Every Discovery. One Living Archive.

Explore Ramanujan's identities, theorems, conjectures, formulas, and notebook entries.

Complete Catalogue

Browse 177 discovery and contribution-family records grouped by field, source, result type and difficulty.

Source-Aware Notes

Each entry distinguishes published papers, notebooks, later names, collaborations and records still needing exact citation review.

Study Path

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177 entries shown from 177 detailed discovery and contribution-family records.

Partition Theory

Partition Generating Function

n = 0 \sum \infty p (n) q^{n} = m = 1 \prod \infty \frac{1}{1 - q ^{m}}

Euler knew the generating function; Ramanujan revealed extraordinary arithmetic properties of its coefficients.

Level: Intermediate
Source: Source catalogue
Status: needs-review

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Partition Theory | 1919

Ramanujan's Partition Congruences

p (5 n + 4) \equiv 0 (mod 5)

Ramanujan discovered striking congruence properties of partition numbers.

Level: Intermediate
Source: Published Papers
Status: source-supplied

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Partition Theory

Higher-Power Partition Congruences

p (5^{k} n + δ_{k}) \equiv 0 (mod 5^{k})

Higher-Power Partition Congruences is catalogued as a contribution family in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory | 1918

Hardy-Ramanujan Asymptotic Formula

p (n) \sim \frac{e ^{π 2 n /3}}{4 n 3}

A collaboration with G. H. Hardy giving the growth of p(n).

Level: Advanced
Source: Hardy-Ramanujan
Status: source-supplied

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Partition Theory

Hardy-Ramanujan Circle Method

Contribution family

Hardy-Ramanujan Circle Method is catalogued as a method in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory

Restricted Partition Identities

Contribution family

Restricted Partition Identities is catalogued as a identity in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory

Partition Rank and Crank Context

Contribution family

Partition Rank and Crank Context is catalogued as a concept in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory

Partition Identities from the Lost Notebook

Contribution family

Partition Identities from the Lost Notebook is catalogued as a contribution family in Partition Theory.

Level: Advanced
Source: Lost Notebook
Status: source-supplied

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Mock Theta Functions | 1919-1920

Mock Theta Functions

f (q) = n = 0 \sum \infty \frac{q ^{n^{2}}}{( - q ; q ) _{n}^{2}}

Mock Theta Functions is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Lost Notebook
Status: source-supplied

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Mock Theta Functions

Third-Order Mock Theta Functions

f (q) = 1 + n = 1 \sum \infty \frac{q ^{n^{2}}}{( 1 + q ) ^{2} \dots ( 1 + q ^{n} ) ^{2}}

Third-Order Mock Theta Functions is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Fifth-Order Mock Theta Functions

Contribution family

Fifth-Order Mock Theta Functions is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Seventh-Order Mock Theta Functions

Contribution family

Seventh-Order Mock Theta Functions is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Tenth-Order Mock Theta Functions

Contribution family

Tenth-Order Mock Theta Functions is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Radial-Limit Behaviour

Contribution family

Radial-Limit Behaviour is catalogued as a concept in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Connection with Harmonic Maass Forms

Contribution family

Connection with Harmonic Maass Forms is catalogued as a concept in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Formulas for pi | 1914

Ramanujan's 1/pi Series

\frac{1}{π} = \frac{2 2}{9801} \sum \dots

Ramanujan's 1/pi Series is catalogued as a series in Formulas for pi.

Level: Advanced
Source: Published Papers
Status: source-supplied

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Formulas for pi

Family of Seventeen 1/pi Series

Contribution family

Family of Seventeen 1/pi Series is catalogued as a contribution family in Formulas for pi.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Formulas for pi

Ramanujan-Sato Series Context

Contribution family

Ramanujan-Sato Series Context is catalogued as a concept in Formulas for pi.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Formulas for pi

Modular-Equation Methods for 1/pi

Contribution family

Modular-Equation Methods for 1/pi is catalogued as a method in Formulas for pi.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

Ramanujan's General Theta Function

f (a, b) = n = - \infty \sum \infty a^{n (n + 1) /2} b^{n (n - 1) /2}

Ramanujan's General Theta Function is catalogued as a function in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

phi(q)

φ (q) = n = - \infty \sum \infty q^{n^{2}}

phi(q) is catalogued as a function in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

psi(q)

