arXiv subjects
Publications and source records attributed to W. D. Banks.
We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $φ(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.
We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.