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Questions tagged [ramanujan]

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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 ...
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On the necessitation of $(-1)^n$ within the series expansion of $f(x)$ concerning the usage of Ramanujan's Master Theorem

Ramanujan's well known Master Theorem states that the series expansion of the transformed function $f(x)$ has to be in form of $$f(x)~=~\sum_{n=0}^{\infty}(-1)^n\frac{\phi(n)}{n!}x^n\tag1$$ ...
17
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1answer
636 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
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3answers
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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 ...
16
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1answer
450 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 ...
39
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2answers
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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 ...
36
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1answer
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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
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1answer
180 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^...
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0answers
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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
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2answers
660 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 ...
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0answers
187 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 {...
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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\...
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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 ...
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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}}=\...
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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)=\...
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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{...
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1answer
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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) &...
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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:...
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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 ...