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- Title
Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach.
- Authors
Darbar, Pranendu; Mukhopadhyay, Anirban
- Abstract
Let Mn$\mathcal {M}_n$ denote the set of monic polynomials of degree n over a finite field Fq$\mathbb {F}_q$ of q elements. For multiplicative functions ψ1,ψ2$\psi _1,\psi _2$, using the recently developed "pretentious method," we establish a "local‐global" principle for correlation functions of the form ∑f∈Mnψ1(f+h1)ψ2(f+h2)asn→∞(qremains fixed),$$\begin{equation*} \sum _{f\in \mathcal {M}_n}\psi _1(f+h_1)\psi _2(f+h_2) \quad \text{ as } n\rightarrow \infty \quad (q \text{ remains fixed}), \end{equation*}$$where h1,h2∈Fq[x]$h_1,h_2\in \mathbb {F}_q[x]$ are fixed polynomials. These results were then applied to find limiting distributions of sums of additive functions. We also study correlations with Dirichlet characters and Hayes characters. As a consequence, we give a new proof of a function field analog of Kátai's conjecture that states that if the average of the first divided difference of a completely multiplicative function whose values lie on the unit circle is zero, then it must be a Hayes character. We further extend this result to pairs and triplets of completely multiplicative functions.
- Subjects
STATISTICAL correlation; ADDITIVE functions; FINITE fields; CIRCLE; POLYNOMIALS
- Publication
Mathematika, 2024, Vol 70, Issue 1, p1
- ISSN
0025-5793
- Publication type
Article
- DOI
10.1112/mtk.12227