Let \[ G(n)=\sum_{a+b=n}\Lambda(a)\Lambda(b) \] and \[ S(x)=\sum_{n\leq x}G(n). \] Then it was shown by G. Bhowmik and J.-C. Schlage-Puchta [Nagoya Math. J. 200, 27–33 (2010; Zbl 1217.11089)] that the Riemann hypothesis yields \(S(x)=\tfrac12 x^2+O(x^{1+\varepsilon})\) for any fixed \(\varepsilon>0\). The present paper shows conversely that if one has \(S(x)=\tfrac12 x^2+O(x^{2-\delta})\) for some positive \(\delta\), then a quasi Riemann hypothesis holds. The paper claims that one may then deduce using work of G. Bhowmik, K. Halupczok, K. Matsumoto and Y. Suzuki [Goldbach representations in arithmetic progressions and zeros of Dirichlet L-functions [to appear, arxiv:1704.06103.]) of G. Bhowmik and J.-C. Schlage-Puchta (cited above) and of A. Granville [Funct. Approximatio, Comment. Math. 37, Part 1, 159–173 (2007; Zbl 1230.11123)] that \(\zeta(s)\not=0\) when \(\operatorname{Re}(s)>1-\delta\). However this would appear to be wrong, if one takes \(\delta=1-\varepsilon\) as allowed by the work cited above.
The theorem is proved using the generating function \[ F(z)=\sum_{n=1}^{\infty}\Lambda(n)z^n. \] The hypothesis gives a good estimate for \(F(z)^2\) when \(z\) is close to 1, and the result then follows from a careful analysis of \[ \int_{| z| =R}F(z)\{z^{-1}+z^{-2}+\ldots +z^{-N}\}dz=\psi(N) \] with \(R\) close to 1, motivated by a simple version of the circle method.