arXiv subjects
Publications and source records attributed to Joshua Zelinsky.
We present upper bounds on certain sums which are related to Artin's primitive root conjecture and are also used in counting ray class characters.
Define $|n|$ to be the complexity of $n$, the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $|n|\ge 3\log_3 n$ for all $n$. Define the defect of $n$, denoted $δ(n)$, to be $|n|-3\log_3 n$; in this paper we present a method for classifying all $n$ with $δ(n)<r$ for a given $r$. From this, we derive several consequences. We prove that $|2^m 3^k|=2m+3k$ for $m\le 21$ with $m$ and $k$ not both zero, and present a method that can, with more computation, potentially prove the same for larger $m$. Furthermore, defining $A_r(x)$ to be the number of $n$ with $δ(n)<r$ and $n\le x$, we prove that $A_r(x)=Θ_r((\log x)^{\lfloor r \rfloor+1})$, allowing us to conclude that the values of $|n|-3\log_3 n$ can be arbitrarily large.