• Corpus ID: 119180081

# On eigenvalues of the kernel ${1\over 2}+\lfloor {1\over xy}\rfloor - {1\over xy}$ ($0 @article{Watt2018OnEO, title={On eigenvalues of the kernel \$\{1\over 2\}+\lfloor \{1\over xy\}\rfloor - \{1\over xy\}\$(\$0},
author={Nigel Watt},
journal={arXiv: Number Theory},
year={2018}
}
• N. Watt
• Published 13 November 2018
• Mathematics
• arXiv: Number Theory
We show that the kernel $K(x,y)={1\over 2}+\lfloor {1\over xy}\rfloor -{1\over xy}$ ($0<x,y\leq 1$) has infinitely many positive eigenvalues and infinitely many negative eigenvalues. Our interest in this kernel is motivated by the appearance of the quadratic form $\sum_{m,n\leq N} K\bigl( {m\over N} , {n\over N}\bigr) \mu(m)\mu(n)$ in an indentity involving the Mertens function.
1 Citations

### On eigenfunctions of the kernel $\frac{1}{2} + \lfloor \frac{1}{xy} \rfloor - \frac{1}{xy}$

The integral kernel $K(x,y) := \frac{1}{2} + \lfloor \frac{1}{xy} \rfloor - \frac{1}{xy}$ ($0<x,y\leq 1$) has connections with the Riemann zeta-function and a (recently observed) connection with the

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