All Questions
Tagged with multiplicative-function analytic-number-theory
37 questions
0
votes
1
answer
66
views
Show that $\sum_{n \leq x} \left( \frac{\phi(n)}{n} \right)^\kappa = c(\kappa)x + O(x^\epsilon)$.
Let $\kappa$ be a fixed real number. Show that
$$\sum_{n \leq x} \left( \frac{\phi(n)}{n} \right)^\kappa = c(\kappa)x + O(x^\epsilon),$$
where
$$c(\kappa) = \prod_p \left( 1 - \frac{1}{p} \left( 1 - (...
- 707
1
vote
1
answer
150
views
Dirichlet series as Euler product
I am trying to show the following proposition:
Let $f$ be a multiplicative function and $P$ be the set of prime numbers. Then the Dirichlet series $\sum_{n=1}^\infty f(n)/n^s$ converges to the value $\...
- 135
3
votes
1
answer
83
views
The "Euler Product formula" for general multiplicative functions
For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$
This is easy to show ...
4
votes
1
answer
204
views
Does this function in $3$b$1$b has a name?
I was watching this video from $3$Blue$1$Brown channel, at minute 21:00 he introduced the following function:
$$
\chi(n)=
\begin{cases}
0 & \text{if } n=2k \\
1 & \text{if } n=4k+1\\
-1 & \...
- 6,267
4
votes
0
answers
156
views
Probability that one random number among many has a unique prime factor
If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M=2^k\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{...
- 679
3
votes
1
answer
125
views
The mean square of $d_k(n)$
Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating
...
- 1,694
0
votes
2
answers
121
views
Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.
Here $f(n)=\begin{cases}
1\ \text{if } n=m^2\text{ for some }m\in\Bbb{N}\\
0\ \text{if otherwise}
\end{cases}$
This is a multiplicative function. At first I define $g:\Bbb{N}\to\Bbb{C}$ be $g(n)$ to ...
- 3,174
0
votes
1
answer
86
views
2.20 Question Introduction to analytic number theory
The following question is on page 48 apostol Introduction to analytic number theory( 20)
Let $P(n)$ be the product of the positive integers which are less than equal to $n$ and relatively prime to $n$...
user775699
0
votes
1
answer
106
views
A question in solution of a problem involving multiplicative functions
This question is question 2.3 of Apostol Introduction to analytic number theory.
It's solution image:
My question: how did in RHS on last line of solution author got $n / \phi(n) $ as $n = p_{1}^{a_1}...
user775699
0
votes
0
answers
125
views
Why does the divisor-counting function appear in bounds for Kloosterman sums?
Given integers $m,n$ and $c \geq 2$, the Kloosterman sum is defined as $S(m,n;c) = \sum_{k \in (\mathbb{Z}/c\mathbb{Z})^{\times}}{e^{\frac{2i\pi}{c}(mk+nk^{-1})}}$, where $k^{-1}$ is the reciprocal of ...
- 36.7k
4
votes
1
answer
395
views
Estimate for $\sum_{n\leq x}2^{\Omega(n)}$
I need some help to find a mistake in my proof.
I have to prove that $\sum_{n\leq x}2^{\Omega(n)}\sim cx\log^2x$ for $x\rightarrow+\infty$, where $\Omega(p_1^{k_1}\cdot\ldots\cdot p_j^{k_j})=k_1+\...
- 197
2
votes
1
answer
147
views
How to generalize Newman's simplification of O-Tauberian theorem?
Can you prove the next theorem:
Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then
$\lim_\limits{...
- 2,376
1
vote
1
answer
345
views
Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.
This is a detailed problem, let me write down the problem and the process I have done:
Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$.
Let $\chi$ be a ...
- 1,877
2
votes
1
answer
212
views
If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$
The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
- 1,877
1
vote
1
answer
592
views
Inverse of completely multiplicative function with respect to dirichlet convolution
Is the inverse of a completely multiplicative function $f(n)$ with respect to Dirichlet convolution again completely multiplicative? I know that for multiplicative functions its true(Apostol's ...
1
vote
1
answer
125
views
Upper bound on coefficients of the logarithmic derivative of a certain Dirichlet series
For a multiplicative arithmetic function $f(n)$, we define $ F(s) = \sum_{n\ge1}^{} \dfrac{f(n)}{n^s}$. We then define the coefficients $\Lambda_f (n)$ by
$$ -\dfrac{F'(s)}{F(s)} = \sum_{n\ge1}^{} \...
14
votes
3
answers
1k
views
Some convincing reasoning to show that to prove that Ramanujan tau function is multiplicative is very difficult
I am curious to know (some words or reasoning about how to justify) why to prove that the so-called Ramanujan $\tau$ is a multiplicative function is (was) very difficult.
In this Wikipedia is showed ...
user243301
2
votes
2
answers
132
views
Lower bound for sum of a multiplicative function based on lower bound on value of primes
Let $f$ be a non-negative multiplicative function. Suppose we have some bound on $\sum_{p \le x} f(p)$, where the summation is over primes. Is it possible to give a lower bound on $\sum_{n \le x} f(n)$...
