All Questions
Tagged with analytic-number-theory limits
69
questions
2
votes
2
answers
215
views
Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
- 928
6
votes
2
answers
279
views
$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$$ where $\{x\}$ denotes the fractional part of $x$ and $n\in\mathbb{N}$.
By definition of ...
- 928
6
votes
4
answers
300
views
Limit of lacunar power series in $1^-$.
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[0,1]$ be ...
- 9,357
6
votes
0
answers
194
views
Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski
I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski.
They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
- 1,145
1
vote
2
answers
106
views
What is the limit of $\frac{\Omega(n!)}{n}$ as $n \to \infty$?
Let $p_i$ be the $i$th prime number, and $\Omega(n) =$ the total number of prime factors of $n$ counting multiplicities. We know that $\Omega(n)$ is completely additive, so:
$$
\frac{\Omega(n!)}{n} ...
2
votes
1
answer
132
views
Analytic Number Theory [closed]
I really need help with two problems that I can't solve.
I have the following function: $$R(N) = \sum_{n=1}^N\exp(2\pi i\sqrt{n})$$
I was able to show that $R(n)=O(\sqrt{n})$, but I am also being ...
- 111
6
votes
0
answers
142
views
How fast does the coprime probability converge to $6/\pi^2$?
It is known that the probability that two positive integers are coprime is $6/\pi^2$. This is an amazing result. I wanted to see experimentally how the probability converges to $6/\pi^2$, but I found ...
- 165k
2
votes
1
answer
106
views
Does $\lim_{k\rightarrow\infty}\frac{\log(P_1(k))}{\log(k)}$ exist?
Wikipedia gives the definition of the Golomb-Dickman constant as $$\lambda=\lim_{n\rightarrow\infty}\frac1n\sum_{k=2}^n\frac{\log(P_1(k))}{\log(k)}$$Where $P_1(x)$ is the largest prime factor of $x$.
...
- 6,549
3
votes
1
answer
346
views
Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Note: Posted in MO since (2) is open in MSE
Let $p_n$ be the $n$-th prime and $p_n < c_n < p_{n+1}$ be the composite number such that $c_n$ has the largest prime factor $l_n$ in this prime gap. ...
- 20.8k
2
votes
1
answer
86
views
Compute limit of a function of the first $n$ prime numbers
Just playing with the result in this answer, I am asking some help for computing:
$$\lim_{n \to \infty}\frac{\prod_{k=1}^{n} p_{k}^{\frac{p_k}{(p_k-1)^2}}}{p_n}$$
where $p_k$ is the $k$-th prime ...
- 2,103
4
votes
2
answers
118
views
What is the limiting mean value of the product of the exponents in the prime factorization of numbers?
Let $n = p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m}$ be the prime factorization of $n$ and let $f(n) = a_1 a_2 \cdots a_m$. Using a heuristic argument I am able to show that the mean value of $\lim_{n \to \...
- 20.8k
0
votes
0
answers
79
views
Sum versus product on Primes
Disclaimer: I had a hard time choosing the title.
Hello MSE!
So in this post I was wondering why $e^\gamma$ has importance in number theory while $\gamma$ and $e$ don't. But I have another question ...
- 6,549
1
vote
1
answer
136
views
Elementary proof that $\lim_{n \to \infty} \prod_{i=1}^n (1-\frac{1}{p_i}) = 0$
Is there an elementary proof that for primes $p_i$,
$$\lim_{n \to \infty} \prod_{i=1}^n \left(1-\frac{1}{p_i}\right)=0\;?$$
This follows from Mertens third theorem which states that
$$ \prod_{i=1}^n \...
- 437
5
votes
2
answers
199
views
Radical of Infinite Product of Primes
I somehow stumbled across this thing.
$$\lim_{n \to \infty}({p_1\times p_2\times p_3\times\cdots \times p_n})^{\frac{1}{p_n}}$$
Where $p_i$ is the $i$-th prime.
I wanted to know if this converges or ...
- 71
2
votes
0
answers
154
views
How is this indefinite integral a holomorphic function?
In an analytic number theory textbook we derived the formula
$$\zeta(s)=\frac{s}{s-1}-s\int_1^\infty \{x\}x^{-s-1}\mathrm{dx}$$
and the textbook says, that the integral on the right side converges ...
- 901