• Corpus ID: 119180081

On eigenvalues of the kernel ${1\over 2}+\lfloor {1\over xy}\rfloor - {1\over xy}$ ($0

  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},
  • 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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