# All questions with tag [math: arithmetic-functions]

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$$\sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n).$$ Is the next asymptotic term recognized? That is, exists a monotonic raising function $f(x)$ such that $$\sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+f(n)+o(f(n))$$ ? (I would certainly presume something like $f(x)=cx$ if $f$ exists.) At the same time, exist monotonic raising features $f... 2022-07-25 20:13:44 0 ## Generalizing a result on sums involving Euler's function Motivation: It's known that there is a constant$0&lt;K$such that for any natural number$N$,$KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$(with$\varphi$being Euler's function). See more details here: On sums involving Euler's totient function My intention is to generalize this result. So my question is: ... 2022-07-15 05:44:52 1 ## Asymptotics for almost all$x$Theorem 2.2 in Shparlinski 2006 says: For all positive integers$n\le x$except possibly$o(x)$of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. The "except possibly$o(x)$of them" part seems to substantially weaken the conclusion: there could be a block of$x^{0.99}$numbers where... 2022-07-14 05:00:15 1 ## On sums involving Euler's totient function I've been having problem with the adhering to case without having the ability to confirm it, so your aid would certainly be very appreciated: Let$\varphi(n)$be Euler is totient function. Show that there is a constant$0&lt;K$such that for any kind of all-natural number$N$,$KN\leq\frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+...+\frac{\varp...
2022-07-14 04:57:45
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## Question about the proof of Theorem 2.19 (Page 38) of the book Introduction to Analytic Number Theory by Apostol

At the last line of the proof: $\lambda^{-1}(n)=\mu(n)\lambda(n)=\mu^2(n)=|\mu(n)|$. Why $\mu(n)\lambda(n)=\mu^2(n)$? Just how to confirm this?
2022-07-01 16:40:41
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## How to prove $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$

For every favorable integer $d$ , we allowed $\tau\left(d\right)$ be the variety of favorable divisors of $d$ . Confirm that \begin{align} \sum_{d|n} \tau^3(d) = \left(\sum_{d|n} \tau (d)\right)^2 \end{align} for each and every favorable integer $n$ , where the amounts vary over all favorable divisors $d$ of $n$ . Currently I just recogni...
2022-06-27 20:33:30
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## Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

I'm attempting to resolve Ireland and also Rosen is A Classical Introduction to Modern Number Theory as I've listened to good ideas concerning it. This is Exercise 12 from Chapter 2. Below $\mu$ is the Moebius function, and also $\phi$ the totient function. Locate solutions for $\sum_{d|n}\mu(d)\phi(d)$, $\sum_{d|n}\mu(d)^2\phi(d)^2$, and als...
2022-06-09 01:02:56