Questions tagged [ramanujan]
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59
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How to find the right path of integration to get the asymptotic partition formula
I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers.
I am ...
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2
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Citation for the ill-posed Ramanujan's problem
I've seen many times the following problem posed by Ramanujan:
$$\sqrt{1+2{\sqrt{1+3{\sqrt{1+\cdots}}}}} = \mbox{?}$$
This problem is also mentioned on Ramanujan's Wikipedia page along with the ...
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Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?
I. Recurrences
(Continued from this post.) In Cooper's 2012 paper, "Sporadic sequences, modular forms and new series for 1/π", he did a computer search for the recurrence relation,
$$(n+1)^3 ...
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Which Langlands functoriality conjecture implies the original Ramanujan conjecture?
I heard that the Langlands functoriality conjecture implies the Ramanujan conjecture for $GL(2)$. especially for the Maass form.
There are various versions of the Langlands functoriality concerning to ...
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Can we calculate the spectral radius of the universal cover for specific graphs?
Background
For a finite graph $G$, let $\tilde{G}$ denote the universal cover of $G$. For a vertex $v$, let $p_{2n}(v)$ denote the number of paths of length $2n$ that start and end at $v$. The ...
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Fourier series of Eisenstein series — elegant and very good approximation
I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...
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Explicit constructions of Ramanujan graphs
I am trying to find a list of all the explicit constructions of Ramanujan graphs. By a Ramanujan graph I mean a $k$-regular multi-graph $G$ such that all the non-trivial eigenvalues $\lambda$ of the ...
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Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers
The classical rational Ramanujan-type series for $1/\pi$ have the following four forms:
\begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1}
\\\sum_{k=0}^\...
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Surprising symmetry in the Ramanujan bound
The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy
$$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$
With a ...
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What will be the smallest value of $k$ such that $P(k)=m$?
Let us suppose that we have a group of order $p^k$, where $p$ is prime.In General,there is one group group of order $p^k$ for each set of positive integers whose sum is $k$(such a set is called ...
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Class numbers of functions fields and spanning trees
In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-...
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Conceptual explanation for the gap in the spectrum of biregular trees
Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval
$$[-2\sqrt{q}, 2\sqrt{q}].$$
The reason for this ...
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Impact of Ramanujan's Note on a set of simultaneous equations
I had been pointed to Ramanujan's 1912 article Note on a set of simultaneous equations in this answer to my former question about the Solvability of a system of polynomial equations.
While the ...
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Ramanujan's type sum
Is it possible to get a good upper bound for $$\sum_{1\leq |h|\leq q}\frac{c_{q}(a-h)}{h}$$ with $(a,q)=1$ and $1\leq a\leq q$.
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Ramanujan and his influence on others
A few years ago I saw a paper where a few important researchers were asked which theorem of Ramanujan impressed them most.
I don't remember details, but I would like to see this paper again.
Details, ...
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A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$
Ramanujan's claim on page 98 in the book Ramanujan's note book part 1 by Bruce C. Berndt, is that
$$
\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}
...
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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 ...
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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 $$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{...
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The unproved formulas of Ramanujan
Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^...
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Ramanujan's infinite sum for pi
Ramanujan's famous pi formula states that
\begin{equation}
\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^4}\frac{26390k+1103}{396^{4k}}
\end{equation}
How can one prove this?...
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Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
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An explicit solution to the congruence $x^2\equiv 14(\frac 3p)-(\frac p3)-12\pmod {p}$?
Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Then
$$14\left(\frac 3p\right)-\left(\frac p3\right)-12=\begin{cases}1&\text{if}\ p\equiv1\pmod{12},
\\-25&\text{if}\ ...
- 13.9k
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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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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 $(\frac{\cdot}p)$ ...
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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, ...
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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 $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...
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Solvability of a system of polynomial equations
What can be said about the solutions of, resp. solving, the following system polynomial equations, in which the $x_i$ and $y_j$ are the variables and $c_{i,j},\,d_i\in\mathbb{R}$ are constants:
$$\...
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Reference request: proof of Ramanujan's Cos/Cosh Identity
The Ramanujan Cos/Cosh Identity, as stated here, is
$$\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}=...
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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 ...
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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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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 ...
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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, ...
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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 gained ...
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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 ...
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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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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 ...
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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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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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How did Ramanujan discover this identity?
Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64 F_6 F_{10}=45 F_8^2$$
This fascinating identity is due to Ramanujan and can be found in "...
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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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