Mertens Sums requiring Fewer Values of the M\"{o}bius function

TL;DR

This paper develops identities involving the Möbius function and Mertens sums to efficiently compute these sums for large N, using spectral analysis and matrix methods, with potential applications in number theory.

Contribution

It introduces new identities for Möbius sums that reduce computational complexity and applies spectral and matrix analysis to understand their properties.

Findings

Identifies eigenvalues and eigenvectors of the matrix A related to Möbius sums.

Provides estimates for traces of A and A^2, aiding in sum approximation.

Discusses spectral decomposition and Perron's formula for sum estimation.

Abstract

We discuss certain identities involving $\mu(n)$ and $M(x)=\sum_{n\leq x}\mu(n)$, the functions of M\"{o}bius and Mertens. These identities allow calculation of $M(N^d)$, for $d=2,3,4,\ldots\ $, as a sum of $O_d \left( N^d(\log N)^{2d - 2}\right)$ terms, each a product of the form $\mu(n_1) \cdots \mu(n_r)$ with $r\leq d$ and $n_1,\ldots , n_r\leq N$. We prove a more general identity in which $M(N^d)$ is replaced by $M(g,K)=\sum_{n\leq K}\mu(n)g(n)$, where $g(n)$ is an arbitrary totally multiplicative function, while each $n_j$ has its own range of summation, $1,\ldots , N_j$. We focus on the case $d=2$, $K=N^2$, $N_1=N_2=N$, where the identity has the form $M(g,N^2) = 2 M(g,N) - {\bf m}^{\rm T} A {\bf m}$, with $A$ being the $N\times N$ matrix of elements $a_{mn}=\sum _{k \leq N^2 /(mn)}\,g(k)$, while ${\bf m}=(\mu (1)g(1),\ldots ,\mu (N)g(N))^{\rm T}$. Our results in Sections 2 and 3 assume, moreover, that $g(n)$ equals $1$ for all $n$. In this case the Perron-Frobenius theorem applies: we find that $A$ has an eigenvalue that is approximately $(\pi^2 /6)N^2$, with eigenvector approximately ${\bf f} = (1,1/2,1/3,\ldots ,1/N)^{\rm T}$, and that, for large $N$, the second-largest eigenvalue lies in $(-0.58 N, -0.49 N)$. Estimates for the traces of $A$ and $A^2$ are obtained. We discuss ways to approximate ${\bf m}^{\rm T} A {\bf m}$, using the spectral decomposition of $A$, or Perron's formula: the latter approach leads to a contour integral involving the Riemann zeta-function. We also discuss using the identity $A = N^{2\,} {\bf f}^{\,} \!{\bf f}^T - \textstyle{1\over 2} {\bf u} {\bf u}^T + Z$, where ${\bf u} = (1,\ldots ,1)^{\rm T}$ and $Z$ is the $N\times N$ matrix of elements $z_{mn} = - \psi(N^2 / (mn))$, with $\psi(x)=x - \lfloor x\rfloor - \textstyle{1\over 2}$.