiv 0\pmod 2,\;n\neq p_1+ p_2\}\ll_\varepsilon X^{1-2\vartheta+ \varepsilon}\quad\text{for every }\varepsilon> 0. \] The proof follows the lines of [A. Perelli and J. Pintz, J. Lond. Math. Soc., II. Ser. 47, 41-49 (1993; Zbl 0806.11042)] and D. R. Heath-Brown’s paper on (MC) and prime number differences [Acta Arith. 41, 85-99 (1982; Zbl 0414.10044)].
The authors also discuss and correct a mistake in a note on Goldbach exceptions by J. Kaczorowski, A. Perelli and J. Pintz [Monatsh. Math. 116, 275-282 (1993; Zbl 0792.11040)].


11P32 Goldbach-type theorems; other additive questions involving primes
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Full Text: DOI


[1] Goldston, Goldbach numbers in short intervals. Glasgow Math. J. 32 pp 285– (1990) · Zbl 0719.11065
[2] Goldston, Proc. Amalfi Conf. Analytic Number Theory pp 115– (1992)
[3] Davenport, Multiplicative Number Theory (1980)
[4] DOI: 10.1112/jlms/s2-25.2.201 · Zbl 0443.10015
[5] Ann. Inst. Fourier 27 pp 1– (1977)
[6] DOI: 10.1112/jlms/s2-47.1.41 · Zbl 0806.11042
[7] DOI: 10.1007/BFb0060851
[8] Montgomery, Proc. A.M.S. Symp. Pure Math. 24 pp 181– (1973)
[9] Languasco, Ann. Inst. Fourier 44 pp 307– (1994) · Zbl 0799.11040
[10] Heath-Brown, Acta Arith. 41 pp 85– (1982)
[11] DOI: 10.1007/BF01301533 · Zbl 0792.11040
[12] DOI: 10.1112/plms/s2-22.1.46 · JFM 49.0127.03
[13] Goldston, Analytic Number Theory and Dioph. Probl pp 183– (1987)
[14] Goldston, Number Theory with an Emphasis on the Markoff Spectrum pp 101– (1993)
[15] Gallagher, J. reine angew. Math. 303 pp 205– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.