ψ (q) = n = 0 \sum \infty q^{n (n + 1) /2}

psi(q) is catalogued as a function in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

f(-q)

f (- q) = n = 1 \prod \infty (1 - q^{n})

f(-q) is catalogued as a function in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

Theta Addition and Subtraction Formulas

Contribution family

Theta Addition and Subtraction Formulas is catalogued as a identity in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

Theta Product Representations

f (a, b) = (- a; ab)_{\infty} (- b; ab)_{\infty} (ab; ab)_{\infty}

Theta Product Representations is catalogued as a identity in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

Theta Dissection Formulas

Contribution family

Theta Dissection Formulas is catalogued as a method in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Theta Functions

Theta-Function Quotients

Contribution family

Theta-Function Quotients is catalogued as a formula in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Representation of Integers by Quadratic Forms

Representations by Squares

r_{2} (n) = 4 d ∣ n \sum χ (d)

Representations by Squares is catalogued as a formula in Representation of Integers by Quadratic Forms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Representation of Integers by Quadratic Forms

Representations by Triangular Numbers

Contribution family

Representations by Triangular Numbers is catalogued as a formula in Representation of Integers by Quadratic Forms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Identities

First Rogers-Ramanujan Identity

n = 0 \sum \infty \frac{q ^{n^{2}}}{( q ; q ) _{n}} = \frac{1}{( q ; q ^{5} ) _{\infty} ( q ^{4} ; q ^{5} ) _{\infty}}

First Rogers-Ramanujan Identity is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Identities

Second Rogers-Ramanujan Identity

n = 0 \sum \infty \frac{q ^{n (n + 1)}}{( q ; q ) _{n}} = \frac{1}{( q ^{2} ; q ^{5} ) _{\infty} ( q ^{3} ; q ^{5} ) _{\infty}}

Second Rogers-Ramanujan Identity is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Identities

Partition Interpretations

Contribution family

Partition Interpretations is catalogued as a concept in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Identities

\sum \frac{q ^{n^{2}}}{( q ; q ) _{n}} = \frac{1}{( q , q ^{4} ; q ^{5} ) _{\infty}}

Rogers-Ramanujan Identities is catalogued as a contribution family in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Continued Fraction

R (q) = q^{1/5} \frac{1}{1 + \frac{q}{1 + \frac{q ^{2}}{1 + \dots}}}

Rogers-Ramanujan Continued Fraction is catalogued as a continued fraction in Rogers-Ramanujan Continued Fraction.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Continued Fraction

Product Form of the Continued Fraction

R (q) = q^{1/5} \frac{( q ; q ^{5} ) _{\infty} ( q ^{4} ; q ^{5} ) _{\infty}}{( q ^{2} ; q ^{5} ) _{\infty} ( q ^{3} ; q ^{5} ) _{\infty}}

Product Form of the Continued Fraction is catalogued as a formula in Rogers-Ramanujan Continued Fraction.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Continued Fraction

Exact Values and Modular Equations

Contribution family

Exact Values and Modular Equations is catalogued as a formula in Rogers-Ramanujan Continued Fraction.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Rogers-Ramanujan Continued Fraction

Golden-Ratio Connections

R (e^{- 2 π}) = \frac{5 + 5}{2} - \frac{1 + 5}{2}

Golden-Ratio Connections is catalogued as a concept in Rogers-Ramanujan Continued Fraction.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Other Continued Fractions

Ramanujan's Cubic Continued Fraction

G (q) = \frac{q ^{1/3}}{1 + \frac{q + q ^{2}}{1 + \frac{q ^{2} + q ^{4}}{1 + \dots}}}

Ramanujan's Cubic Continued Fraction is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Other Continued Fractions

Ramanujan-Gollnitz-Gordon Continued Fraction

Contribution family

Ramanujan-Gollnitz-Gordon Continued Fraction is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Gamma-Function Identities

Continued Fractions for Gamma Ratios

Contribution family

Continued Fractions for Gamma Ratios is catalogued as a continued fraction in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Other Continued Fractions

Continued Fractions for Theta Quotients

Contribution family

Continued Fractions for Theta Quotients is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Other Continued Fractions

Continued Fractions Associated with Modular Equations

Contribution family

Continued Fractions Associated with Modular Equations is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Other Continued Fractions

Continued Fractions for Exponential and Trigonometric Functions

Contribution family

Continued Fractions for Exponential and Trigonometric Functions is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

