The goal of this article is twofold. First, we extend a result of Murty and Saradha (2007) related to the digamma function at rational arguments. Further, we extend another result of the same authors (2008) about the nature of p-adic Euler-Lehmer constants.

We obtain new bounds for the integer Chebyshev constant of intervals [p/q, r/s] where p, q, r and s are non-negative integers such that qr - ps = 1. As a consequence of the methods used, we improve the known lower bound for the trace of totally positive algebraic integers.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

We study the function $M_{\theta }\left(n\right)=\lfloor 1/{\theta }^{1/n}\rfloor$, where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_{\theta }$, that if log θ is rational, then for all but finitely many positive integers n, $M_{\theta }\left(n\right)=\lfloor n/log\theta -1/2\rfloor$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy $M_{\theta }\left(n\right)=\lfloor n/log\theta -1/2\rfloor$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...