Sums related to Euler's totient function
Artyom Radomskii
TL;DR
This work studies upper bounds for sums of the form $\sum_{n\le N} (a_n/\varphi(a_n))^s$ where the $a_n$ are positive integers with controlled divisibility, and derives tail bounds for the large values of $a_n/\varphi(a_n)$. The approach reduces the problem to a sieve-like decomposition over square-free moduli with small prime factors, using a multiplicative majorant $g(n)$ and a Radomskii-type bound for $n/\varphi(n)$ to obtain a product bound over primes $p\le y$, yielding a general estimate $\sum_{n\le N} (a_n/\varphi(a_n))^s \le K (c/\alpha)^s \prod_{p\le y} \left(1+\frac{(1+p^{-1})^s-1}{g(p)}\right)$ and, under suitable hypotheses, $\sum_{n\le N} (a_n/\varphi(a_n))^s \le \exp( s\log\log(s+2) + C s)\,N$ with a double-exponential tail bound $\#\{n\le N: a_n/\varphi(a_n)>t\} \le c_1 \exp(-\exp(c_2 t))\,N$. The results extend to polynomial inputs $f(n)$ and to values at primes $f(p)$, including applications to linear forms and polynomials such as $f(x)=x-1$ and $f(x)=x^2+1$, with analogous exponential-type bounds. These contributions sharpen prior bounds and provide tools for analyzing the distribution of $a_n/\varphi(a_n)$ in sieve- and polynomial-analytic contexts.
Abstract
We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some restrictions. As applications, for any $t>0$, we obtain an upper bound for the number of $n\in [1,N]$ such that $a_{n}/ \varphi (a_{n})> t$.