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Joint distribution in residue classes of families of polynomially-defined additive functions

Akash Singha Roy

TL;DR

The paper develops a framework for joint equidistribution of families of additive functions g_i defined by polynomial values on primes, extending Delange’s criteria to moduli q that vary with x. Under a near-optimal linear-independence condition on the defining polynomials' derivatives, it proves joint equidistribution modulo q uniformly for q up to a small power of log x, within a q-set determined by the polynomials. It then strengthens these results to complete uniformity for general moduli and squarefree moduli, while detailing optimality limits for input restrictions and highlighting the necessity of the linear-independence hypothesis. The work employs a combination of Delange-type reductions, a main-term-anatomy decomposition into convenient inputs, and exponential-sum bounds (Weil/Cochrane–Zheng) to control auxiliary counting sets, culminating in Siegel–Walfisz-type theorems for values of additive functions instead of primes. These results broaden the scope of uniform distributions of additive functions and connect to prior single-function and fixed-modulus work, with implications for understanding the joint distribution in polynomially-defined arithmetic settings.

Abstract

Let $g_1, \dots , g_M$ be additive functions for which there exist nonconstant polynomials $G_1, \dots , G_M$ satisfying $g_i(p) = G_i(p)$ for all primes $p$ and all $i \in \{1, \dots , M\}$. Under fairly general and nearly optimal hypotheses, we show that the functions $g_1, \dots , g_M$ are jointly equidistributed among the residue classes to moduli $q$ varying uniformly up to a fixed but arbitrary power of $\log x$. Thus, we obtain analogues of the Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes replaced by values of such additive functions. Our results partially extend work of Delange from fixed moduli to varying moduli, and also generalize recent work done for a single additive function.

Joint distribution in residue classes of families of polynomially-defined additive functions

TL;DR

The paper develops a framework for joint equidistribution of families of additive functions g_i defined by polynomial values on primes, extending Delange’s criteria to moduli q that vary with x. Under a near-optimal linear-independence condition on the defining polynomials' derivatives, it proves joint equidistribution modulo q uniformly for q up to a small power of log x, within a q-set determined by the polynomials. It then strengthens these results to complete uniformity for general moduli and squarefree moduli, while detailing optimality limits for input restrictions and highlighting the necessity of the linear-independence hypothesis. The work employs a combination of Delange-type reductions, a main-term-anatomy decomposition into convenient inputs, and exponential-sum bounds (Weil/Cochrane–Zheng) to control auxiliary counting sets, culminating in Siegel–Walfisz-type theorems for values of additive functions instead of primes. These results broaden the scope of uniform distributions of additive functions and connect to prior single-function and fixed-modulus work, with implications for understanding the joint distribution in polynomially-defined arithmetic settings.

Abstract

Let be additive functions for which there exist nonconstant polynomials satisfying for all primes and all . Under fairly general and nearly optimal hypotheses, we show that the functions are jointly equidistributed among the residue classes to moduli varying uniformly up to a fixed but arbitrary power of . Thus, we obtain analogues of the Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes replaced by values of such additive functions. Our results partially extend work of Delange from fixed moduli to varying moduli, and also generalize recent work done for a single additive function.
Paper Structure (9 sections, 13 theorems, 114 equations)

This paper contains 9 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminary Discussion: Delange's equidistribution criteria and consequences for polynomially-defined additive functions
  3. Preparation for Theorems \ref{['thm:UnrestrictedInput']}, \ref{['thm:restrictediput_generalq']} and \ref{['thm:restrictediput_squarefreeq']}: Obtaining the main term
  4. Joint equidistribution without input restriction: Proof of Theorem \ref{['thm:UnrestrictedInput']}
  5. Optimality of range of $q$ in Theorem \ref{['thm:UnrestrictedInput']}
  6. Complete uniformity for general moduli: Proof of Theorem \ref{['thm:restrictediput_generalq']}
  7. Complete uniformity in squarefree moduli: Proof of Theorem \ref{['thm:restrictediput_squarefreeq']}
  8. Optimality in the input restrictions in Theorem \ref{['thm:restrictediput_squarefreeq']}:
  9. Necessity of the linear independence hypothesis: Proof of Theorem \ref{['thm:LIHNecess']}

Key Result

Theorem 1.1

Fix $K \ge 1$, $\delta \in (0, 1]$ and an integer $M \ge 1$. Let $g_1, \dots , g_M$ be additive functions defined by the polynomials $G_1, \dots , G_M$ such that the polynomials $\{G_i'\}_{1 \le i \le M} \subset \mathbb{Z}[T]$ are $\mathbb{Q}$-linearly independent. Then $g_1, \dots , g_M$ are jointl

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more