Counterexamples of Friedlander--Iwaniec dual sums conjecture

2026/07/18 by Khai-Hoan Nguyen-Dang
#math.NT

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Abstract

Let a(n) and b(n) be arithmetic sequences, and A(s)=∑_n≥1a(n)n^-s, B(s)=∑_n≥1b(n)n^-s, be the two Dirichlet series related by a certain functional equation. Let m be the 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_ℓ,D(x,N) := ∑_\substackn∈\mathbb N
n≤ N b(n)n^-β_m cos( 2πm((nx)/(D))^1/m +(πℓ)/(4) ), where D≥1 is the conductor, β_m:=(m+1)/(2m), and ℓ=m-3-2k is determined by the archimedean weight k of the functional equation. Their Conjecture 1 predicts that, for every ε>0, \mathcal B_ℓ,D(x,N) ≪_{ε,\boldsymbolκ} (DNx)^ε, 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≥ 4 where ζ(s):=∑_n≥1n^-s (Res>1) is the Riemann zeta function.

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