Watt, Nigel On eigenvalues of the kernel \(\frac{1}{2} +\lfloor \frac{1}{xy} \rfloor -\frac{1}{xy}\). (English. French summary) Zbl 1454.11015 J. Théor. Nombres Bordx. 31, No. 3, 653-662 (2019). For \( 0< x, y\le 1 \), define the kernel \[ K(x,y)=\dfrac{1}{2}-\left\{\dfrac{1}{xy}\right\}, \] where \( \{t\} \in [0,1) \) denotes the fractional part of \( t\in \mathbb{R} \). It is well-known that every such a kernel has at least one eigenvalue \( \lambda \). That is, there exists a number \( \lambda \ne 0 \), and an associated eigenfunction \(\phi(x)\), with \( 0< \int_0^1 |\phi(x)|^{2}\,dx < \infty\), satisfying \[ \phi(x)=\lambda \int_{0}^{1}K(x,y)\phi(y) \,dy \] almost everywhere, with respect to Lebesgue measure, in \( [0,1] \). Since, the kernel \( K \) is symmetric, that is, \( K(x,y)=K(y,x) \), all eigenvalues of \( K \) are real and so there is no essential loss of generality in considering those eigenfunctions of \( K \) that are real valued.In the paper under review, the author considers the set \[ \mathcal{S}(K) = \{ \lambda: \lambda ~ \text{is an eigenvalue of}~ K\}. \] From the general theory, it is well-known that the set \( \mathcal{S}(K) \) is countable, and it is not hard to see that \( \mathcal{S} (K) \) cannot be a finite set. The main result of the author is the following.Theorem 1. The sets \( \mathcal{S}(K) \cap (-\infty, 0) \) and \( \mathcal{S}(K) \cap (0, \infty) \) are infinite.The interest of the author in the kernel \( K \) is motivated by the appearance of the quadratic form \[\sum_{m=1}^{N}\mu (m) \sum_{n=1}^{N}\mu(n)K\left( \frac{m}{N}, \frac{n}{N}\right)\]in an identity involving the Mertens function \( M(x)=\sum_{m\le x} \mu(n)\), where \( \mu(n) \) is the well-known Möbius function. The proof of Theorem 1 follows from a clever combination of techniques in analytic number theory and some results from the Hilbert-Schmidt theory. Reviewer: Mahadi Ddamulira (Kampala) MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11A07 Congruences; primitive roots; residue systems Keywords:Mertens function; eigenvalue; symmetric kernel Citations:Zbl 1441.15006 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Huxley, Martin N.; Watt, Nigel, Mertens sums requiring fewer values of the Möbius function, Chebyshevskiĭ Sb., 19, 3, 20-34 (2018) · Zbl 1441.15006 · doi:10.22405/2226-8383-2018-19-3-19-34 [2] Mertens, Franz, Über eine zahlentheoretische Function, Wien. Ber., 106, 761-830 (1897) · JFM 28.0177.01 [3] Tricomi, Francesco G., Integral Equations (1985), Dover Publications · Zbl 0078.09404 [4] Watt, Nigel, The kernel \(\frac{1}{2} +\lfloor \frac{1}{xy}\rfloor - \frac{1}{xy}\, (0<x,y\le 1)\) and Mertens sums (2018) [5] Weyl, Hermann, Ueber die asymptotische Verteilung der Eigenwerte, Gött. Nachr., 1911, 110-117 (1911) · JFM 42.0432.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.