Intereting Posts

Prove that $e^{-A} = (e^{A})^{-1}$
How to compute $2^{2475} \bmod 9901$?
Exploding (a.k.a open-ended) dice pool
$\int_0^1 {\frac{{\ln (1 – x)}}{x}}$ without power series
Exponent and abelian groups
Expansion of $(1+\sqrt{2})^n$
Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?
Least squares problem: find the line through the origin in $\mathbb{R}^{3}$
Help solving $\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}dx}$
Get location of vector/circle intersection?
A basis for the dual space of $V$
Finding the circles passing through two points and touching a circle
Show that $ \cap \mathbb{Q}$ is not compact in $$
Homology groups of the Klein bottle
help with $\lim\limits_{(x,y) \to (0,0)} f(x,y) = {\cos(x) -1 – {x^2/2} \over x^4 + y^4}$

You might have read about the fortuitous meeting between Montgomery and Dyson. The background is that the nontrivial zeros of the Riemann zeta function, when normalized to have unit spacing on average, (seem to) have the pair correlation function $1-\mathrm{sinc}^2(x)$, where $\mathrm{sinc}$ is the normalized function $\sin(\pi x)/ (\pi x)$. It’s still a conjecture but it has good numerical support.

So what about prime numbers? Let $\Sigma(x,u)$ be the number of pairs of primes $p,q\le x$ which satisfy the inequality $0\le p-q\le u\,x/\pi(x)$, where $\pi(x)$ is the prime counting function. This inequality is chosen because multiplying primes by $\pi(x)/x$ will ensure the gaps between consecutives is exactly unity (hence they are normalized). Then what might

$$g(u)=\frac{d}{du}\left(\lim_{x\to\infty}\frac{\Sigma(x,u)}{\pi(x)}\right)$$

- What are dirichlet characters?
- How many elements in a number field of a given norm?
- What does this $\asymp$ symbol mean? (subject: analytic number theory)
- Representing a number as a sum of at most $k$ squares
- Are there any Combinatoric proofs of Bertrand's postulate?
- A convergence problem： splitting a double sum

end up looking like? This basically asks, “what is the density of normalized primes around so-and-so apart from each other?” (You can see the original Montgomery conjecture as equation 12 here. I’ve adapted it to prime numbers by essentially changing the asymptotic number of zeta zeros to the prime counting function instead.)

I have also posted this question on MathOverflow here. I figured this question might be at more of a researcher’s level than Math.StackExchange is used to and so would be beneficial to cross-post. MO user David Roberts informs me this is something of a malpractice, as it makes collecting answers awkward and a delay should be allowed for one site (presumably SE) to have a go at it. I’ll keep this in mind in the future.

- there exist infinite many $n\in\mathbb{N}$ such that $S_n-<\frac{1}{n^2}$
- Find all natural numbers $n$ such that $21n^2-20$ is a perfect square.
- Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$
- Reason for LCM of all numbers from 1 .. n equals roughly $e^n$
- The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$
- Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
- Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
- Integral solutions to $56u^2 + 12 u + 1 = w^3$
- Gaps between numbers of the form $pq$
- $t^3$ is never equal to

The answer is posted here on MathOverflow.

The purpose of this answer is to redirect others, and also so that the question does not remain “unanswered”.

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- The Area of an Irregular Hexagon
- Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
- Ordinals definable over $L_\kappa$
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- A place to learn about math etymology?
- computing the orbits for a group action
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