Mertens sums requiring fewer values of the M\"obius function

M. Huxley ; N. Watt

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 20-34.

Résumé

We discuss certain identities involving $\mu(n)$ and $M(x) = \sum _{n \leq x}\mu (n)$, the functions of Möbius and Mertens. These allow calculation of $M(N^d)$, for $d=1,2,3,\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, $1,\ldots , N_j$. This is not new, except perhaps in that $N_1,\ldots , N_d$ are arbitrary, but our proof (inspired by an identity of E. Meissel, 1854) is new. We are mainly interested in the case $d=2$, $K=N^2$, $N_1=N_2=N$, where the identity has the form $M(g, N^2) = 2 M(g,N) - \mathbf{m}^{\mathrm{T}} A \mathbf{ m}$, with $A$ being the $N\times N$ matrix of elements $a_{mn}=\sum _{k \leq N^2 /(mn)}\,g(k)$, while $\mathbf{ m}=(\mu (1)g(1),\ldots ,\mu (N)g(N))^{\mathrm{T}}$. Our results in Sections 2 and 3 of the paper assume that $g(n)$ equals $1$ for all $n$. The Perron-Frobenius theorem applies in this case: we find that $A$ has one large positive eigenvalue, approximately $(\pi^2 /6)N^2$, with eigenvector approximately $\mathbf{f} = (1,1/2,1/3,\ldots ,1/N)^{\mathrm{T}}$, and that, for large $N$, the second-largest eigenvalue lies in $(-0.58 N, -0.49 N)$. Section 2 includes estimates for the traces of $A$ and $A^2$ (though, for $\mathrm{Tr}(A^2)$, we omit part of the proof). In Section 3 we discuss ways to approximate $\mathbf{ m}^{\mathrm{T}} A \mathbf{ m}$, using the spectral decomposition of $A$, or (alternatively) 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 $\mathbf{u} = (1,\ldots ,1)^{\mathrm{T}}$ and $Z$ is the $N\times N$ matrix of elements $z_{mn} = - \psi(N^2 / (mn))$, with $\psi(x)=x - \lfloor x\rfloor - \frac{1}{2}$.

Détail

Keywords: Möbius function, Mertens function, completely multiplicative function, Meissel, Linnik's identity, Vaughan's identity, symmetric matrix, Perron-Frobenius, eigenvalue, eigenvector, Perron's formula, Riemann zeta-function.

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