1
vote
1answer
38 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
6
votes
1answer
103 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
8
votes
2answers
165 views

Is $\lim_{x\rightarrow\infty}\frac{x}{\pi(x)}-\ln(x) =-1$?

$\pi(x)$ is the number of primes not exceeding $x$. The prime number theorem states that $\lim_{x\rightarrow \infty} \frac{\pi(x)}{x/\ln(x)} = 1.$ So I, naïvely, inferred that ...
1
vote
1answer
103 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
1
vote
1answer
125 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
4
votes
2answers
132 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
4
votes
2answers
130 views

Estimating $\sum_{p_2 \leq x} (\log p_2)^2$

This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
-1
votes
1answer
77 views

$\log$ transform of the fundamental theorem of arithmetic? [closed]

Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this? Note: ...
0
votes
1answer
118 views

Smallest Mersenne prime with 100 million digits?

As some of you are probably aware, the Great Internet Mersenne Prime Search (GIMPS) is managing the search for the largest Mersenne primes of the form $M_p=2^p-1$, where $p$ is itself prime (GIMPS ...
5
votes
1answer
120 views

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = ...
36
votes
1answer
1k views

Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$

Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
8
votes
0answers
411 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
1
vote
1answer
89 views

$x \sim y \implies \pi(x) \sim \pi(y) $ and repeated applications of PNT

Let $\sim$ mean if $a \sim b$ then $\lim_{x \to \infty} \frac{a}{b} =1.$ The following is a threshold question. It seems that $x \sim y \implies \pi(x) \sim \pi(y).$ Pf. $\pi(x) \sim \frac{x}{\log ...
2
votes
1answer
91 views

Solving $ f(\log x)$

A generalization of the conjecture $$\pi(x+x^{\theta}) - \pi(x) \sim \frac{x^\theta}{\log x} $$ (Ingham, 1937 or earlier) might be $$\Delta \pi_k = \pi_k((x+1)^2) - \pi_k(x^2)\sim \frac{x}{\log ...
3
votes
1answer
392 views

Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
1
vote
1answer
111 views

Logarithms and prime factors

Let a-f be integers g.t. 2 with $a < b < c < d < e < f$. Let $$\ln def - \ln a b c = \alpha.$$ Let $\{p_i\}$ be the set of prime factors (with repetitions) in a,b,c. Let $\{q_i\}$ be ...
10
votes
2answers
251 views

The Fermat prime 257 and binomial sum $\sum_{n=0}^\infty \frac{(-1)^n}{\binom {8n}{4n}}$?

We have, $\begin{aligned} \sum_{n=0}^\infty \frac{(-1)^n}{\binom n{n/2}} &= \frac{4}{27}(9-\pi\sqrt{3}\,)\\[2.5mm] \sum_{n=0}^\infty \frac{(-1)^n}{\binom {2n}n} &= \frac{4}{5} - ...
1
vote
1answer
186 views

Numbering primes within a range.

$$n\ln n + n\ln\ln n−n < p_n < n\ln n+n\ln\ln n \mbox{ for } n\geq 6$$ This is the range where the $n$-th prime must lie. However sieving within this range generates a large number of primes. ...
14
votes
3answers
449 views

Are the logarithms in number theory natural?

I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them? Has it maybe ...
3
votes
2answers
106 views

Limit with prime sequence and inverse logintegral

I found formula below$$\lim_{n\to\infty}\frac{\operatorname{li^{-1}}(n)}{p_n}=1$$ where $\operatorname{li^{-1}}(n)$ is inverse logintegral function and $p_n$ is prime number sequence. Can anyone ...
1
vote
0answers
70 views

lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...