The Kernel ${1\over 2} +\lfloor {1\over xy}\rfloor  {1\over xy}\,$ ($0<x,y\leq 1$) and Mertens Sums
Abstract
Let $K(x,y) = {1\over 2} + \lfloor {1\over x y}\rfloor  {1\over x y}\,$ ($0<x,y\leq 1$). A nontrivial identity connects the sum $S(N):=\sum_{m=1}^N \sum_{n=1}^N K\left({m\over N}, {n\over N}\right) \mu(m) \mu(n)$ (where $\mu(n)$ is the Möbius function) with the Mertens sum $\mu(1) + \mu(2) +\cdots + \mu(N^2)$. Motivated by this, we consider two families of degenerate integral kernels that are stepfunction approximations to $K$. We obtain new results concerning the distribution of the relevant reciprocal eigenvalues: those of the approximations to $K$, and those of $K$ itself. The proofs utilise both the HilbertSchmidt theory of linear integral equations and Weyl's inequalities for reciprocal eigenvalues of kernels $U,V,W$ satisfying $U=V+W$. We show also that $S(N)$ is reasonably well approximated by sums involving just $\mu(1),\ldots,\mu(N)$, a limited number of reciprocal eigenvalues of $K$ and the values that the corresponding eigenfunctions have at the points $x={1\over N},{2\over N},\ldots,{N\over N}$. Machine computations have informed much of our work: we discuss some numerical results concerning the least and the greatest of the reciprocal eigenvalues of $K$.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.01039
 Bibcode:
 2018arXiv181201039W
 Keywords:

 Mathematics  Number Theory;
 11A25 (Primary) 47B06;
 47B35;
 4704;
 45C05;
 454P05;
 45H05 (Secondary)
 EPrint:
 55 Pages, Plain TeX, LaTeX version submitted to Experimental Mathematics