Counterexamples of Friedlander--Iwaniec dual sums conjecture

Abstract

Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum _{n\ge 1}a(n)n^{-s},B(s)=\sum _{n\ge 1}b(n)n^{-s},$$ 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 $${\mathcal{B}}_{\ell ,D}(x,N):=\sum _{\begin{array}{c}n\in \mathbb{N}\\ n\le N\end{array}}b(n)n^{-{\beta }_m}\cos \left(2\pi m{\left(\frac{nx}{D}\right)}^{1/m}+\frac{\pi \ell }{4}\right),$$ where $D\ge 1$ is the conductor, ${\beta }_m:=\frac{m+1}{2m}$, and $\ell =m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\epsilon >0$, $${\mathcal{B}}_{\ell ,D}(x,N){\ll }_{\epsilon ,\mathbf{\kappa }}(DNx{)}^{\epsilon },$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=\zeta (s{)}^m,m\ge 4$$ where $$\zeta (s):=\sum _{n\ge 1}{n}^{-s}(\mathrm{Re}s>1)$$ is the Riemann zeta function.

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