Ramanujan's 1 psi 1 Summation

_{1} ψ_{1} (a; b; q, z) = n = - \infty \sum \infty \frac{( a ; q ) _{n}}{( b ; q ) _{n}} z^{n}

Ramanujan's 1 psi 1 Summation is catalogued as a theorem in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

Bilateral Basic Hypergeometric Series

Contribution family

Bilateral Basic Hypergeometric Series is catalogued as a series in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

Partial Theta Functions

Contribution family

Partial Theta Functions is catalogued as a function in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

False Theta Functions

Contribution family

False Theta Functions is catalogued as a function in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

q-Binomial Identities

\frac{1}{( z ; q ) _{\infty}} = n = 0 \sum \infty \frac{z ^{n}}{( q ; q ) _{n}}

q-Binomial Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

q-Series Dissections

Contribution family

q-Series Dissections is catalogued as a method in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

Root-of-Unity Identities

Contribution family

Root-of-Unity Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Infinite Products

Infinite-Product Transformations

(a; q)_{\infty} = k = 0 \prod \infty (1 - a q^{k})

Infinite-Product Transformations is catalogued as a identity in Infinite Products.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Tau Function | 1916

Ramanujan's Tau Function

Δ (q) = n = 1 \sum \infty τ (n) q^{n}

Ramanujan's Tau Function is catalogued as a function in Ramanujan's Tau Function.

Level: Advanced
Source: Published Papers
Status: source-supplied

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Ramanujan's Tau Function

Tau Multiplicativity

τ (mn) = τ (m) τ (n) ((m, n) = 1)

Tau Multiplicativity is catalogued as a conjecture in Ramanujan's Tau Function.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Tau Function

Tau Prime-Power Recurrence

τ (p^{r + 1}) = τ (p) τ (p^{r}) - p^{11} τ (p^{r - 1})

Tau Prime-Power Recurrence is catalogued as a formula in Ramanujan's Tau Function.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Tau Function

Ramanujan-Petersson Conjecture

∣ τ (p) ∣ \leq 2 p^{11/2}

The modern theorem and naming developed after Ramanujan's original conjectural bounds.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Tau Function

Tau-Function Congruences

τ (n) \equiv σ_{11} (n) (mod 691)

Tau-Function Congruences is catalogued as a formula in Ramanujan's Tau Function.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Sums

c_{q} (n) = 1 \leq a \leq q (a, q) = 1 \sum e^{2 π ian / q}

Ramanujan Sums is catalogued as a formula in Ramanujan Sums.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Sums

Ramanujan Expansions

f (n) \sim q = 1 \sum \infty a_{q} c_{q} (n)

Ramanujan Expansions is catalogued as a method in Ramanujan Sums.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Highly Composite Numbers | 1915

Highly Composite Numbers

m < n \Rightarrow d (m) < d (n)

Highly Composite Numbers is catalogued as a concept in Highly Composite Numbers.

Level: Intermediate
Source: Published Papers
Status: source-supplied

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Highly Composite Numbers

Superior Highly Composite Numbers

Contribution family

Superior Highly Composite Numbers is catalogued as a concept in Highly Composite Numbers.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Divisor Functions

σ_{k} (n) = d ∣ n \sum d^{k}

Divisor Functions is catalogued as a function in Divisor Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Divisor Functions

Divisor-Sum Convolution Identities

Contribution family

Divisor-Sum Convolution Identities is catalogued as a identity in Divisor Functions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Prime-Number Theory | 1919

Ramanujan Primes

π (x) - π (x /2) \geq n

Ramanujan Primes is catalogued as a concept in Prime-Number Theory.

Level: Intermediate
Source: Source catalogue
Status: needs-review

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Approximation to the Prime-Counting Function

Prime-Counting Approximations

π (x) \sim Li (x)

Prime-Counting Approximations is catalogued as a approximation in Approximation to the Prime-Counting Function.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Master Theorem | 1918

Ramanujan's Master Theorem

\int_{0}^{\infty} x^{s - 1} ϕ (x) d x = \frac{π}{sin π s} ϕ (- s)

Ramanujan's Master Theorem is catalogued as a theorem in Ramanujan's Master Theorem.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan's Factorial Approximation

n! \sim π (\frac{n}{e})^{n} 6 8 n^{3} + 4 n^{2} + n + \frac{1}{30}

Ramanujan's Factorial Approximation is catalogued as a approximation in Ramanujan's Factorial Approximation.

