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Chebyshevskii Sbornik, 2020, Volume 21, Issue 3, Pages 196–214
DOI: https://doi.org/10.22405/2226-8383-2018-21-3-196-214
(Mi cheb935)
 

On the mean value of functions related to the divisors function in the ring of polynomials over a finite field

V. V. Iudelevich

Lomonosov Moscow State University (Moscow)
References:
Abstract: Let $ g: \mathbb{F}_q[T] \rightarrow \mathbb{R} $ be a multiplicative function which values at the degrees of the irreducible polynomial, depends only on the exponent, such that $g(P^k)=d_k$ polynomial $P$ and for some arbitrary sequence of reals $\{d_k\}_{k=1}^{\infty}$. This paper regards the sum
$$ T (N) = \sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}{g (F)}, $$
where $ F $ ranges over polynomials of degree $ N $ with leading coefficient equal to 1 (unitary polynomials). For the sum $ T (N) $, an exact formula is found, and various asymptotics are calculated in cases of $ q \to \infty; \ q \to \infty, \ N \to \infty; \ q ^ N \to \infty $. In particular, the following asymptotic formulas are obtained
$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\tau(F^k)=\binom{k+N}{N}q^N+O_{N,k}\left(q^{N-1}\right),\ \ N\ge 1,\ q\to\infty; $$

$$ \sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=\dfrac{q^N}{4^N}\left(\binom{2N}{N}-\dfrac{2}{3}\binom{2N-4}{N-2}q^{-1}+O\left(\ \dfrac{4^N}{\sqrt{N}}q^{-2}\right)\right),\ N\to\infty,\ q\to\infty; $$

$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=C_1\cdot\dfrac{\binom{2N}{N}}{4^N}q^N+O\left(\dfrac{q^{N-0.5}}{N^{1.5}}\right),\ \ C_1=\prod_{l=1}^{+\infty}\left(\sqrt{q^{2l}-q^{l}}\ln\dfrac{q^l}{q^l-1}\right)^{\pi_q(l)},\ q^N\to\infty;$$
where $\tau(F)$ is a number of monic divisors of $F$, and $\pi_q(l)$ is a number of monic irreducible polynomials of degree $l$. The second and third equalities are analogous for polynomials over a finite field of one of Ramanujan's results
$$\sum_{n\leq x}{\dfrac{1}{d(n)}}=\dfrac{x}{\sqrt{\ln x}}\left(a_0+\dfrac{a_1}{\ln{x}}+\ldots+\dfrac{a_N}{(\ln{x})^N}+O_N\left(\dfrac{1}{(\ln{x})^{N+1}}\right)\right),$$
where $d(n)$ is a classical divisor function, and $a_i$ are some constants. In particular,
$$a_0=\dfrac{1}{\sqrt{\pi}}\prod\limits_{p}\ln\dfrac{p}{p-1}\sqrt{p(p-1)}.$$
Keywords: the ring of polynomials over a finite field, divisor function.
Received: 03.02.2020
Accepted: 22.10.2020
Document Type: Article
UDC: 511.321
Language: Russian
Citation: V. V. Iudelevich, “On the mean value of functions related to the divisors function in the ring of polynomials over a finite field”, Chebyshevskii Sb., 21:3 (2020), 196–214
Citation in format AMSBIB
\Bibitem{Iud20}
\by V.~V.~Iudelevich
\paper On the mean value of functions related to the divisors function in the ring of polynomials over a finite field
\jour Chebyshevskii Sb.
\yr 2020
\vol 21
\issue 3
\pages 196--214
\mathnet{http://mi.mathnet.ru/cheb935}
\crossref{https://doi.org/10.22405/2226-8383-2018-21-3-196-214}
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