A note on primes in reciprocal arithmetic progressions (Q1379861)

The author deals with the limiting distributions \(F(u,q,a)\) and \(\widetilde F(u,q,a)\) of certain normalized remainder terms related with \[ \sum_{n\leq x \atop n\equiv a \pmod q} \Lambda(n) \quad \text{and} \quad \sum_{n\leq x \atop n\equiv a\pmod q} {\Lambda (n) \over n},\quad (a,q) =1. \] Assuming GRH, the author proves that \(F(u,q,a)\) and \(\widetilde F(u,q,a)\) exist and for almost all \(u\in\mathbb{R}\) one has \[ F(u,q,a)= 1-\widetilde F(-u,q,\overline a), \quad a\overline a \equiv 1 \pmod q. \] Moreover, the exceptional set consists of discontinuity points of such distributions and is therefore countable. The main feature of this interesting paper is that the result is obtained assuming only GRH. In fact, \textit{C. Hooley} [J. Lond. Math. Soc., II. Ser. 16, 1-8 (1977; Zbl 0377.10023)] proved a more precise result on \(F(u,q,a)\) assuming, in addition to GRH, the simplicity of the nontrivial zeros of certain Dedekind zeta functions and the linear independence over \(\mathbb{Q}\) of their imaginary parts. A multidimensional version of such results was obtained by \textit{M. Rubinstein} and \textit{P. Sarnak} [Exp. Math. 3, No. 3, 173-197 (1994; Zbl 0823.11050)]. The proof is based on the analysis of certain boundary functions related with the \(k\)-functions, introduced by the author in a series of papers in Acta Arithmetica. The details are quite condensed.