In my course on Modular forms (Lemma 11.5, p. 31) I use the same argument as Keith and zeb but in a different group theoretic context.

How much the Fourier analysis and the theory of Laplace transforms would be different if we assumed Dirac Delta to be a function $\overline{\delta}(x)$ rather than distribution $\delta(x)$, in other words, a function such that at zero it takes some infinitely-large value (say, $\omega/\pi$, equiv...

"One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large gaps in the primes." So begins the paper by Gethner, Wagon, and Wick, "A Stroll Through the Gaus...

Ramanujan observed the congruence $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$, where $\tau$ is the Ramanujan $\tau$-function. Does anybody know how he proved it, or would anybody venture an educated guess? I know there is a proof in https://faculty.math.illinois.edu/~berndt/articles/pt.pdf, access...

I have two questions about q-continued fractions, but a little intro first. Given Ramanujan's theta function, $$f(a,b) = \sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$ then the following, $$A(q) = q^{1/8} \frac{f(-q,-q^3)}{f(-q^2,-q^2)}$$ $$B(q) = q^{1/5} \frac{f(-q,-q^4)}{f(-q^2,-q^3)}$$...

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on Ramanujan, there is a link to a collection of problems posed by him. The page has a collection of about sixty proble...