Tagged Questions
4
votes
1answer
29 views
Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function
In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$.
I have a question about Theorem 2.27 on page 22.
My ...
1
vote
2answers
41 views
showing that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$
i can't see that $H_n \leq \prod_{n \leq N}{(1-p^{-1})^{-1}}$
and i can't see that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$
p is prime and $H_n$ is harmonic series
2
votes
1answer
43 views
A number-theoretical estimation-inequality
I have some trouble understanding the following number-theoretical estimation:
$$\sum_{k\le \sqrt{n}} (1-k^2/n)^{1+o_n(1)}=n^{1/2+o(1)} \
(n\to\infty),$$ where $o_n(1)$ denotes a $o(1)$ function ...
6
votes
1answer
309 views
Showing that $\log(\log(N+1)) \leq 1+\sum\limits_{p \leq N} \frac{1}{p}$
I can't see how you get this.
I want to show that
$$\log(\log(N+1)) \leq \sum_{p \leq N} \frac{1}{p}+1$$
Can't see how it follows from this. So you show that
$$0 \lt -\log(1-x)-x \lt ...
12
votes
1answer
245 views
Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$
How do I prove the following:
$$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$
I tried induction, but I didn't know how to go on because I don't have a look at all numbers.
...
3
votes
1answer
284 views
Where can I find the paper by Guy Robin?
\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...
2
votes
1answer
332 views
Proof of Chebyshev's theorem
(a) Show that $\int_2^x\frac{\pi(t)}{t^2}dt=\sum_{p\leq x }\frac{1}{p}+o(1)\sim\log\log x.$
(b) Let $\rho(x)$ be the ratio of the two functions involved in the prime number theorem:
...
