I was reading Hoffstein¹, page 152. They define the counter part of the famous big-$\mathcal{O}$ notation: the big-$\Omega$,
Similarly, we say that $f$ is big-$\Omega$ of g and write, $$f(X)=\Omega(g(X))$$ if there are positive constants $c$ and $C$ such that $$f (X) \geq cg(X) \quad\text{for all} \quad X \geq C.$$
Which reads pretty standard. But, they have a footnote saying,
Note: Big-$\Omega$ notation as used by computer scientists and cryptographers does not mean the same thing as the big-$\Omega$ notation of mathematicians. In mathematics, especially in the field of analytic number theory, the expression $f(n) = \Omega(g(n))$ means that there is a constant $c$ such that there are infinitely many integers $n$ such that $f (n) \geq cg(n)$. In this book we use the computer science definition.
Errmm... buh how is that any different than what they defined as the computer scientist definition? The only thing I can imagine is that in case of the mathematician definition as noted in the footnote, the integers $n$ do not have to be consecutive whereas in the C. S. definition, $X \geq C$ are consecutive? Even then I think the C. S. definition is implied by the mathematician definition. Am I right? Or am I missing some other distinction that the book wants to make?
¹Silverman, J. H., Pipher, J., & Hoffstein, J. (2008). An introduction to mathematical cryptography. Springer New York.
The math definition is that there exists some infinite subsequence $I=\{n_i:i\geq 1\}$of the natural numbers and a positive constant $c$ such that for all $n\in I,$ $$ f(n)\geq c g(n). $$
One could have $I=\{100,110,120,\ldots \}=\{100+10k: k\geq 0\}$ for example and a function $f(n)$ which is equal to $n^2$ when $n\in I$ and zero otherwise would satisfy this definition.
The CS definition is that there is some positive $N$ and a positive constant $c$ such that for all $n\geq N,$ $$ f(n)\geq c g(n). $$
So if $f$ satisfies the CS definition with $N=10^{100}$ it is clear that the math definition will also be satisfied since the intersection $$ I \cap \{n: n\geq 10^{100}\}=\{10^{100}+k:k \geq 0\} $$ is itself an infinite subsequence of the natural numbers.