bibkey: bergryanyoung2023sextictwists
authors: Jennifer Berg; Nathan C. Ryan; Matthew P. Young
year: 2023
title: "Vanishing of Quartic and Sextic Twists of L-functions"
doi: null
url: https://arxiv.org/abs/2301.05329v2
claim: "Corollary 3.23 supplies equidistribution of normalized Gauss-sum squares for totally sextic primitive Dirichlet characters; full-family sextic equidistribution is Conjecture 3.24. The mixed-family discussion does not supply the project's coupled Hecke estimate."
strata_touched: []
license: citation-only
triage: anchor

Composite-even Gauss sums and the totally sextic boundary

The source is arXiv:2301.05329v2, updated 23 January 2023. It studies quartic and sextic Dirichlet twists of elliptic-curve $L$-functions. Definition 3.1, Lemma 3.21, Corollary 3.23, display (3.42) and Conjecture 3.24 were inspected in the 31-page primary PDF. Its full proofs have not been independently audited or Lean compiled here. The statements below are attributed source results and conjectures.

Proved subfamily and conjectured full family

Definition 3.1 calls a primitive character totally order six when every prime-power conductor component has exact order six. Corollary 3.23, attributed to Patterson, states that

$$\frac{\tau (\chi {)}^2}{q},q=\mathrm{cond}(\chi ),$$

is equidistributed on the unit circle in this totally sextic subfamily. Section 3.1 also cites Patterson’s equidistribution of the normalized Gauss sum itself and twists by a fixed nonzero integer $k$. The twisted phase $\arg (\tau (\chi )\chi (k))$ is considered only where $(k,q)=1$; otherwise $\chi (k)=0$ and the phase is undefined. These statements do not specify uniformity for a growing twist or the project’s pair-dependent weights.

For all primitive sextic characters, equidistribution of the squares is Conjecture 3.24. Display (3.42) gives asymptotics for selected sixth-power moments on conductors $q\equiv 1(\mathrm{mod}4)$; those moments do not settle every nonzero Fourier mode required for full-family equidistribution. Thus the paper’s discussion of slower full-family convergence must not be cited as an unconditional full-family equidistribution theorem. Its elliptic-curve vanishing asymptotics in Conjecture 1.1 are also conjectural.

The source already supplies the sextic Gauss factorization

Under Lemma 3.21’s hypotheses, write $\chi ={\chi }_2{\chi }_3$, with quadratic ${\chi }_2$ and cubic ${\chi }_3$ both of conductor $q$, and take the cubic parameter $\beta$ from Lemma 3.4, so $\beta \overline{\beta }=q$. Equation (3.35), printed pp.21–22, is

$$\tau (\chi )=\mu (q){\chi }_3(2)\tau ({\chi }_2)\tau ({\chi }_3)\frac{\overline{\beta }}{q}.$$

This is an existing Gauss–Jacobi identity. In this source’s Dirichlet family, multiplying by $\mu (q)$ cancels its displayed conductor sign. It leaves a coupled quadratic Gauss factor, cubic Gauss factor and angular factor; it supplies no independence theorem for them.

Relation to the current contour gap

The project’s complementary kernel contains a moving quadratic twist coupled to a cubic Gauss factor. Its joint estimate must retain these factors on the same arithmetic realization. Replacing it by separate quadratic and cubic averages requires an additional correlation estimate; neither Corollary 3.23 nor Conjecture 3.24 supplies one.

The source’s conductor $q$ and $\mu (q)$ are rational-integer data. Their identification with the original Hecke ratio’s conductor ideal, its Möbius coefficient and fixed bad-prime conventions requires a separate bridge. In particular, a field or norm substitution cannot be inferred from the matching order six. The source does not bound the original signed kernel, establish the order-six residue identity needed for (J), provide detector length coverage, or imply Robin’s inequality or RH.