Counterexamples of Friedlander--Iwaniec dual sums conjecture

Abstract

Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-β_m} \cos\left( 2πm\left(\frac{nx}{D}\right)^{1/m} +\frac{π\ell}{4} \right),$$ where $D\ge1$ is the conductor, $β_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbolκ} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=ζ(s)^m,\; m\geq 4$$ where $$ζ(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.