[Paper Review] Formulas for Positive, Negative and Zero Values of the Möbius Function

R. M. Abrarov, Sanjar M. Abrarov|ArXiv.org|May 4, 2009

Mathematics and Applications6 references3 citations

TL;DR

This paper derives explicit formulas for counting positive, negative, and zero values of the Möbius function up to any real $x$, using recursive identities based on the Möbius function values at $\sqrt{x}$. It employs Dirac and Kronecker delta functions to express the Mertens function and related arithmetic functions in integral and discrete forms, revealing a structural mechanism for how these functions generate their outputs through number-theoretic identities.

ABSTRACT

We obtained the formulas for the quantities of positive, negative and zero values of the Mobius function for any real x in terms of the Mobius function values for square root of x - similar to the identities we found earlier for the Mertens function [1]. Using the Dirac delta function approach [3, 2] we propose the equations showing how the Mertens and related functions can generate the output values.

Motivation & Objective

• To derive exact formulas for $N_+(x)$, $N_-(x)$, and $N_0(x)$, the counts of natural numbers $\leq x$ with $\mu(n) = +1$, $-1$, and $0$ respectively.
• To express these counts recursively in terms of the Möbius function values at $\sqrt{x}$, extending the recursive structure of the Mertens function.
• To reformulate the Mertens function and related arithmetic functions using Dirac and Kronecker delta functions to eliminate explicit floor functions and reveal underlying generating mechanisms.
• To provide a deterministic, identity-based framework for analyzing the distribution of Möbius function values and their frequencies.

Proposed method

• Derives $N_0(x)$ via inclusion-exclusion over square factors, leading to $N_0(x) = -\sum_{i=2}^{\sqrt{x}} \mu_i \left[\frac{x}{i^2}\right]$.
• Uses the identity $M(x) = 2M(\sqrt{x}) - \sum_{i,j=1}^{\sqrt{x}} \mu_i \mu_j \left[\frac{x}{ij}\right]$ to express the Mertens function recursively.
• Applies the Dirac delta function $\delta^D$ and Kronecker delta $\delta^K$ to replace floor functions, expressing $\left[\frac{x}{ij}\right]$ as integrals or sums of delta functions.
• Transforms the Mertens function identity into a double sum over $k$ and $i,j$ using $\delta^K\left(\frac{k}{ij}\right)$, yielding $M(x) = 2 - \sum_{k=1}^x \sum_{i,j=1}^{\sqrt{k}} \mu_i \mu_j \delta^K\left(\frac{k}{ij}\right)$.
• Derives analogous delta-function representations for $Q(x) = N_+(x) + N_-(x)$, $N_+(x)$, and $N_-(x)$, using the same transformation technique.
• Establishes the identity $\mu_x = -\sum_{i,j=1}^{\sqrt{x}} \mu_i \mu_j \delta^K\left(\frac{x}{ij}\right)$ for $x \geq 2$, showing how individual Möbius values are generated.

Experimental results

Research questions

• RQ1 How can the counts of positive, negative, and zero values of the Möbius function be expressed recursively in terms of values at $\sqrt{x}$?
• RQ2 Can the Mertens function and related arithmetic functions be reformulated using Dirac or Kronecker delta functions to eliminate floor functions?
• RQ3 What structural mechanism underlies the generation of Möbius function values through summatory identities?
• RQ4 How do the frequencies of $\mu(n) = +1$, $-1$, and $0$ relate to the Mertens function and squarefree number counts?
• RQ5 What is the role of inclusion-exclusion and multiplicative identities in deriving exact formulas for $N_0(x)$ and $Q(x)$?

Key findings

• The number of integers $\leq x$ with $\mu(n) = 0$ is given by $N_0(x) = -\sum_{i=2}^{\sqrt{x}} \mu_i \left[\frac{x}{i^2}\right]$, derived via inclusion-exclusion over square factors.
• The count of squarefree numbers $Q(x) = N_+(x) + N_-(x)$ is expressed as $Q(x) = \sum_{i=1}^{\sqrt{x}} \mu_i \left[\frac{x}{i^2}\right]$, linking it directly to Möbius values at $\sqrt{x}$.
• The counts $N_+(x)$ and $N_-(x)$ are derived as $N_+(x) = M(\sqrt{x}) - \frac{1}{2}\sum_{i,j=1}^{\sqrt{x}} \mu_i \mu_j \left[\frac{x}{ij}\right] + \frac{1}{2}Q(x)$ and similarly for $N_-(x)$, using the Mertens function and $Q(x)$.
• Using delta functions, the paper derives $M(x) = 2 - \sum_{k=1}^x \sum_{i,j=1}^{\sqrt{k}} \mu_i \mu_j \delta^K\left(\frac{k}{ij}\right)$, replacing floor functions with discrete delta functions.
• The identity $\mu_x = -\sum_{i,j=1}^{\sqrt{x}} \mu_i \mu_j \delta^K\left(\frac{x}{ij}\right)$ for $x \geq 2$ shows how individual Möbius values are generated by the recursive structure.
• The delta-function formulation reveals a deterministic, constructive mechanism for generating the values of the Möbius function and its summatory functions through number-theoretic identities.