1
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1answer
132 views

Ramanujan's prime counting inequality

This article says that Ramanujan claimed that $$ \pi^2(x) < {ex \over \log x} \pi(x/e), $$ and that it is indeed true for $x$ sufficiently large. They really glaze over the proof though, and I'm ...
5
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1answer
147 views

How to prove this inequality $\pi(x) > \log x - 1$ involving the prime counting function?

Problem Prove that $\pi(x) > \log x - 1$. Progress Based on a hint and very elementary methods, I got that $$ \prod_{p \leq x} (1-p^{-1})^{-1} \leq \prod_{k=2}^{\pi(x)+1} (1-k^{-1})^{-1}. $$ The ...
-1
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1answer
96 views
0
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1answer
198 views

Regarding Chebyshev's theta function

It is known that $x\sim \theta \left ( x \right )$, where $$\theta \left ( x \right )= \sum_{p\leqslant x}\log p.$$ For all values of x for which it has been calculated, $x> \theta \left ( x ...
0
votes
0answers
91 views

estimate $\sum_{x<p\le x+y} \log{p}/p$

In his paper the prime number theorem via the large sieve, A. Hildebrand made use of the following inequality $$\sum_{x<p\le x+y} \frac{\log{p}}{p} \le (2+o(1))\log{\frac{x+y}{x}}$$ where $x\ge y$ ...
4
votes
1answer
69 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
60 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
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1answer
47 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
380 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 ...
11
votes
3answers
1k views

Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
12
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1answer
276 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. ...
4
votes
1answer
439 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
671 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: ...