Questions tagged [ramanujan]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
1answer
96 views

What is the collection of series that amount to $\gamma$ deduced by Ramanujan?

On p. 20 of an article by Borwein et al., it is stated that Ramanujan could generalize the following formula due to Glaisher $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...
8
votes
1answer
2k views

Reference request: proof of Ramanujan's Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity, as stated here, is $$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...
10
votes
1answer
339 views

Closed Form for $\sum\limits_{n=-2a}^\infty(n+a){2a\choose-n}^4,~a\not\in\mathbb Z$

Do either $~S_4^+(a)~=~\displaystyle\sum_{n=0}^\infty(n+a){2a\choose n}^4~$ or $~S_4^-(a)~=~\displaystyle\sum_{n=-2a}^\infty(n+a){2a\choose-n}^4~$ possess a meaningful closed form expression1 in terms ...
5
votes
0answers
189 views

On rational Ramanujan-type series for $1/\pi$

A Ramanujan-type series for $1/\pi$ is a series of the following form $$ \sum_{n=0}^{\infty} \frac{(\frac12)_n(\frac{1}{s})_n(1-\frac{1}{s})_n}{(1)_n^3} (bn+a) z^n=\frac{1}{\pi}, $$ where $(c)_n=c(c+1)...
1
vote
0answers
44 views

Non-vanishing Taylor coefficients and Poincaré series

I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had. The table is found in the book "The 1-2-3 of Modular Forms" by Jan ...
3
votes
2answers
637 views

Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
3
votes
0answers
74 views

Maass forms associated with Ramanujan's mock theta functions

If you perform a fulltext search on the L-functions and modular forms database for "Ramanujan", you get entries related to the $\tau$ function and the modular discriminant $\Delta$. If you want the ...
8
votes
0answers
222 views

Searching for hypergeometric motives that split

Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...
1
vote
1answer
86 views

Solutions to Diophantine equation for Ramanujan graph construction

I am working with so-called Ramanujan graphs which have the property that a lower bound on their girth can be stated. I am reading about those in paper On the Construction of Turbo Code Interleavers ...
8
votes
1answer
2k views

Percentage of Ramanujan's conjectures that were proven correct

Today I read the following brief but insightful account of Ramanujan's approach to mathematics: https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf and while reading this I wondered whether we ...
17
votes
1answer
680 views

Congruences Ramanujan-style

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating ...
11
votes
3answers
423 views

Evaluating the sum of $kl^2$ over $p,q,k,l$ such that $pk +ql = n$

I have come across the following sum: $$\sum_{\substack{p, q, k, l \in \mathbb{N} \\ k > l \\ pk + ql = n}}kl^2$$ and I am trying to simplify it, hoping to get a nice formula in terms of $n$ and ...
18
votes
1answer
517 views

On a pattern for upside-down Ramanujan pi formulas

Define, $$\lambda_n =\frac{(\tfrac12)_n}{(1)_n} =\frac{(\tfrac12)_n}{n!} =\frac{\tbinom{2n}{n}}{2^{2n}} =\binom{n-\tfrac12}{n}$$ with Pochhammer symbol $(x)_n$ and binomial $\tbinom{n}{k}$. I noticed ...
40
votes
2answers
10k views

What did Ramanujan get wrong?

Quoting his Wikipedia page (current revision): compiled nearly 3,900 results Nearly all his claims have now been proven correct Which of his claims have been disproven, can any insight be ...
37
votes
1answer
3k views

A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives

The question below is the follow-up of this question on MathOverflow. Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
1
vote
1answer
181 views

How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold?

While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is \begin{equation} \prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^...
6
votes
0answers
229 views

Ramanujan Pi formula and idoneal numbers

The following question arises when I went through an old table of Heinrich Martin Weber. It is obvious that Weber was trying to calculate class invariants related to idoneal numbers, where the ...
6
votes
2answers
822 views

12th grade - Ramanujan Partition theory

I've been really trying to prove Ramanujan Partition theory, and different sources give me different explanations. Can someone please explain how Ramanujan (and Euler) found out the following ...
2
votes
0answers
190 views

Is there a proof that all the following expressions for Ramanujan's summation are equal?

$$\sum _{x\ge0}^\Re f(x)= -\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(0)}{k!}$$ where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$ $$\sum _{x\ge0}^\Re f(x)=\sum _{k=1}^\infty \frac {...
4
votes
0answers
370 views

A sum of Ramanujan sums

I have the following question about Ramanujan sums. (All vectors and matrices here will be understood to have integer entries.) Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...
2
votes
0answers
202 views

Elliptic functions and “complex multiplication” [closed]

I have before me a copy of "The Indian Mathematician Ramanujan", by G. H. Hardy (not actually related to me, as far as I know), which appeared in volume 44, number 3 (March 1937) of The American ...
3
votes
0answers
242 views

A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form $$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots (x)_{k_{r-1}-k_r}(x)_{k_r}}=\...
2
votes
0answers
165 views

Modular forms related to $G(q)$ and $H(q)$

If $G(q),H(q)$ are the functions appearing in Rogers-Ramanujan identities $$G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}$$ and $$H(q)=\...
13
votes
3answers
2k views

Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: \begin{align}\frac{1}{2\pi\sqrt{2}} &= \frac{1103}{99^{2}} + \frac{...
20
votes
1answer
2k views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$. 1. Define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) &...
21
votes
0answers
2k views

Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation $$ [m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}. $$ Consider the following trigonometric numbers:...
25
votes
2answers
3k views

Potential modularity and the Ramanujan conjecture

A little background: Let $f(z)=\sum_{n=1}^{\infty} a(n) e^{2\pi i nz}$ be a classical holomorphic cuspidal eigenform on $\Gamma_1(N)$, of weight $k \geq 2$ normalized with $a(1)=1$. The Ramanujan ...