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The Landau-Selberg-Delange method for products of Dirichlet $L$-functions, and applications, I

Akash Singha Roy

TL;DR

This work extends the Landau-Selberg-Delange framework to products of Dirichlet $L$-functions, delivering sharp, uniform partial-sum asymptotics for $\sum_{n\le x} a_n$ in regimes where the modulus $q$ varies widely and where Siegel zeros may occur. The core tools are an inner-contour shift, averaged-growth hypotheses, and a precise control of the logarithmic derivatives of $\mathcal F(s\nu)$, enabling effective outer-contour analysis even in the presence of multiple potential singularities. The paper then applies this LSD machinery to a suite of distribution problems: integers with restricted prime factors in progressions, and the invariant- and primary-factor structures of multiplicative groups, as well as Sathe–Selberg-type local laws in arithmetic progressions for $\omega_a(n)$ and $\Omega_a(n)$ with moduli varying up to Siegel-Walfisz ranges and beyond under conjectural hypotheses. Conditional results under Siegel-zero scenarios extend the valid $q$-ranges, while unconditional statements recover and improve classical outcomes (Landau, Chang–Martin, Martin–Nguyen) in a unified analytic-automorphic framework. The techniques yield explicit main-terms and error bounds with clear dependence on $q$ via parameters like $\lambda_q$ and $B(q)$, offering a versatile toolkit for uniform distribution questions in arithmetic settings.

Abstract

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations often arise when the Dirichlet series behaves like a product of complex powers of several Dirichlet $L$-functions to a modulus $q$. In such situations, one often requires sharp asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ that apply in much wider ranges of $q$ than permitted by known forms of the Landau-Selberg-Delange method. In this manuscript, we address this problem, giving new estimates on $\sum_{n \le x} \, a_n$ in ranges of $q$ that are (in most applications) much wider than attainable from previous results. Our results also weaken certain hypotheses on the size of $\{a_n\}_n$. As applications of our main theorems, we extend Landau's classical results on the distribution of integers with prime factors restricted to progressions, and improve upon Chang, Martin and Nguyen's work on the distributions of the least invariant factors and least primary factors of multiplicative groups. We also extend the classical Sathe-Selberg theorem and study the local laws of the functions $Ω_a(n)$ and $ω_a(n)$, that count (with and without multiplicity, respectively), the number of prime divisors of $n$ lying in the progression $a$ mod $q$.

The Landau-Selberg-Delange method for products of Dirichlet $L$-functions, and applications, I

TL;DR

This work extends the Landau-Selberg-Delange framework to products of Dirichlet -functions, delivering sharp, uniform partial-sum asymptotics for in regimes where the modulus varies widely and where Siegel zeros may occur. The core tools are an inner-contour shift, averaged-growth hypotheses, and a precise control of the logarithmic derivatives of , enabling effective outer-contour analysis even in the presence of multiple potential singularities. The paper then applies this LSD machinery to a suite of distribution problems: integers with restricted prime factors in progressions, and the invariant- and primary-factor structures of multiplicative groups, as well as Sathe–Selberg-type local laws in arithmetic progressions for and with moduli varying up to Siegel-Walfisz ranges and beyond under conjectural hypotheses. Conditional results under Siegel-zero scenarios extend the valid -ranges, while unconditional statements recover and improve classical outcomes (Landau, Chang–Martin, Martin–Nguyen) in a unified analytic-automorphic framework. The techniques yield explicit main-terms and error bounds with clear dependence on via parameters like and , offering a versatile toolkit for uniform distribution questions in arithmetic settings.

Abstract

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums of a Dirichlet series that behaves like a complex power of the Riemann zeta function. However, situations often arise when the Dirichlet series behaves like a product of complex powers of several Dirichlet -functions to a modulus . In such situations, one often requires sharp asymptotic formulas for the partial sums that apply in much wider ranges of than permitted by known forms of the Landau-Selberg-Delange method. In this manuscript, we address this problem, giving new estimates on in ranges of that are (in most applications) much wider than attainable from previous results. Our results also weaken certain hypotheses on the size of . As applications of our main theorems, we extend Landau's classical results on the distribution of integers with prime factors restricted to progressions, and improve upon Chang, Martin and Nguyen's work on the distributions of the least invariant factors and least primary factors of multiplicative groups. We also extend the classical Sathe-Selberg theorem and study the local laws of the functions and , that count (with and without multiplicity, respectively), the number of prime divisors of lying in the progression mod .

Paper Structure

This paper contains 21 sections, 23 theorems, 155 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The main results: General statements
  3. The distribution of integers with prime factors restricted to progressions
  4. Distributions of the least invariant factors of multiplicative groups
  5. Distributions of the least primary factors of multiplicative groups
  6. The Sathe--Selberg Theorem in arithmetic progressions
  7. Key analytic inputs: Logarithmic Derivatives, Auxiliary Functions, and the "Inner Contour Shift"
  8. Analytic Continuations
  9. Analysis of Logarithmic Derivatives
  10. Bounds on ${\mathcal{H}}(s)$ and $\mathcal{F}(s\nu)$.
  11. The LSD method for $L$--functions under average growth conditions: Proofs of Theorem \ref{['thm:LFuncLSDVariant1']} and Corollary \ref{['cor:LFuncLSDVariant1LogPowerInterval']}
  12. Perron's Formula: Error--term control via averaging
  13. When $\boldsymbol{1-c_0/10\lambda_q\log q < \eta_e < 1-3\nu/\log x}$: Proof of Theorem \ref{['thm:LFuncLSDVariant1Gen']}(1)
  14. When $\boldsymbol{\eta_e}$ does not exist or $\boldsymbol{\eta_e \le 1-c_0/10\lambda_q\log q}$: Proof of Theorem \ref{['thm:LFuncLSDVariant1Gen']}(2)
  15. When the stronger growth condition \ref{['eq:Variant1GrowthConditionSpecial']} is available
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $\{a_n\}_n$ has property $\mathcal{P}(\nu, \{\alpha_\chi\}_\chi; c_0, \Omega)$, and that for all $x>1$, we have Fix any $K_0>0$. The following hold uniformly in all $x \ge q \ge e^{4+5/3\nu}$, $h \in (0, x/2]$, $N \in \mathbb{Z}_{\ge 0}$, and in $\{\alpha_\chi\}_\chi \subset \mathbb C$ with $\max\{|\alpha_{\chi_0}|, |\alpha_{\chi_e}|\} \le K_0$; the implied constants depend only on $c

Figures (4)

  • Figure 1: The Contour $\omega_M$
  • Figure 2: Contour $\boldsymbol{\Gamma_0}$ for Theorem \ref{['thm:LFuncLSDVariant1Gen']}(1), i.e. when ${\boldsymbol{1-\frac{c_0}{10\lambda_q\log q} < \eta_e < 1-\frac{3\nu}{\log x}}}$.
  • Figure 3: The truncated Hankel contour $\mathcal{W}$ after the substitution ${w=\left(s-\frac{1}{\nu}\right)\log X}$.
  • Figure 4: Contour $\boldsymbol{\Gamma_0}$ when $\boldsymbol{{\eta_e \le 1-\frac{c_0}{10\lambda_q\log q}}}$. Here ${\boldsymbol{\sigma_\nu(t) = \frac{1}{\nu}\left(1-\frac{c_0}{80\lambda_q\,\mathcal{L}_q(t)}\right)}}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 24 more