## 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 $$L(s, \chi)$$

### Citations:

Zbl 0268.10023; Zbl 0806.11042; Zbl 0414.10044; Zbl 0792.11040
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### References:

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