Note on sums involving the Euler function

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordell\`es et al. and Wu.Comment: Bull. Aust. Math. Soc., accepte

On the 'Reality' of Observable Properties

This note contains some initial work on attempting to bring recent developments in the foundations of quantum mechanics concerning the nature of the wavefunction within the scope of more logical and structural methods. A first step involves generalising and reformulating a criterion for the reality of the wavefunction proposed by Harrigan & Spekkens, which was central to the PBR theorem. The resulting criterion has several advantages, including the avoidance of certain technical difficulties relating to sets of measure zero. By considering the 'reality' not of the wavefunction but of the observable properties of any ontological physical theory a novel characterisation of non-locality and contextuality is found. Secondly, a careful analysis of preparation independence, one of the key assumptions of the PBR theorem, leads to an analogy with Bell locality, and thence to a proposal to weaken it to an assumption of `no-preparation-signalling' in analogy with no-signalling. This amounts to introducing non-local correlations in the joint ontic state, which is, at least, consistent with the Bell and Kochen-Specker theorems. The question of whether the PBR result can be strengthened to hold under this relaxed assumption is therefore posed.Comment: 8 pages, re-written with new section

Unlimited parity alternating partitions

We introduce a new type of partitions that consists of partitions whose different parts alternate in parity (e.g., $3+2+2+1+1$). Various properties of this partition function are studied. In particular, we obtain its asymptotic behavior by employing Ingham's Tauberian theorem