All Questions
Tagged with analytic-number-theory arithmetic-functions
107
questions
2
votes
1
answer
52
views
Number of primitive Dirichlet characters of certain order and of bounded conductor
Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that
$$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$
My ...
user1296122
2
votes
0
answers
32
views
Important Subgroups of Arithmetical Functions [closed]
I am taking a course in Analytic Number Theory. The main object of study is arithmetical functions. Moreover, if we look at the arithmetical functions which do not vanish at $1$, then they form a ...
- 439
4
votes
0
answers
205
views
Relationship between two types of partition functions
After downvoting my previous thread, here is a more detailed explanation of my question. For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \...
1
vote
1
answer
78
views
How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.
Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
- 937
1
vote
1
answer
120
views
A problem on von Mangoldt function.
Suppose $n$ is odd natural number. Define $r(n)=\sum_{n_1+n_2+n_3=n} \Lambda(n_1)\Lambda(n_2)\Lambda(n_3)$ and
$r'(n)=\sum_{p_1+p_2+p_3=n}(\log p_1)(\log p_2)(\log p_3)$ where $p_1,p_2,p_3$ are ...
1
vote
0
answers
50
views
Characterization of Möbius-monotonicity
We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
- 1,810
1
vote
1
answer
135
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
- 299
1
vote
1
answer
105
views
On the second moment of prime divisor function
Let $\omega(n)$ denote the number of distinct prime divisors of $n$. I learned from Cojocaru & Murty that
$$
\sum_{n\le x}\omega(n)^2=x(\log\log x)^2+O(x\log\log x).\tag1
$$
I wonder whether it is ...
- 7,193
1
vote
2
answers
169
views
$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\frac xd\right\rfloor$ and $\sum_{d|n}\varphi\left(\frac xd,\frac nd\right)=\lfloor x\rfloor$
If $x$ is real, $x\ge 1$, let $\varphi(x,n)$ denote the number of positive integers less than or equal to $x$ that are relatively prime to $n$. Prove that
$$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\...
- 9,592
2
votes
1
answer
550
views
Product form of Mobius Inversion formula: $g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$
Product form of the Möbius inversion formula: If $f(n)>0$ for all $n$ and if $a(n)$ is real, $a(1)\neq 0$, prove that
$$g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$$
where $b=a^...
- 9,592
5
votes
2
answers
302
views
Prove that $\sum_{d|n}\mu(d)\log^m d=0$
Prove that
$$\sum_{d|n}\mu(d)\log^m d=0$$
if $m\ge 1$ and $n$ has more than $m$ distinct prime factors.
I tried using Induction and kind of succeeded in the sense that if we write down the case $m=2$ ...
- 9,592
4
votes
2
answers
170
views
Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?
Define $f(n)$ to be:
$$
\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}
$$
But $\sigma_0(d) = 2^{\omega(d)}$ for any $d \mid n\#$ a primorial, so:
$$
f(n) = \prod_{p \text{ prime} \\ p \leq n} ...
2
votes
0
answers
70
views
Different ways to average an arithmetic function
Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$:
Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$
Factorized: $\displaystyle\...
- 152k
2
votes
1
answer
146
views
Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$
I'm investigating the behavior of the following function as $x\to \infty$:
$$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$
where $J_k(n)$ is the Jordan's totient function
$$J_k(n):=n^k\prod_{p\mid n}(1-...
- 5,217
0
votes
1
answer
234
views
Asymptotics of the sum of squares of Von Mangoldt function values.
This question was asked in my quiz of number theory and I was not able to make any progress in it.
Question: Show that
$$\sum_{ n\leq x}{\Lambda(n)}^2 = x\log x- x+o(x),$$
where $\Lambda(n)$ is the ...
user775699