What does the symbol $\gtrsim$ mean?

Question

What does the symbol $\gtrsim$ mean exactly? I know $f(x) \sim g(x)$ ($x \to x_0)$ means $\lim_{x \to x_0} \frac{f(x)}{g(x)}=1 $.

does $\gtrsim$ mean $\lim_{x \to x_0} \frac{f(x)}{g(x)}\geq 1 ? $ Thanks!

Answer 1

The use I make in analysis is:

$$f\gtrsim g\iff \exists + \infty>C>0 \text{ s.t } f\geq C\cdot g$$

Answer 2

I have a different answer from that of b00n heT's. That interpretation is something I have seen in harmonic analysis but it is not the convention in analytic number theory (which would use the notation $f \gg g$ to describe the same situation).

When I write $f \gtrsim g$, it means that $f$ is greater than $g$ up to an asymptotically small error. That is, there exists a function $h(x)$ such that $f(x) \ge g(x) - h(x)$ and $\lim_{x\to\infty} h(x)/g(x) = 0$.

Note that this definition is still sensible even if $g$ is negative (in contrast to the definition proposed in the OP), and it doesn’t require the limit $f/g$ to exist. It also means the constant matters, so you couldn't say that $x \gtrsim 2x$.