no. 3
On eigenvalues of the kernel 1 2+⌊1 xy⌋-1 xy
Watt, Nigel1
1 45 Charles Way Limekilns, Fife, KY11 3LH, United Kingdom
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 653-662.

We show that the kernel K(x,y)=1 2+⌊1 xy⌋-1 xy (0<x,y≤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 ∑ m=1 N μ(m)∑ n=1 N μ(n)⁢K(m/N,n/N) in an identity involving the Mertens function.

Nous montrons que le noyau K(x,y)=1 2+⌊1 xy⌋-1 xy (0<x,y≤1) possède une infinité de valeurs propres positives et une infinité de valeurs propres négatives. Notre intérêt pour ce noyau est motivé par l’apparition de la forme quadratique ∑ m=1 N μ(m)∑ n=1 N μ(n)K(m/N,n/N) dans une identité pour la fonction de Mertens.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1099
Classification: 11A25, 45C05, 11A07
Keywords: Mertens function, eigenvalue, symmetric kernel
Watt, Nigel 1

1 45 Charles Way Limekilns, Fife, KY11 3LH, United Kingdom 
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Watt, Nigel. On eigenvalues of the kernel $\protect \frac{1}{2} +\protect \lfloor \protect \frac{1}{xy}\protect \rfloor - \protect \frac{1}{xy}$. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 653-662. doi : 10.5802/jtnb.1099. http://archive.numdam.org/articles/10.5802/jtnb.1099/

[1] Huxley, Martin N.; Watt, Nigel Mertens sums requiring fewer values of the Möbius function, Chebyshevskiĭ Sb., Volume 19 (2018) no. 3, pp. 20-34 | DOI

[2] Mertens, Franz Über eine zahlentheoretische Function, Wien. Ber., Volume 106 (1897), pp. 761-830 | Zbl

[3] Tricomi, Francesco G. Integral Equations, Dover Publications, 1985

[4] Watt, Nigel The kernel 1 2+⌊1 xy⌋-1 xy (0<x,y≤1) and Mertens sums (2018) (https://arxiv.org/abs/1812.01039)

[5] Weyl, Hermann Ueber die asymptotische Verteilung der Eigenwerte, Gött. Nachr., Volume 1911 (1911), pp. 110-117 | Zbl

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