The variance and correlations of the divisor function in \(\mathbb{F}_q[T]\), and Hankel matrices. (English) Zbl 07809835
For \(q\geqslant 2\) the classical \(k\)-th divisor function is defined by equality
\[
d_k(n)=\#\big\{(a_1,\ldots,a_k)\in\mathbb{N}^k: a_1\ldots a_k=n\big\}, n\in\mathbb{N}.
\]
In the polynomial case
\[
d_k(N)=\#\big\{(A_1,\ldots,A_k)\in\mathcal{M}^k: A_1\ldots A_k=N\big\}, N\in\mathcal{M},
\]
where \(\mathcal{M}\) is the set of monic polynomials in the polynomial ring \(\mathbb{F}_q\) with a prime power \(q\).
The properties of the functions \(d_k:\mathcal{M}\rightarrow \mathbb{N}\) and related quantities are considered in this paper. For instance, the author gets the exact expression for the correlation \[ \frac{1}{q^{n+h}}\sum_{A\in\mathcal{M}_n}\sum_{B\in\mathcal{A}_{<h}} d(A)d(A+B), \] where \(d=d_2\), \(h\leqslant n\), \(\mathcal{M}_n\) is the set of monic polynomials with degree equal \(n\), and \(\mathcal{A}_{<h}\) is the set of polynomials from \(\mathbb{F}_q\) with degree \(<h\).
The properties of the functions \(d_k:\mathcal{M}\rightarrow \mathbb{N}\) and related quantities are considered in this paper. For instance, the author gets the exact expression for the correlation \[ \frac{1}{q^{n+h}}\sum_{A\in\mathcal{M}_n}\sum_{B\in\mathcal{A}_{<h}} d(A)d(A+B), \] where \(d=d_2\), \(h\leqslant n\), \(\mathcal{M}_n\) is the set of monic polynomials with degree equal \(n\), and \(\mathcal{A}_{<h}\) is the set of polynomials from \(\mathbb{F}_q\) with degree \(<h\).
Reviewer: Jonas Šiaulys (Vilnius)
MSC:
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
| 11N64 | Other results on the distribution of values or the characterization of arithmetic functions |
| 11T24 | Other character sums and Gauss sums |
| 11T55 | Arithmetic theory of polynomial rings over finite fields |
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
Keywords:
divisor function; variance; correlations; Hankel matrices; additive characters; function fieldsReferences:
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