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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
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1answer
122 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 $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
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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
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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 ...
2
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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 ...
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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: ...