Dedekind harmonic numbers

Abstract

The harmonic sums \(\sum_{r=k+1}^n \frac{1}{r}\) are not integers for any \(k \geq 1\). One way of proving this uses the Bertrand postulate that there is always a prime strictly between \(k\) and \(2k\) if \(k \geq 2\). The paper under review considers the more general analogous sums \(h_K(n) := \sum_{r=1}^n \frac{a_r}{r}\) where \(a_r\) is the number of ideals in the ring of integers of a number field \(K\) whose norm is \(r\). Using the analytic properties of the Dedekind zeta function \(\zeta_K\) of \(K\), it follows that \(h_K(n) \sim c_K \log n\) where \(c_K\) is the constant \(\lim_{s \rightarrow 1^+} (s-1) \zeta_K(s)\). From this, it is immediate that \(h_K(n) \rightarrow \infty\) as \(n \rightarrow \infty\). Thus, it is an interesting question as to whether \(h_K(n)\) is ever an integer for some \(n>1\). The first theorem asserts: There exists a constant \(n_K > 0\) such that \(h_K(n)\) is not an integer whenever \(n \geq n_K\). The second theorem that the authors prove is an explicit, uniform version for quadratic fields \(\mathbb{Q}(\sqrt{d})\). Interestingly, the bound \(n_K = 4\) works when \(d \not\equiv 1, 17\) mod \(24\) and these two cases are more subtle. In fact, in these two families, some Sage-Math computations suggest that there may not be a uniform bound independent of \(d\) for the \(p\)-adic valuation of \(h_K(n)\) for a specific prime \(p\). The third theorem addresses the difference \(h_K(n)-h_K(m)\) for \(n > m \geq 1\) and any number field \(K\). Under the extended Riemann Hypothesis for \(\zeta_K\), the authors obtain the non-integrality of this difference when \(n-m\) is bounded below by certain constants depending on \(K\). We remark that the proof of the first theorem is ultimately based on the fact that for large enough \(x\), there is a prime number \(p\) between \(x\) and \(2x\) which is the norm of an ideal in \(O_K\).