Level: Advanced
Source: Source catalogue
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Ramanujan Summation of Divergent Series

Ramanujan Summation

1 + 2 + 3 + \dots = - \frac{1}{12}

Ramanujan Summation is catalogued as a method in Ramanujan Summation of Divergent Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Gamma-Function Identities

Γ (z + 1) = z Γ (z)

Gamma-Function Identities is catalogued as a identity in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Definite Integrals and Integral Transforms

Definite Integrals

Contribution family

Definite Integrals is catalogued as a formula in Definite Integrals and Integral Transforms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Definite Integrals and Integral Transforms

Mellin Transforms

M {f} (s) = \int_{0}^{\infty} f (x) x^{s - 1} d x

Mellin Transforms is catalogued as a method in Definite Integrals and Integral Transforms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Asymptotic Expansions

Contribution family

Asymptotic Expansions is catalogued as a method in Asymptotic Expansions.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Odd Zeta Values

Odd Zeta-Value Transformations

Contribution family

Odd Zeta-Value Transformations is catalogued as a formula in Odd Zeta Values.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Odd Zeta Values

Formula for zeta(3)

ζ (3) = n = 1 \sum \infty \frac{1}{n ^{3}}

Formula for zeta(3) is catalogued as a formula in Odd Zeta Values.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Bernoulli Numbers

Bernoulli-Number Identities

\frac{x}{e ^{x} - 1} = n = 0 \sum \infty B_{n} \frac{x ^{n}}{n !}

Bernoulli-Number Identities is catalogued as a identity in Bernoulli Numbers.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Ramanujan Polynomials

Contribution family

Ramanujan Polynomials is catalogued as a concept in Ramanujan Polynomials.

Level: Advanced
Source: Source catalogue
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Diophantine Equations | 1913

Ramanujan-Nagell Equation

x^{2} + 7 = 2^{n}

Ramanujan-Nagell Equation is catalogued as a equation in Diophantine Equations.

Level: Intermediate
Source: Source catalogue
Status: needs-review

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Named Constants Connected with Ramanujan

Hardy-Ramanujan Number 1729

1729 = 1^{3} + 1 2^{3} = 9^{3} + 1 0^{3}

Hardy-Ramanujan Number 1729 is catalogued as a concept in Named Constants Connected with Ramanujan.

Level: Accessible
Source: Source catalogue
Status: needs-review

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Nested Radicals

Infinite Nested Radical

1 + 2 1 + 3 1 + \dots = 3

Infinite Nested Radical is catalogued as a formula in Nested Radicals.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Nested Radicals

General Nested-Radical Identities

Contribution family

General Nested-Radical Identities is catalogued as a contribution family in Nested Radicals.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Representation of Integers by Quadratic Forms

Sums of Squares

Contribution family

Sums of Squares is catalogued as a formula in Representation of Integers by Quadratic Forms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Representation of Integers by Quadratic Forms

Quadratic Forms

Contribution family

Quadratic Forms is catalogued as a contribution family in Representation of Integers by Quadratic Forms.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Named Constants Connected with Ramanujan

Landau-Ramanujan Constant

K = \frac{1}{2} p \equiv 3 (4) \prod (1 - p^{- 2})^{- 1/2}

Landau-Ramanujan Constant is catalogued as a concept in Named Constants Connected with Ramanujan.

Level: Advanced
Source: Source catalogue
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Named Constants Connected with Ramanujan

Ramanujan-Soldner Constant

Contribution family

Ramanujan-Soldner Constant is catalogued as a concept in Named Constants Connected with Ramanujan.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Near-Integer Phenomena

Near-Integer e^(pi sqrt 163)

e^{π 163} \approx 262537412640768743.99999999999925

Near-Integer e^(pi sqrt 163) is catalogued as a approximation in Near-Integer Phenomena.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Magic Squares

Ramanujan's Magic Square

22 + 12 + 18 + 87 = 139

Ramanujan's Magic Square is catalogued as a concept in Magic Squares.

