Questions tagged [ramanujan]

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5
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1answer
219 views

Ramanujan and his influence on others

A few years ago I saw a paper where a few important researchers were asked which theorem of Ramanujan impressed them most. I don't remember details, but I would like to see this paper again. Details, ...
1
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1answer
255 views

A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

Ramanujan's claim on page 98 in the book Ramanujan's note book part 1 by Bruce C. Berndt, is that $$ \sum_{k=1}^{\infty} (−1)^{k−1}\frac{x^{pk}}{k(k!)^p}∼p\ln (x)+p\gamma,\quad p>0 \label{1}\tag{1} ...
6
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1answer
168 views

Statement of classical Ramanujan-Petersson conjecture

I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...
3
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0answers
107 views

A formula in Ramanujan's lost notebook and its connection with Chudnovsky series for $1/\pi$

While studying Berndt's Ramanujan's Lost Notebook Vol. 2, page 369 (chapter on Springerlink), I found that Ramanujan gave values of a certain expression $$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{...
29
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5answers
1k views

The unproved formulas of Ramanujan

Are there any formulas due to Ramanujan that have still not been proved—or disproved? If so, what are they? I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^...
1
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0answers
118 views

Ramanujan's infinite sum for pi

Ramanujan's famous pi formula states that \begin{equation} \frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^4}\frac{26390k+1103}{396^{4k}} \end{equation} How can one prove this?...
4
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2answers
329 views

Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

This question is related to the last question about van der Pol's identity for the sum of divisors. In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
1
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0answers
129 views

An explicit solution to the congruence $x^2\equiv 14(\frac 3p)-(\frac p3)-12\pmod {p}$?

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Then $$14\left(\frac 3p\right)-\left(\frac p3\right)-12=\begin{cases}1&\text{if}\ p\equiv1\pmod{12}, \\-25&\text{if}\ ...
11
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1answer
663 views

What is the roadblock in the discovery of new taxicab numbers?

The $n$-th taxicab number, denoted $\text{Ta}(n)$, is the smallest integer that can be expressed as a sum of two positive integer cubes in $n$ different distinct ways. $\text{Ta}(1) = 2 = 1^3 + 1^3$ ...
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0answers
654 views

Two curious series for $1/\pi$

On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have $$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$ where $(\frac{\cdot}p)$ ...
9
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1answer
805 views

What were Ramanujan's standard tricks/approaches to solving problems?

While trying to formulate an answer to this question, I realized I really have no idea how Ramanujan came up with his formulas. Bruce Berndt has a number of great expository articles, e.g., this one, ...
2
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1answer
114 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 $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...
9
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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}=...
11
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1answer
369 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
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0answers
211 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)...
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0answers
46 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 ...
7
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3answers
1k 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
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0answers
88 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 ...
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0answers
240 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
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1answer
97 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
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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
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1answer
694 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
463 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 ...
19
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1answer
569 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 ...
43
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2answers
11k 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 gained ...
38
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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
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1answer
191 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
239 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
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2answers
884 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
192 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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0answers
384 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\...
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0answers
207 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 ...
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0answers
246 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}}=\...
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0answers
171 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)=\...
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1answer
441 views

A partition congruence modulo 13

In the paper "Note on certain modular relations considered by Messrs Ramanujan, Darling and Rogers" (Proceedings of London Mathematical Society (1922) s2-20 (1): 408-416) Mordell gives proofs of the ...
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4answers
3k 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{...
22
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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) &...
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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:...
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2answers
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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 ...