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

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,eachaproductoftheform$\mu(n_1) \cdots \mu(n_r)$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 $M(g,K)={\sum }_{n\le K}\mu (n)g(n)$, 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 $M(g,N^2) = 2 M(g,N) - {\bf m}^{\rm T} A {\bf m}$,with$A$beingthe$N\times N$matrixofelements$a_{mn}=\sum {k \leq N^2 /(mn)},g(k)$,while${\bf m}=(\mu (1)g(1),\ldots ,\mu (N)g(N))^{\rm T}$\mathrm{.}OurresultsinSections2and3assume,moreover,that$g(n)$equals$1$forall$n$. In this case the Perron-Frobenius theorem applies: we find that $A$ has an eigenvalue that is approximately $(\pi^2 /6)N^2$,witheigenvectorapproximately${\bf f} = (1,1/2,1/3,\ldots ,1/N)^{\rm T}$,andthat,forlarge$N$,thesecond-largesteigenvalueliesin$(-0.58 N, -0.49 N)$\mathrm{.}Estimatesforthetracesof$A$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 $A = N^{2,} {\bf f}^{,} !{\bf f}^T - \textstyle{1\over 2} {\bf u} {\bf u}^T + Z$,where${\bf u} = (1,\ldots ,1)^{\rm T}$and$Z$isthe$N\times N$matrixofelements$z{mn} = - \psi(N^2 / (mn))$,with$\psi(x)=x - \lfloor x\rfloor - \textstyle{1\over 2}$.