On eigenvalues of the kernel $\frac{1}{2}+\lfloor \frac{1}{xy}\rfloor -\frac{1}{xy}$, II

Abstract

We study the eigenvalues ${\lambda }_1,{\lambda }_2,{\lambda }_3,\dots$ (ordered by modulus) of the integral kernel $K(x,y):=\frac{1}{2}+\lfloor \frac{1}{xy}\rfloor -\frac{1}{xy}$ ($0<x,y\le 1$). This kernel is of interest in connection with an identity of F. Mertens involving the Möbius function. We establish that ${\sum }_{m=1}^{\infty }|{\lambda }_m{|}^{-1}=\infty$, and prove that $|{\lambda }_m|>m{\log }^{-3/2}m$ for all but finitely many positive integers $m$. The first of these results is an application of the theory of Hankel operators; the proof of the second result utilises a family of degenerate kernels $k_3,k_4,k_5,\dots$ that are step-function approximations to $K$. Through separate computational work on eigenvalues of $k_N$ ($N=2^{21}$) we obtain numerical bounds, both upper and lower, for specific eigenvalues of $K$. Further computational work, on eigenvalues of $k_N$ ($N\in \{2^{10},2^{11},\dots ,2^{21}\}$), leads us to formulate a quite precise conjecture concerning where on the real line the eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{767}$ are located: we discuss how this conjecture could (if it is correct) be viewed as supportive of certain interesting general conjectures concerning the eigenvalues of $K$.

arXiv commentv1 rejected by Exp. Math.; v2 has much less on the Mertens function (hence the change of title), but more on numerical work and applying Hankel operator theory; Theorems 1,2,3,5 and 7 of v1 have been omitted; Theorems 4 and 6 of v1 have been replaced with slightly weaker results (Proposition 1.2, Theorem 1.1), so proofs could be shortened; 88 Pages (including 5 appendices), 4 Figures; PDFLaTeX