What exactly does \gg and \ll mean?

Question

For example, $f(T)\ll_T 1$ where $T$ is a positive number.

Answer 1

I like Frank Thorne's comment-answer, but I would like to stress a detail he mentions in passing, but which does not seem adressed at all on Wikipedia.

What exactly $\ll$ means is not completly uniform even if on just takes the meaning(s) common in Analytic Number Theory (and ignores those informal ones used in some other context like "a lot less than" or "negligible").

One meaning is this: For maps $f: D \to \mathbb{C}$, and $g: D \to \mathbb{R}_{\ge 0}$ one says $f = O(g)$ or equivalenlty $f \ll g$ if there exists some constant $C$ such that $|f(x)| \le C g(x)$ for all $x \in D$.

In particular, this notation is not in itself asymptotic.

Another meaning is this:

For a map $f: D \to \mathbb{C}$, and a function $g: D \to \mathbb{R}_{\ge 0}$ and an accumulation point $x_0$ (possibly infinity) of $D$ one says $f = O(g)$ as $x\to x_0$ or equivalenlty $f \ll g$ as $x \to x_0$ if for each neighborhood $U$ of $x_0$ there exists some constant $C$ such that $|f(x)| \le C g(x)$ for all $x \in D \cap U$.

And not infrequently $f = O(g)$ and $f \ll g$ is used with the second meaning implictly assuming that $x_0$ is "infinity." In particular this is common if the domain of the function is discrete, as it seems to be in you case (positive integers).

A detailed discussion of this can be found for instance in the introduction of Iwaniec and Kowalski's book, who use the first meaning.

Now, after this general discussion for the subscripts. No matter what precise of the two meaning one chooses one always has some constant $C$ called the implied constant.

In practise, it can arise that one uses the above notation not just for one map at a time but rather for a family of maps at the same time, or the map depends on some parameter one choose earlier.

Say, consider $f_t (x) = t x^2/(1+x^2)$ with $D= \mathbb{R}$ and $t$ a real, then for fixed $t$ it is true that there exists a constant $C$ for example $|t|$, such that
$|f_t(x)| \le C \cdot 1$ for all $x$ (or of course also just as $x \to \infty$).

So if $t$ is fixed one has $f_t \ll 1$. However the constant depends on $t$, and you cannot find a constant that works for all $t$ simultaneously. To express this more conveniently and clearly one typically writes this as $f_t \ll_t 1$.

More generally, a subscript or also multiple subscripts of $\ll$ indicate quantities on which the implied constant will (or can) depend.

Finally, the exact expression you give is somewhat unusual as Frank Thorne already explained. I share his opinion, and just would like to over an additional interpretation. It could be that somebody uses the subscripts in a different sense then the one I mentioned namely to indicate which quantitly tends to infinity. So that this could also mean $f(T) \ll 1$ as $T \to \infty$.

I hope this explication sheds some light on the matter, as opposed to causing additional confusion. As Todd Trimble said without additional context it will be impossible to decide what exactly is meant, since the usage is not uniform as Gerald Edgar stressed.