Level: Accessible
Source: Source catalogue
Status: needs-review

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Eisenstein and Modular Series

Eisenstein Series E2

E_{2} (q) = 1 - 24 n = 1 \sum \infty σ_{1} (n) q^{n}

Eisenstein Series E2 is catalogued as a series in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Eisenstein and Modular Series

Eisenstein Series E4

E_{4} (q) = 1 + 240 n = 1 \sum \infty σ_{3} (n) q^{n}

Eisenstein Series E4 is catalogued as a series in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Eisenstein and Modular Series

Eisenstein Series E6

E_{6} (q) = 1 - 504 n = 1 \sum \infty σ_{5} (n) q^{n}

Eisenstein Series E6 is catalogued as a series in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
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Eisenstein and Modular Series

Ramanujan Differential Equations

q \frac{d E _{2}}{d q} = \frac{E _{2}^{2} - E _{4}}{12}

Ramanujan Differential Equations is catalogued as a formula in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
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Singular Moduli and Class Invariants

Singular Modulus Evaluations

Contribution family

Singular Modulus Evaluations is catalogued as a formula in Singular Moduli and Class Invariants.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Singular Moduli and Class Invariants

Singular Moduli

Contribution family

Singular Moduli is catalogued as a concept in Singular Moduli and Class Invariants.

Level: Advanced
Source: Source catalogue
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Singular Moduli and Class Invariants

Class Invariants

Contribution family

Class Invariants is catalogued as a concept in Singular Moduli and Class Invariants.

Level: Advanced
Source: Source catalogue
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Infinite Products

Eta Quotients

η (τ) = q^{1/24} n = 1 \prod \infty (1 - q^{n})

Eta Quotients is catalogued as a formula in Infinite Products.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Modular Equations and Elliptic Functions

Modular Transformations

τ \mapsto \frac{a τ + b}{c τ + d}

Modular Transformations is catalogued as a method in Modular Equations and Elliptic Functions.

Level: Advanced
Source: Source catalogue
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Broader Named Concepts Resulting from His Work

Ramanujan Graphs

Contribution family

Ramanujan graphs were developed after his death and named through their connection with the Ramanujan conjecture.

Level: Advanced
Source: Source catalogue
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Broader Named Concepts Resulting from His Work

Ramanujan Complexes

Contribution family

Ramanujan Complexes is catalogued as a concept in Broader Named Concepts Resulting from His Work.

Level: Advanced
Source: Source catalogue
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Combinatorial Identities

Contribution family

Combinatorial Identities is catalogued as a identity in Combinatorial Identities.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Approximations to Constants

Contribution family

Approximations to Constants is catalogued as a approximation in Approximations to Constants.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Infinite Series

Infinite Series Transformations

Contribution family

Infinite Series Transformations is catalogued as a series in Infinite Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Infinite Products

Contribution family

Infinite Products is catalogued as a contribution family in Infinite Products.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory

Partition Dissection Formulas

Contribution family

Dissections of generating functions are a recurring tool in partition congruence work.

Level: Advanced
Source: Source catalogue
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Partition Theory

5-, 7-, and 11-Dissections of Partition Series

Contribution family

5-, 7-, and 11-Dissections of Partition Series is catalogued as a contribution family in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Partition Theory

Partition Congruences Modulo Prime Powers

p (25 n + 24) \equiv 0 (mod 25)

Partition Congruences Modulo Prime Powers is catalogued as a contribution family in Partition Theory.

Level: Advanced
Source: Source catalogue
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Partition Theory

q-Series for Partition Identities

Contribution family

q-Series for Partition Identities is catalogued as a series in Partition Theory.

Level: Advanced
Source: Source catalogue
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Partition Theory

Refinements of Partition Asymptotics

Contribution family

Refinements of Partition Asymptotics is catalogued as a contribution family in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Mock Theta Functions

Mock Theta Function f_0(q)

Contribution family

Mock Theta Function f_0(q) is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
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Mock Theta Functions

Mock Theta Function f_1(q)

Contribution family

Mock Theta Function f_1(q) is catalogued as a function in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
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Mock Theta Functions

Mock Theta Function omega(q)

Contribution family

Mock Theta Function omega(q) is catalogued as a function in Mock Theta Functions.

Level: Advanced
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Mock Theta Functions

Mock Theta Function nu(q)

Contribution family

Mock Theta Function nu(q) is catalogued as a function in Mock Theta Functions.

Level: Advanced
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Mock Theta Functions

Transformation Behaviour of Mock Theta Functions

Contribution family

Transformation Behaviour of Mock Theta Functions is catalogued as a concept in Mock Theta Functions.

Level: Advanced
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Mock Theta Functions

Asymptotics of Mock Theta Functions

Contribution family

Asymptotics of Mock Theta Functions is catalogued as a contribution family in Mock Theta Functions.