- 1,565
1
vote
0
answers
210
views
On arithmetic functions whose Dirichlet series has a special kind of abscissa of absolute convergence
For a function $f: \mathbb N \to \mathbb C$ , let $\sigma_c(f) , \sigma_a(f)$ denote the abscissa of convergence and the abscissa of absolute convergence respectively of the Dirichlet series $\sum_{n=...
- 4,476
1
vote
1
answer
166
views
Two Dirichlet characters $\chi$, $\chi'$ are equal if $\chi(p) = \chi'(p)$ for almost all primes
I want to prove the following: Let $\chi, \chi'$ be two primitive Dirichlet-characters of conductor $N$. Suppose that $\chi(p) = \chi'(p)$ for all but a finite number of primes $p$. Then $\chi = \chi'$...
- 4,701
4
votes
0
answers
180
views
Minimal value of summatory function of completely multiplicative functions taking values -1 and 1
Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below.
Define the Liouville ...
- 321
6
votes
2
answers
786
views
Dirichlet series associated to squared Möbius
I would like to estimate the Dirichlet series of a multiplicative function. Consider the following:
$$\sum_{m \leqslant X} \frac{\mu^2(m)}{m^s}$$
When does it converges when $X$ grows? What is an ...
- 1,521
0
votes
1
answer
246
views
Dirichlet character modulo $k$
Let $f$ be a completely multiplicative function. If there exists $k$ such that $$f(n+k) = f(n)$$ for any $n \in \mathbb{N}$, then there exists $k_0$ such that $f$ is a Dirichlet character modulo $k_0$....
- 1,045
2
votes
1
answer
1k
views
show that $ \sum_{p \leq x} \log p \big( \log x - \log p\big) = O(x) $
Show that:
$$ \sum_{p \leq x} \log p \big( \log x - \log p\big) = O(x) $$
This is not an exercise. This is implied in one line of proof by Atle Selberg. Additionally the paper asks to show:
$$ \...
- 24.8k
4
votes
2
answers
2k
views
Formula for square of the number of divisors $\sum_{r\mid n} d(r^2) = d^2(n)$
I am trying to prove or disprove the following statement:
$$\sum_{r|n} d(r^2) = d^2(n),$$
where $d$ is the number of divisors function.
Computing it for small numbers yields equality, so I at least ...
- 698
3
votes
1
answer
104
views
Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question
I would like receive hints at least for one of the following problems, these are going from experiments.
Can you provide to me hints for at least one of the following problems? I will try put the ...
user243301
6
votes
1
answer
309
views
Average Order of $\frac{1}{\mathrm{rad}(n)}$
Again a question about $\mathrm{rad}(n).$
Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, $$\mathrm{rad}(...
user261687
15
votes
3
answers
406
views
Series of the totient function
Good morning,
I wonder if : $$\sum_{n} \frac{(-1)^n}{\varphi (n)}$$ converges or not.
where $\varphi (n)$ is the Euler function.
Do you have any idea ?
user279923
8
votes
1
answer
762
views
Average order of $\mathrm{rad}(n)$
Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
user261687
7
votes
1
answer
325
views
What is known about these arithmetical functions?
Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and
$$
\alpha_N(n)=\prod_p p^{c_p \bmod N}.
$$
The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
- 18.6k
6
votes
1
answer
260
views
Bounding this arithmetic sum
I am interesting in bounding the arithmetic sum
$$ \sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)}$$
(The motivation is that this is a sum that comes up a lot in sieving primes, in particular in the ...
- 6,848
2
votes
2
answers
108
views
Orthogonality de Möbius
Does anyone know how prove that $$\sum_{n\leqslant x}\mu(n)\xi(n) =o(x)$$ when $\xi(n)$ is a multiplicative functions? I found one commentary that exist a connection of this problem with the Theory of ...
- 65
5
votes
1
answer
274
views
To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$
How may we estimate $$\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$$ where for every positive integer $m$ , $d(m)$ denotes the number of positive divisors of $m$ ?
- 8,427
6
votes
1
answer
622
views
Dirichlet Series and Average Values of Certain Arithmetic Functions
If an arithmetic function $f(n)$ has Dirichlet series $\zeta(s) \prod_{i,j = 1} \frac{\zeta(a_i s)}{\zeta(b_j s)}$, for which values of $a_{i}$ and $b_{j}$ is the following true? That
\begin{align}
\...
- 17.3k
14
votes
1
answer
2k
views
Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function
Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
- 1,223
15
votes
3
answers
2k
views
On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
There is a second part of the problem posted in Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$, from Apostol's book, but I can't figure it out. It ...
- 161
8
votes
2
answers
3k
views
Asymptotic formula for $\sum_{n\leq x}\mu(n)[x/n]^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$
I would like to show (for $x \ge 2$) that $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\right]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x)).$$
I already have the identity $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\...
- 12.6k