17
votes
1answer
480 views

What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
6
votes
3answers
1k views

Sum of the sum-of-divisors function

I was looking at the abstract of a paper [1] which claims that [2] and [3] prove $$ \sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x). $$ But I cannot find the above—or indeed, ...
12
votes
0answers
525 views

Where might I find a scanned handwritten copy of Ramanujan's second letter to Hardy?

I am giving a lecture to undergraduates on the lovely identity $$1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}.$$ Ramanujan wrote in his second letter to Hardy (courtesy Wikipedia), "Dear Sir, I am very ...
5
votes
5answers
2k views

Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...
2
votes
1answer
495 views

Variants of Grönwall's theorem

Except the original Grönwall's theorem that $$\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is ...
9
votes
4answers
1k views

Who first proved that there are at least n^(1-ε) primes up to n?

It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any ...
11
votes
5answers
1k views

Historical question in analytic number theory

The analytic continuation and functional equation for the Riemann zeta function were proved in Riemann's 1859 memoir "On the number of primes less than a given magnitude." What is the earliest ...