Averages of diagonal Elliott-Halberstam problem twisted by Möbius function with Sobolev and Hölder-Zygmund weights

Abstract

Recalling that the so-called Elliott-Halberstam conjecture twisted by the Möbius function $\hat{I}¼(n)$ claims that \[ \sum_{q\leq N^θ}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{\scriptstyle n\equiv a\,\mod\,q}{n\leq y}}Λ(n)μ\left(N-n\right)-\frac{1}{φ\left(q\right)}\sum_{n\leq y}Λ(n)μ\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}} \] for every $A>0$, where $0<\hat{I}¸<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $\hat{I}¸$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the Hölder-Zygmund spaces ${\mathcal{C}}^{\hat{I}}´$ for suitable range of $\hat{I}´$, the bound of the average is consistent with the bound of the ``diagonal versions'' of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mathrm{mod}q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<\hat{I}¸<1$ and, in the case of weights in the Hölder-Zygmund class ${\mathcal{C}}^{\hat{I}}´$, for $\hat{I}¸$ that depends on the choice of $\hat{I}´$ but still not below the $1/2-2\epsilon$ threshold.