All questions with tag [math: arithmetic-functions]


Next asymptotic term of the average order of sigma

$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n). $$ 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 17:13:44

Generalizing a result on sums involving Euler's function

Motivation: It's known that there is a constant $0<K$ 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 02:44:52

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 02:00:15

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 $0<K$ 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 01:57:45

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 13:40:41

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 17:33:30

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...
2022-06-08 22:02:56