For questions on arithmetic functions, a real or complex valued function $f(n)$ defined on the set of natural numbers.
8
votes
1answer
113 views
Comparing average values of an arithmetic function
Suppose $f(n)$ is a positive real-valued arithmetic function such that
$$
\frac1n\sum_{k=1}^nf(k)\sim g(n)
$$
for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
7
votes
1answer
96 views
Is there a complex variant of Möbius' function?
When you're dealing with arithmetic functions, you might have come across the classical Möbius' function
$$
\mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
5
votes
2answers
156 views
Sum of $n \sigma(n)$
What is known about the asymptotic behavior of
$$
-\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ?
$$
It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.
4
votes
1answer
225 views
On sums involving Euler's totient function
I've been struggling with the following claim without being able to prove it, so your help would be highly appreciated:
Let $\varphi(n)$ be Euler's totient function. Show that there is a constant ...
4
votes
2answers
58 views
Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$
Prove: $\sum_{k<n, (k,n)=1}k = \frac{1}{2}n \varphi (n)$
I have had strep throat and missed the lecture discussing properties of the Euler function. Any help in solving this is appreciated. ...
4
votes
1answer
88 views
Asymptotics for almost all $x$
Theorem 2.2 in Shparlinski 2006 says:
For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound
$$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$
holds.
...
4
votes
2answers
190 views
Elements of finite order in the group of arithmetic functions under Dirichlet convolution.
Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
4
votes
0answers
182 views
Numbers $n$ such that Mertens' function is zero.
OEIS (A028442) lists the
Numbers n such that Mertens' function
$$
M(n)=\sum_{k=1}^n\mu(k)
$$
is zero:
2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, ...
4
votes
0answers
114 views
Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?
Let me start with the following on elementary symmetric polynomials:
The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity
...
3
votes
1answer
91 views
Minimal $x$ for which $\phi(k) > n$ for all $k > x$
It's well-known that
$$
\liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma}
$$
and there exists an effective version
$$
\varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}}
$$
valid for ...
2
votes
1answer
253 views
Chebyshev's first $\vartheta(x)$ function question
This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
1
vote
2answers
66 views
Is Wiki wrong on Dirichlet Chararcters Modulo $10$?
Wiki says:
Modulus 10
There are $\phi(10)=4$ characters modulo $10$.
Note that $χ$ is wholly determined by $\chi(3)$, since $3$ generates the group of units modulo $10$.
I can ...
1
vote
1answer
149 views
How can the Möbius function be applied to a series?
Given a series $p_n(s)=\sum_{k=1}^n a_k $. I'd like to get $\hat{p}_n(s)=\sum_{k=1}^n \mu(k)a_k $. Think of $a_k=k^{-s}$ for example. If you let $n$ go to $\infty$, you'll see the well-known relation ...
1
vote
1answer
54 views
Function that counts the number of divisors of a natural number?
Let Function $f(n)$ be formally defined for natural numbers such that it gives number of distinct divisors of the number n (n and 1 included) For example, $f (12)=6$, then what is a quick way to ...
1
vote
1answer
82 views
Next asymptotic term of the average order of sigma
$$
\sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n).
$$
Is the next asymptotic term known? That is, is there a monotonic increasing function $f(x)$ such that
$$
...
1
vote
1answer
57 views
how to prove $f$ is an arithmetic function with this property $\sum_{d\mid n} f(d)=n^2$
how to prove $f$ is an arithmetic function with this property
$$\sum_{d\mid n} f(d)=n^2$$
Arithmetic function
1
vote
1answer
69 views
Solving $ f(\log x)$
A generalization of the conjecture
$$\pi(x+x^{\theta}) - \pi(x) \sim \frac{x^\theta}{\log x} $$ (Ingham, 1937 or earlier) might be
$$\Delta \pi_k = \pi_k((x+1)^2) - \pi_k(x^2)\sim \frac{x}{\log ...
1
vote
0answers
57 views
Approximate how the Numbers $n$ such that Mertens' function is zero grow.
Is it possible to approximate how the "Numbers $n$ such that Mertens' function is zero" grow?
1
vote
0answers
84 views
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 ...
0
votes
1answer
50 views
Converting loop to a closed form expression? [duplicate]
Possible Duplicate:
How to convert this loop into a closed form expression?
I have the following code in Python
...
