Intereting Posts

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How to evaluate limiting value of sums of a specific type
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Proof by Induction:
please solve a 2013 th derivative question?
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Is trace of regular representation in Lie group a delta function?
showing that $\lim_{x\to b^-} f(x)$ exists.

The arithmetic mean of prime gaps around $x$ is $\ln(x)$.

What is the geometric mean of prime gaps around $x$ ?

Does that strongly depend on the conjectures about the smallest and largest gap such as Cramer’s conjecture or the twin prime conjecture ?

- What is to geometric mean as integration is to arithmetic mean?
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- Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior
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- Prove inequality of generalized means
- Dirichlet Series and Average Values of Certain Arithmetic Functions

- Proving $p\nmid \dbinom{p^rm}{p^r}$ where $p\nmid m$
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- $\lfloor a n\rfloor \lfloor b n\rfloor \lfloor c n\rfloor = \lfloor d n\rfloor \lfloor e n\rfloor \lfloor f n\rfloor$ for all $n$
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- Reading the mind of Prof. John Coates (motive behind his statement)
- If $m=\operatorname{lcm}(a,b)$ then $\gcd(\frac{m}{a},\frac{m}{b})=1$
- Status of a conjecture about powers of 2

In 1976 Gallagher proved, under the assumption of a uniform version of the Hardy-Littlewood $k$-tuples conjecture, that for any fixed $\lambda>0$ and integer $k$ $$\#\{\text{ integers } x\leq X\ :\ \pi(x+\lambda \log x)-\pi(x)=k\}\sim e^{-\lambda}\frac{\lambda^k}{k!}X,$$ that is it follows a Poisson distribution.

Since the waiting times for a Poisson distribution is an exponential distribution, Gallagher’s work also yields (on the assumption of a uniform Hardy-Littlewood conjecture) that for fixed $\alpha,\beta$ $$\frac{1}{\pi(x)}\#\{n\leq \pi(x):\ g_n\in \left(\alpha \log x, \beta \log x\right)\}\sim \int_\alpha^\beta e^{-t}.$$ Thus the geometric mean of the $g_n$ asymptotically will equal $$\exp\left(\frac{1}{\pi(x)}\sum_{n\leq \pi(x)} \log (g_n)\right)\sim \exp\left(\log \log x+\int_0^\infty \log t e^{-t}dt\right).$$

Since $\int_0^\infty \log t e^{-t}dt=-\gamma$ where $\gamma$ is the Euler-Mascheroni constant, and we find that the geometric mean is

$$\sim e^{-\gamma}\log x.$$

I thought Hardy-Littlewood might come into it.

Here is some numerical data following Erics great answer:

x-axis: N

y-axis: Geometric mean of the first 10000 prime gaps following $10^N$ divided by $\ln 10^N$.

$e^{-\gamma} \approx 0.56146$.

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