Mertens Sums requiring Fewer Values of the Möbius function

Abstract

We discuss certain identities involving $\mu (n)$ and $M(x)={\sum }_{n\le x}\mu (n)$, the functions of Möbius and Mertens. These identities allow calculation of $M(N^d)$, for $d=2,3,4,\dots$, 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 $r\le d$ and $n_1,\dots ,n_r\le N$. We prove a more general identity in which $M(N^d)$ is replaced by $M(g,K)={\sum }_{n\le K}\mu (n)g(n)$, where $g(n)$ is an arbitrary totally multiplicative function, while each $n_j$ has its own range of summation, $1,\dots ,N_j$. We focus on the case $d=2$, $K=N^2$, $N_1=N_2=N$, where the identity has the form $M(g,N^2)=2M(g,N)-{\mathbf{m}}^{\mathrm{T}}A\mathbf{m}$, with $A$ being the $N\times N$ matrix of elements $a_{mn}={\sum }_{k\le N^2/(mn)}g(k)$, while $\mathbf{m}=(\mu (1)g(1),\dots ,\mu (N)g(N){)}^{\mathrm{T}}$. 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 $\mathbf{f}=(1,1/2,1/3,\dots ,1/N{)}^{\mathrm{T}}$, and that, for large $N$, the second-largest eigenvalue lies in $(-0.58N,-0.49N)$. Estimates for the traces of $A$ and $A^2$ are obtained. We discuss ways to approximate ${\mathbf{m}}^{\mathrm{T}}A\mathbf{m}$, 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 $A=N^2{\mathbf{f}}^{}{\mathbf{f}}^T-\frac{1}{2}\mathbf{u}{\mathbf{u}}^T+Z$, where $\mathbf{u}=(1,\dots ,1{)}^{\mathrm{T}}$ and $Z$ is the $N\times N$ matrix of elements $z_{mn}=-\psi (N^2/(mn))$, with $\psi (x)=x-\lfloor x\rfloor -\frac{1}{2}$.

14 Pages, Plain TeX, submitted to Chebyshevskiĭ Sb