What do following asymptotic symbols mean?

Question

What do these symbols mean? I see them in analytic number theory.

$$\ll$$ $$\gg$$ $$\ll_\epsilon$$ $$\gg_\epsilon$$ $$\asymp$$ $$\sim$$

All these appear in here http://www.math.uiuc.edu/~lenfuchs/Binaryrep.pdf.

Answer 1

These are all completely standard notations, and are used often in analytic number theory. (I've used them all at some point or another.) I would say that mathematicians working in analytic number theory would all be pretty comfortable with these notations and would not expect them to be defined in a paper. In a textbook, on the other hand, they'd probably be written down somewhere: for example, I believe there is a list of such definitions at the beginning of Montgomery and Vaughan's book.

We say that $f(x) \ll g(x)$ (implicitly understood as $x \to \infty$) if $f(x) = O(g(x))$, or equivalently if \[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)} = C < \infty\] for some nonnegative constant $C$. Note that $g(x)$ is usually implicitly assumed to be a positive function, at least for all $x$ sufficiently large.

Similarly, $f(x) \gg g(x)$ (again implicitly understood as $x \to \infty$) if $g(x) = O(f(x))$, or equivalently if \[\limsup_{x \to \infty} \frac{g(x)}{f(x)} = C < \infty\] for some nonnegative constant $C$. Here both $f(x)$ and $g(x)$ are implicitly understood to be positive functions.

The notation $\ll_{\varepsilon}$ implies that the implicit constant $C$ depends on the parameter $\varepsilon$. For example, if we let $f(x) = \sum_{n \leq x} \mu(n)$, the Mertens function, and $g(x) = x^{1/2 + \varepsilon}$, which depends on the parameter $\varepsilon$, then the Riemann hypothesis is equivalent to the statement $f(x) \ll_{\varepsilon} g(x)$. This is also written as $f(x) = O_{\varepsilon}(g(x))$. This means that \[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)} = C < \infty\] for some nonnegative constant $C = C(\varepsilon)$ that is allowed to depend on the parameter $\varepsilon$.

The notation $\gg_{\varepsilon}$ is defined analogously.

The notation $f(x) \asymp g(x)$ means that both $f(x) \gg g(x)$ and $f(x) \ll g(x)$ hold simultaneously. For an example, take $f(x) = x + \sin x$ and $g(x) = x$.

The notation $f(x) \sim g(x)$ is occasionally also written as $f(x) = (1 + o(1)) g(x)$ or $f(x) = g(x) + o(g(x))$. It simply means that $f(x) - g(x) = o(g(x))$, or equivalently that \[\limsup_{x \to \infty} \frac{|f(x) - g(x)|}{g(x)} = 0.\]

The one other common definition that you don't mention is $\Omega$. We write $f(x) = \Omega(g(x))$ if \[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)}> 0.\] It may be the case that this limit is $\infty$, and this is certainly allowed. For example, take $f(x) = x^2$ and $g(x) = x$. One can be more precise and write $f(x) = \Omega_+(g(x))$ if \[\limsup_{x \to \infty} \frac{f(x)}{g(x)} > 0,\] and similarly $f(x) = \Omega_-(g(x))$ if \[\liminf_{x \to \infty} \frac{f(x)}{g(x)} < 0.\]

Answer 2

My guess would be the following: $$ f \ll g \iff f \in \mathcal{O}(g) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | < \infty$$ The next I'm not sure, but I'd guess he means $$ f \ll_{\epsilon} g(\epsilon) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | = M(\epsilon) < \infty $$ Then $$ f \asymp g \iff f \ll g\quad \& \quad g \ll f$$ Lastly $$ f \sim g \iff \lim_{X \to \infty} \frac{f}{g} = 1 $$

Answer 3

The authors should have explained their notation, so shame on Bourgain and Fuchs. I've never seen something like $\leq_{\epsilon}$.