Level: Advanced
Source: Source catalogue
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Modular Equations and Elliptic Functions

Elliptic-Integral Modular Parameters

Contribution family

Elliptic-Integral Modular Parameters is catalogued as a concept in Modular Equations and Elliptic Functions.

Level: Advanced
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Modular Equations and Elliptic Functions

Modular Equations of Degree 3

Contribution family

Modular Equations of Degree 3 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
Source: Source catalogue
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Modular Equations and Elliptic Functions

Modular Equations of Degree 5

Contribution family

Modular Equations of Degree 5 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
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Modular Equations and Elliptic Functions

Modular Equations of Degree 7

Contribution family

Modular Equations of Degree 7 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
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Modular Equations and Elliptic Functions

Alpha-Beta Modular Relations

Contribution family

Alpha-Beta Modular Relations is catalogued as a formula in Modular Equations and Elliptic Functions.

Level: Advanced
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Modular Equations and Elliptic Functions

Elliptic-Function Transformations

Contribution family

Elliptic-Function Transformations is catalogued as a method in Modular Equations and Elliptic Functions.

Level: Advanced
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Formulas for pi

Elliptic-Modular Approximations to pi

Contribution family

Elliptic-Modular Approximations to pi is catalogued as a method in Formulas for pi.

Level: Advanced
Source: Source catalogue
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Formulas for pi

Chudnovsky-Ramanujan pi Context

Contribution family

Later algorithms for pi build on the modular world opened by Ramanujan's 1/pi series rather than being his own work.

Level: Advanced
Source: Source catalogue
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Formulas for pi

Class Invariants in 1/pi Series

Contribution family

Class Invariants in 1/pi Series is catalogued as a concept in Formulas for pi.

Level: Advanced
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Ramanujan Theta Functions

Ramanujan Theta Identities

Contribution family

Ramanujan Theta Identities is catalogued as a identity in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
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Ramanujan Theta Functions

Theta-Function Modular Transformations

Contribution family

Theta-Function Modular Transformations is catalogued as a method in Ramanujan Theta Functions.

Level: Advanced
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Ramanujan Theta Functions

Theta-Function Representation Formulas

Contribution family

Theta-Function Representation Formulas is catalogued as a formula in Ramanujan Theta Functions.

Level: Advanced
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Ramanujan Theta Functions

Jacobi Triple Product Context

f (a, b) = (- a; ab)_{\infty} (- b; ab)_{\infty} (ab; ab)_{\infty}

Jacobi Triple Product Context is catalogued as a concept in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
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Rogers-Ramanujan Identities

Schur Partition Theorem Context

Contribution family

Schur Partition Theorem Context is catalogued as a concept in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
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Rogers-Ramanujan Identities

Rogers-Ramanujan Product Identities

Contribution family

Rogers-Ramanujan Product Identities is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
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Rogers-Ramanujan Identities

Rogers-Ramanujan Modular Relations

Contribution family

Rogers-Ramanujan Modular Relations is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
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Other Continued Fractions

Ramanujan-Selberg Continued Fractions

Contribution family

This is a historically associated named continued-fraction family; exact attribution needs source review.

Level: Advanced
Source: Source catalogue
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Other Continued Fractions

Continued Fractions for Eulerian Functions

Contribution family

Continued Fractions for Eulerian Functions is catalogued as a continued fraction in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
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Other Continued Fractions

Exact Values of Ramanujan Continued Fractions

Contribution family

Exact Values of Ramanujan Continued Fractions is catalogued as a formula in Other Continued Fractions.

Level: Advanced
Source: Source catalogue
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q-Series and Basic Hypergeometric Series

q-Pochhammer Product Formulas

(a; q)_{n} = k = 0 \prod n - 1 (1 - a q^{k})

q-Pochhammer Product Formulas is catalogued as a formula in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
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q-Series and Basic Hypergeometric Series

Basic Hypergeometric Transformations

Contribution family

Basic Hypergeometric Transformations is catalogued as a method in q-Series and Basic Hypergeometric Series.

Level: Advanced
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q-Series and Basic Hypergeometric Series

Bilateral Series Transformations

Contribution family

Bilateral Series Transformations is catalogued as a method in q-Series and Basic Hypergeometric Series.

