A question about prime gaps

Question

I have been reading the Wikipedia article about prime gaps (http://en.wikipedia.org/wiki/Prime_gap) and I came across the following:

Hoheisel was the first to show that there exists a constant $\theta<1$ such that $$\pi(x+x^\theta)-\pi(x)\sim\frac{x^\theta}{\log x}$$ as $x$ tends to infinity.

I have a few questions:

1. What's the meaning of the $\sim$ symbol?

2. Does the symbol $\sim$ mean that $\pi(x+x^\theta)-\pi(x)$ is asymptotic to $x^\theta/\log x$? If this is the case, does this mean that $\pi(x+x^\theta)-\pi(x)$ is always greater than $x^\theta/\log x$ at least for sufficiently large $x$? Is $\pi(x+x^\theta)-\pi(x)$ the same as $\pi[x,x+x^\theta]$, that is to say, the amount of primes in the interval $[x,x+x^\theta]$?

I'm not sure what the above expression means, and I'm not sure what it means when they say that a certain function is "asymptotic" to another function or when they say that a certain asymptotic formula holds.

3. Also, in the above expression, is $\log$ the natural logarithm or the logarithm in base $10$?

Answer 1

$f(x)\sim g(x)$ just means that $\lim_{x\to\infty}f(x)/g(x)=1.$ Informally, $g(x)$ is a good approximation for $f(x)$ (and vice versa) when $x$ is large.

$\pi(x+x^\theta)-\pi(x)\sim x^\theta/\log x$ mean that

1. $\pi(x+x^\theta)-\pi(x)$ is greater than $0.99x^\theta/\log x$ for sufficiently large $x$ , and this would remain true (for a different value of "large enough") if you replace 0.99 with any number less than 1.
2. $\pi(x+x^\theta)-\pi(x)$ is less than $1.01x^\theta/\log x$ for sufficiently large $x$, and this would remain true (for a different value of "large enough") if you replace 1.01 with any number greater than 1.

And yes, $\log$ is the natural logarithm, and $\pi(x+x^\theta)-\pi(x)$ is the number of primes in the interval $(x,x+x^\theta]$.