Intereting Posts

Find the limit of $(2\sin x-\sin 2x)/(x-\sin x)$ as $x\to 0$ without L'Hôpital's rule
Intriguing polynomials coming from a combinatorial physics problem
If $a$ is a quadratic residue modulo every prime $p$, it is a square – without using quadratic reciprocity.
A finite set always has a maximum and a minimum.
Differential Equations: solve the system
Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime
How to comprehend $E(X) = \int_0^\infty {P(X > x)dx} $ and $E(X) = \sum\limits_{n = 1}^\infty {P\{ X \ge n\} }$ for positive variable $X$?
A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
How to write well in analysis (calculus)?
Radius of convergence of product
Fundamental group of the connected sum of manifolds
Predicting digits in $\pi$
Uniform convergence with two sequences of functions
showing a function defined from an integral is entire
Possible fake proof of $1= -1$

I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a very helpful step towards proving the prime number theorem, although I don’t know how, primarily because I have not gone through the proof of the theorem even once.

Anyway, I was trying to prove the result $\lim \frac{p_n}{p_{n+1}}=1$ by really elementary methods, as is always my habit, and quite recently I learned of a theorem that the sum of the reciprocals of the primes diverges (I am only a beginner). The idea that suddenly struck me is that this theorem implies $\limsup\frac{p_n}{p_{n+1}}=1$, by a simple application of the ratio test and the fact that $\frac{p_n}{p_{n+1}} < 1$ for all $n$. So, is there any simple way to show that $\liminf\frac{p_n}{p_{n+1}}=1$ or at least $\ge1$, so that the result is proved?

Another beautiful result that came to me, follows from the theorem on divergence of $\sum\frac{1}{p}$, that given any $h > 1$ there are infinitely many $n$ such that $p_n < h^n$. Are there any other results with really simple proofs (easy to understand even for beginners like me) having deep consequences in the theory of the distribution of primes? I’d really like to hear them!

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- Riemann zeta function at odd positive integers
- Proving $\sqrt{2}\in\mathbb{Q_7}$?

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- Problem with infinite product using iterating of a function: $ \exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots $
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- Generating functions and the Riemann Zeta Function
- How to use the method of “Hensel lifting” to solve $x^2 + x -1 \equiv 0\pmod {11^4}$?
- A local-global problem concerning roots of polynomials
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- Is there a rational surjection $\Bbb N\to\Bbb Q$?
- If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
- Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?

Was just about to ask/confirm the same question, this is a good place to put my thoughts then. So, from Rosser’s theorem

$$\ln(n) + \ln(\ln(n)) -1< \frac{p_{n}}{n}<\ln(n) + \ln(\ln(n))$$

$$\frac{1}{\ln(n+1) + \ln(\ln(n+1))} < \frac{n+1}{p_{n+1}} < \frac{1}{\ln(n+1) + \ln(\ln(n+1)) -1}$$

Or

$$\frac{\ln(n) + \ln(\ln(n))-1}{\ln(n+1)+\ln(\ln(n+1))}< \frac{n+1}{n}\cdot \frac{p_{n}}{p_{n+1}}<\frac{\ln(n)+\ln(\ln(n))}{\ln(n+1)+\ln(\ln(n+1))-1}$$

From which:

$$\lim_{n \to \infty } \frac{p_{n}}{p_{n+1}}=\lim_{n \to \infty } \frac{\ln(n)}{\ln(n+1)}=1$$

Also, these bounds apply $1 > \frac{p_{n}}{p_{n+1}} > \frac{1}{2}$.

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