Level: Advanced
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q-Series and Basic Hypergeometric Series

q-Continued Fractions

Contribution family

q-Continued Fractions is catalogued as a continued fraction in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
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q-Series and Basic Hypergeometric Series

Partial Theta Identities

Contribution family

Partial Theta Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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q-Series and Basic Hypergeometric Series

False Theta Identities

Contribution family

False Theta Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
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Ramanujan's Tau Function

Generating Function for tau(n)

Δ (q) = q n = 1 \prod \infty (1 - q^{n})^{24}

Generating Function for tau(n) is catalogued as a formula in Ramanujan's Tau Function.

Level: Advanced
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Ramanujan's Tau Function

Tau Congruence Modulo 691

τ (n) \equiv σ_{11} (n) (mod 691)

Tau Congruence Modulo 691 is catalogued as a formula in Ramanujan's Tau Function.

Level: Advanced
Source: Source catalogue
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Ramanujan's Tau Function

Further Tau Congruences

Contribution family

Further Tau Congruences is catalogued as a contribution family in Ramanujan's Tau Function.

Level: Advanced
Source: Source catalogue
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Divisor Functions

Divisor-Function Congruences

Contribution family

Divisor-Function Congruences is catalogued as a identity in Divisor Functions.

Level: Advanced
Source: Source catalogue
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Divisor Functions

Sums of Powers and Divisor Sums

Contribution family

Sums of Powers and Divisor Sums is catalogued as a identity in Divisor Functions.

Level: Advanced
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Divisor Functions

Sigma-Function Identities

σ_{k} (n) = d ∣ n \sum d^{k}

Sigma-Function Identities is catalogued as a identity in Divisor Functions.

Level: Advanced
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Prime-Number Theory

Ramanujan's Prime-Interval Theorem

Contribution family

Ramanujan's Prime-Interval Theorem is catalogued as a theorem in Prime-Number Theory.

Level: Advanced
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Approximation to the Prime-Counting Function

Prime-Counting Inequalities

Contribution family

Prime-Counting Inequalities is catalogued as a contribution family in Approximation to the Prime-Counting Function.

Level: Advanced
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Approximation to the Prime-Counting Function

Logarithmic-Integral Corrections

Contribution family

Logarithmic-Integral Corrections is catalogued as a approximation in Approximation to the Prime-Counting Function.

Level: Advanced
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Ramanujan's Master Theorem

Mellin-Transform Context of the Master Theorem

Contribution family

Mellin-Transform Context of the Master Theorem is catalogued as a method in Ramanujan's Master Theorem.

Level: Advanced
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Definite Integrals and Integral Transforms

Ramanujan Integral Evaluations

Contribution family

Ramanujan Integral Evaluations is catalogued as a contribution family in Definite Integrals and Integral Transforms.

Level: Advanced
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Definite Integrals and Integral Transforms

Frullani-Type Integral Evaluations

Contribution family

Frullani-Type Integral Evaluations is catalogued as a formula in Definite Integrals and Integral Transforms.

Level: Advanced
Source: Source catalogue
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Gamma-Function Identities

Gamma and Beta Integral Identities

Contribution family

Gamma and Beta Integral Identities is catalogued as a identity in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
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Gamma-Function Identities

Gamma-Ratio Identities

Contribution family

Gamma-Ratio Identities is catalogued as a identity in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
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Ramanujan's Factorial Approximation

Stirling-Series Refinements

Contribution family

Stirling-Series Refinements is catalogued as a approximation in Ramanujan's Factorial Approximation.

Level: Advanced
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Ramanujan's Factorial Approximation

Ramanujan Constant in Factorial Approximation

Contribution family

Ramanujan Constant in Factorial Approximation is catalogued as a concept in Ramanujan's Factorial Approximation.

Level: Advanced
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Ramanujan Summation of Divergent Series

Ramanujan Summation Formula

Contribution family

Ramanujan Summation Formula is catalogued as a method in Ramanujan Summation of Divergent Series.

Level: Advanced
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Ramanujan Summation of Divergent Series

Regularized Series Values

Contribution family

Regularized Series Values is catalogued as a contribution family in Ramanujan Summation of Divergent Series.

Level: Advanced
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Bernoulli Numbers

Bernoulli-Number Congruences

Contribution family

Bernoulli-Number Congruences is catalogued as a identity in Bernoulli Numbers.

Level: Advanced
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Bernoulli Numbers

Bernoulli-Polynomial Identities

Contribution family

Bernoulli-Polynomial Identities is catalogued as a identity in Bernoulli Numbers.

