Let q be a prime power, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> the finite field with q elements, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f \in \mathbb {F}_q[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> a monic polynomial. Write <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega (f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the number of monic irreducible factors of f, counted with multiplicity. We obtain asymptotic estimates for the distribution of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$z^{\Omega (f)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>z</mi> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> by developing a function-field analogue of the Selberg–Delange method, then use it to count monic polynomials of a given degree and a given value of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. This approach sharpens and extends earlier results in the literature concerning this function.

On the Selberg-Delange method in function fields

期刊论文
发表日期：

2026年7月6日

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Sourabhashis Das,
Wentang Kuo,
Yu-Ru Liu,
Owen Sharpe

摘要

Let q be a prime power, $\mathbb {F}_q$ $F_{q}$ the finite field with q elements, and $f \in \mathbb {F}_q[t]$ $f \in F_{q} [t]$ a monic polynomial. Write $\Omega (f)$ $Ω (f)$ for the number of monic irreducible factors of f, counted with multiplicity. We obtain asymptotic estimates for the distribution of $z^{\Omega (f)}$ $z^{Ω (f)}$ by developing a function-field analogue of the Selberg–Delange method, then use it to count monic polynomials of a given degree and a given value of $\Omega $ $Ω$ . This approach sharpens and extends earlier results in the literature concerning this function.