How to obtain an upper bound for $\sum_{x < X} \frac{\mu^2(x) \tau_k(x)}{\phi(x)}$?

Question

Let $\mu$ be the Mobius function, $\tau_k(x)$ the number of ways to write $x$ as a product of $k$ natural numbers and $\phi$ the Euler totient function. I would like to obtain an upper bound for $$ \sum_{x < X} \frac{\mu^2(x) \tau_k(x)}{\phi(x)}. $$ In the paper I am reading, this is bounded by $$ \ll (\log X)^k $$ without explanation. I would greatly appreciate any explanation. Thank you.

ps I can see that $$ \sum_{x< X} \frac{\mu^2(x)}{\phi(x)} \ll \log X $$ so I think I just have to bound $\tau_k(x)$...

Answer 1

We can obtain an explicit upper bound using the identity (where $p$ is restricted to primes) $$\frac{n}{\phi(n)}=\prod_{p\mid n}\left(1+\frac{1}{p-1}\right)=\sum_{d\mid n}\frac{\mu^2(d)}{\phi(d)}.$$ For $X\geq 1$, the above identity implies that \begin{align*}\sum_{n\leq X}\frac{\tau_k(n)}{\phi(n)} &=\sum_{n\leq X}\frac{\tau_k(n)}{n}\sum_{d\mid n}\frac{\mu^2(d)}{\phi(d)}\\ &=\sum_{d\leq X}\frac{\mu^2(d)}{\phi(d)} \sum_{m\leq X/d}\frac{\tau_k(dm)}{dm}\\ &\leq\sum_{d\leq X}\frac{\mu^2(d)\tau_k(d)}{d\phi(d)} \sum_{m\leq X/d}\frac{\tau_k(m)}{m}\\ &<\left(\sum_{d=1}^\infty\frac{\mu^2(d)\tau_k(d)}{d\phi(d)}\right) \left(\sum_{m\leq X}\frac{\tau_k(m)}{m}\right). \end{align*} On the right hand side, \begin{align*}\sum_{d=1}^\infty\frac{\mu^2(d)\tau_k(d)}{d\phi(d)} &=\prod_p\left(1+\frac{k}{p(p-1)}\right)\\ &\leq\prod_p\left(1+\frac{1}{p(p-1)}\right)^k\\ &=\left(\prod_p\frac{1-p^{-6}}{(1-p^{-2})(1-p^{-3})}\right)^k\\ &=\left(\frac{\zeta(2)\zeta(3)}{\zeta(6)}\right)^k, \end{align*} while it is straightforward that $$\sum_{m\leq X}\frac{\tau_k(m)}{m}\leq \left(\sum_{m\leq X}\frac{1}{m}\right)^k\leq (1+\log X)^k.$$ We conclude that $$\sum_{n\leq X}\frac{\tau_k(n)}{\phi(n)}<\left(\frac{\zeta(2)\zeta(3)}{\zeta(6)}\right)^k(1+\log X)^k.$$

P.S. Of course many other approaches are available and an asymptotic formula can also be proved. My goal was to give a fully explicit upper bound.

Answer 2

One should always consider Rankin's method: for $\varepsilon>0$, $$ \sum_{n\le x} \frac{\mu^2(n)\tau_k(n)}{\phi(n)} \le \sum_{n=1} ^\infty \frac{\mu^2(n)\tau_k(n)}{\phi(n)} \frac{x^\varepsilon}{n^\varepsilon} = x^\varepsilon \prod_p \bigg( 1 + \frac k{(p-1)p^\varepsilon} \bigg), $$ assuming $\varepsilon$ is chosen so that the series/product converges. To show that the right-hand side is $\ll(\log x)^k$, it suffices to prove that $$ \varepsilon \log x + \sum_p \log\bigg( 1 + \frac k{(p-1)p^\varepsilon} \bigg) \le k\log\log x+O(1). $$ We note that \begin{align*} \varepsilon \log x + \sum_p \log\bigg( 1 + \frac k{(p-1)p^\varepsilon} \bigg) &\le \varepsilon \log x + \sum_p \frac k{(p-1)p^\varepsilon} \\ &\le \varepsilon \log x + \sum_p \frac k{p\cdot p^\varepsilon} + O\bigg( \sum_p \frac1{p^2\cdot p^\varepsilon} \bigg), \end{align*} and the sum in the error is $O(1)$ uniformly for $\varepsilon\ge0$; therefore it suffices to show that $$ \varepsilon \log x + \sum_p \frac k{p\cdot p^\varepsilon} \le k\log\log x+O(1). $$ If we choose for example $\varepsilon = 1/\log x$, the left-hand side is bounded above by \begin{align*} 1 + &\sum_{p\le x} \frac kp + \sum_{j=0}^\infty \sum_{x^{2^j} < p \le x^{2^{j+1}}} \frac k{p\cdot (x^{2^j})^{1/\log x}} \\ &= k\log\log x+O(1) + k\sum_{j=1}^\infty \frac1{e^{2^j}} \sum_{x^{2^j} < p \le x^{2^{j+1}}} \frac1p \\ &= k\log\log x+O(1) + k\sum_{j=1}^\infty \frac1{e^{2^j}} \bigg( \log\log x^{2^{j+1}} - \log \log x^{2^j} + O\bigg( \frac1{\log x^{2^j}} \bigg) \bigg) \\ &= k\log\log x+O(1) + k\sum_{j=1}^\infty \frac1{e^{2^j}} \bigg( \log 2 + O\bigg( \frac1{2^j\log x} \bigg) \bigg) \\ &= k\log\log x+O(1) \end{align*} as needed.

Answer 3

Note that $\tau_{k} (u)= \frac{k^{\omega(u)}}{k!}$ for squarefree $u$ with $\omega (u)\ge k$. Thus $$\sum_{u<R} \frac{\mu (u)^{2} \tau_{k} (u)}{\phi (u)} \ll \sum_{u<R} \frac{\mu (u)^{2}k^{\omega(u)}}{\phi (u)} < \prod_{p<R}(1+\frac{k}{\phi (p)})<\prod_{p<R}(1+\frac{1}{\phi (p)})^{k}=\frac{1}{\prod_{p<R}(1-\frac{1}{p})^{k}}$$ Then, by mertens' third theorem, we get $$y_{max}\sum_{u<R} \frac{\mu (u)^{2} \tau_{k} (u)}{\phi (u)} \ll y_{max} (logR)^{k}$$