Level: Advanced
Source: Source catalogue
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Odd Zeta Values

Zeta-Function Transformation Formulas

Contribution family

Zeta-Function Transformation Formulas is catalogued as a formula in Odd Zeta Values.

Level: Advanced
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Odd Zeta Values

Odd-Zeta Modular Relations

Contribution family

Odd-Zeta Modular Relations is catalogued as a concept in Odd Zeta Values.

Level: Advanced
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Ramanujan Polynomials

Ramanujan Polynomial Identities

Contribution family

Ramanujan Polynomial Identities is catalogued as a identity in Ramanujan Polynomials.

Level: Advanced
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Ramanujan Polynomials

Ramanujan Polynomial Recurrences

Contribution family

Ramanujan Polynomial Recurrences is catalogued as a formula in Ramanujan Polynomials.

Level: Advanced
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Diophantine Equations

Diophantine Equation Families

Contribution family

Diophantine Equation Families is catalogued as a contribution family in Diophantine Equations.

Level: Advanced
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Named Constants Connected with Ramanujan

Taxicab Number Context

1729 = 1^{3} + 1 2^{3} = 9^{3} + 1 0^{3}

Taxicab Number Context is catalogued as a concept in Named Constants Connected with Ramanujan.

Level: Advanced
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Nested Radicals

Nested Radical Continued Forms

Contribution family

Nested Radical Continued Forms is catalogued as a formula in Nested Radicals.

Level: Advanced
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Near-Integer Phenomena

Modular Explanation of Near-Integer Phenomena

Contribution family

Modular Explanation of Near-Integer Phenomena is catalogued as a concept in Near-Integer Phenomena.

Level: Advanced
Source: Source catalogue
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Near-Integer Phenomena

Class-Number Near Integers

Contribution family

Class-Number Near Integers is catalogued as a concept in Near-Integer Phenomena.

Level: Advanced
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Combinatorial Identities

Combinatorial Sums

Contribution family

Combinatorial Sums is catalogued as a identity in Combinatorial Identities.

Level: Advanced
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Combinatorial Identities

Finite Combinatorial Identities

Contribution family

Finite Combinatorial Identities is catalogued as a identity in Combinatorial Identities.

Level: Advanced
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Magic Squares

Birthday Magic Square Construction

22 + 12 + 18 + 87 = 139

Birthday Magic Square Construction is catalogued as a concept in Magic Squares.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Eisenstein and Modular Series

Eisenstein-Series Congruences

Contribution family

Eisenstein-Series Congruences is catalogued as a identity in Eisenstein and Modular Series.

Level: Advanced
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Eisenstein and Modular Series

Quasimodular Transformations of E2

Contribution family

Quasimodular Transformations of E2 is catalogued as a concept in Eisenstein and Modular Series.

Level: Advanced
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Eisenstein and Modular Series

Ramanujan's Modular Differential System

q \frac{d E _{4}}{d q} = \frac{E _{2} E _{4} - E _{6}}{3}

Ramanujan's Modular Differential System is catalogued as a formula in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Singular Moduli and Class Invariants

Class Equations for Singular Moduli

Contribution family

Class Equations for Singular Moduli is catalogued as a contribution family in Singular Moduli and Class Invariants.

Level: Advanced
Source: Source catalogue
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Singular Moduli and Class Invariants

Ramanujan Class Invariants G_n and g_n

Contribution family

Ramanujan Class Invariants G_n and g_n is catalogued as a concept in Singular Moduli and Class Invariants.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Infinite Products

Eta-Product Identities

Contribution family

Eta-Product Identities is catalogued as a identity in Infinite Products.

Level: Advanced
Source: Source catalogue
Status: needs-review

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Infinite Products

Eta-Transformation Formulas

Contribution family

Eta-Transformation Formulas is catalogued as a formula in Infinite Products.

Level: Advanced
Source: Source catalogue
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Broader Named Concepts Resulting from His Work

Ramanujan-Moonshine Context

Contribution family

Ramanujan-Moonshine Context is catalogued as a concept in Broader Named Concepts Resulting from His Work.

Level: Advanced
Source: Source catalogue
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Broader Named Concepts Resulting from His Work

Quantum Modular Forms Context

Contribution family

Quantum modular forms are a modern development influenced by q-series and mock modular themes.

Level: Advanced
Source: Source catalogue
Status: needs-review

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