A Novel Generalization of the Liouville Function $λ(n)$ and a Convergence Result for the Associated Dirichlet Series

TL;DR

The paper introduces a novel arithmetic function $w(n)$, a generalization of the Liouville function, defined via an Euler-product Dirichlet series whose coefficients encode the distribution of prime factors. It proves that the associated Dirichlet series $F(s)$ converges for $\Re(s)$ in $(\tfrac12,1)$ and that $w(n)$ yields a dense distribution on the unit circle, supported by a density argument and Hughes' Corollary. It then extends to a parametrized family $w_m(n)$ with $F_m(s)$ converging uniformly in $m$ to $W(s)=\sum_n \lambda(n)/n^s=\zeta(2s)/\zeta(s)$, implying the same strip of convergence for the classical Dirichlet series and a nonvanishing result for the zeta function in that region. Together, these results connect intrinsic prime-distribution encoding with classical zeta-function behavior, offering a new perspective on Dirichlet-series convergence and prime-factor structure.

Abstract

We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $λ(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors among natural numbers, $w(n)$ allows results to be obtained which rely intrinsically on the distribution of primes without having direct knowledge of that distribution. We prove some properties of the distribution of $w(n)$ and then provide a result on the convergence of its Dirichlet series. A parametrized family of functions $w_m(n)$ is defined of which $w(n)$ is a special case. We show that each function $w_m(n)$ injectively maps $\mathbb{N}$ into a dense subset of the unit circle in $\mathbb{C}$ and that each $F_m(s) = \sum_n \frac{w_m(n)}{n^s}$ converges for all $s$ with $\Re(s)\in\left(\frac{1}{2},1\right)$. Finally, we show that the family of functions $w_m(n)$ converges to $λ(n)$ and that $F_m(s)$ converges uniformly in $m$ to $\sum_n \frac{λ(n)}{n^s}$, implying convergence of that series in the same region and thereby proving an interesting property about a closely related function.

Figures (1)

• Figure 1: Flowchart for Selection Rounds

Theorems & Definitions (20)

• Lemma 1
• proof
• Theorem 1: Density Theorem
• proof : Sketch of proof
• Corollary 1: Hughes' Corollary
• proof
• Corollary 2
• proof
• Theorem 2: Convergence of $F_N(s)$ for $\Re(s)\in\left(\frac{1}{2},1\right)$
• proof