• 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

References

SHOWING 1-10 OF 17 REFERENCES

Mertens Sums requiring Fewer Values of the M\"{o}bius function.

We discuss certain identities involving $\mu(n)$ and $M(x)=\sum_{n\leq x}\mu(n)$, the functions of M\"{o}bius and Mertens. These identities allow calculation of $M(N^d)$, for $d=2,3,4,\ldots\ $, as a

HANKEL OPERATORS OF CLASS $ \mathfrak{S}_p$ AND THEIR APPLICATIONS (RATIONAL APPROXIMATION, GAUSSIAN PROCESSES, THE PROBLEM OF MAJORIZING OPERATORS)

A criterion is given for a Hankel operator , where is the orthogonal projection of onto ) to belong to the Schatten-von Neumann class in terms of its symbol . Various applications are considered: a

Integral Equations, Dover Publications

  • New York
  • 1985

Representation theorems for holomorphic and harmonic functions in L[P] . The molecular characterization of certain Hardy spaces

© Société mathématique de France, 1980, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les

The macmillan company.