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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)].

MSC:

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