The Formula Archive

A searchable collection of Ramanujan's equations, identities, series and mathematical transformations.

61 formulas indexed

Bernoulli-Number Identities formula

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

A central formula or notation associated with Bernoulli-Number Identities.

Bernoulli NumbersidentitySource catalogue

Open Formula

Birthday Magic Square Construction formula

22 + 12 + 18 + 87 = 139

A central formula or notation associated with Birthday Magic Square Construction.

Magic SquaresconceptSource catalogue

Open Formula

Divisor Functions formula

σ_{k} (n) = d ∣ n \sum d^{k}

A central formula or notation associated with Divisor Functions.

Divisor FunctionsfunctionSource catalogue

Open Formula

Eisenstein Series E2 formula

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

A central formula or notation associated with Eisenstein Series E2.

Eisenstein and Modular SeriesseriesSource catalogue

Open Formula

Eisenstein Series E4 formula

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

A central formula or notation associated with Eisenstein Series E4.

Eisenstein and Modular SeriesseriesSource catalogue

Open Formula

Eisenstein Series E6 formula

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

A central formula or notation associated with Eisenstein Series E6.

Eisenstein and Modular SeriesseriesSource catalogue

Open Formula

Eta Quotients formula

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

A central formula or notation associated with Eta Quotients.

Infinite ProductsformulaSource catalogue

Open Formula

f(-q) formula

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

A central formula or notation associated with f(-q).

Ramanujan Theta FunctionsfunctionSource catalogue

Open Formula

First Rogers-Ramanujan Identity formula

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

Rogers discovered the identities earlier; Ramanujan independently rediscovered them and deepened their significance.

Rogers-Ramanujan IdentitiesidentitySource catalogue

Open Formula

Formula for zeta(3) formula

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

A central formula or notation associated with Formula for zeta(3).

Odd Zeta ValuesformulaSource catalogue

Open Formula

Gamma-Function Identities formula

Γ (z + 1) = z Γ (z)

A central formula or notation associated with Gamma-Function Identities.

Gamma-Function IdentitiesidentitySource catalogue

Open Formula

Generating Function for tau(n) formula

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

A central formula or notation associated with Generating Function for tau(n).

Ramanujan's Tau FunctionformulaSource catalogue

Open Formula

Golden-Ratio Connections formula

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

A central formula or notation associated with Golden-Ratio Connections.

Rogers-Ramanujan Continued FractionconceptSource catalogue

Open Formula

Hardy-Ramanujan Asymptotic Formula formula

p (n) \sim \frac{1}{4 n 3} e^{π 2 n /3}

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

Partition TheoryformulaHardy-Ramanujan

Open Formula

Hardy-Ramanujan Number 1729 formula

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

Ramanujan did not discover 1729 during the taxi conversation; he immediately recognized its special property.

Named Constants Connected with RamanujanconceptSource catalogue

Open Formula

Higher-Power Partition Congruences formula

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

A central formula or notation associated with Higher-Power Partition Congruences.

Partition Theorycontribution-familySource catalogue

Open Formula

Highly Composite Numbers formula

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

A central formula or notation associated with Highly Composite Numbers.

Highly Composite NumbersconceptPublished Papers

Open Formula

Infinite Nested Radical formula

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

A central formula or notation associated with Infinite Nested Radical.

Nested RadicalsformulaSource catalogue

Open Formula

Infinite-Product Transformations formula

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

A central formula or notation associated with Infinite-Product Transformations.

Infinite ProductsidentitySource catalogue

Open Formula

Jacobi Triple Product Context formula

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

Ramanujan used and transformed theta-product ideas in his own notation; the classical theorem predates him.

Ramanujan Theta FunctionsconceptSource catalogue

Open Formula

Landau-Ramanujan Constant formula

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

The Landau-Ramanujan theorem is primarily due to Landau, while Ramanujan obtained related ideas.

Named Constants Connected with RamanujanconceptSource catalogue

Open Formula

Mellin Transforms formula

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

A central formula or notation associated with Mellin Transforms.

Definite Integrals and Integral TransformsmethodSource catalogue

Open Formula

Mock Theta Functions formula

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

Their modern harmonic-Maass-form interpretation came much later.

Mock Theta FunctionsfunctionLost Notebook

Open Formula

Modular Transformations formula

τ \mapsto \frac{a τ + b}{c τ + d}

A central formula or notation associated with Modular Transformations.

Modular Equations and Elliptic FunctionsmethodSource catalogue

Open Formula

Near-Integer e^(pi sqrt 163) formula

e^{π 163} \approx 262537412640768743.99999999999925

This near-integer is also historically associated with Charles Hermite.

Near-Integer PhenomenaapproximationSource catalogue

Open Formula

Partition Congruences Modulo Prime Powers formula

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

A central formula or notation associated with Partition Congruences Modulo Prime Powers.

Partition Theorycontribution-familySource catalogue

Open Formula

Partition Generating Function formula

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.

Partition TheoryformulaSource catalogue

Open Formula

phi(q) formula

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

A central formula or notation associated with phi(q).

Ramanujan Theta FunctionsfunctionSource catalogue

Open Formula

Prime-Counting Approximations formula

π (x) \sim Li (x)

A central formula or notation associated with Prime-Counting Approximations.

Approximation to the Prime-Counting FunctionapproximationSource catalogue

Open Formula

Product Form of the Continued Fraction formula

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}}

A central formula or notation associated with Product Form of the Continued Fraction.

Rogers-Ramanujan Continued FractionformulaSource catalogue

Open Formula

psi(q) formula

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

A central formula or notation associated with psi(q).

Ramanujan Theta FunctionsfunctionSource catalogue

Open Formula

q-Binomial Identities formula

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

A central formula or notation associated with q-Binomial Identities.

q-Series and Basic Hypergeometric SeriesidentitySource catalogue

Open Formula

q-Pochhammer Product Formulas formula

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

A central formula or notation associated with q-Pochhammer Product Formulas.

q-Series and Basic Hypergeometric SeriesformulaSource catalogue

Open Formula

Ramanujan Differential Equations formula

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

A central formula or notation associated with Ramanujan Differential Equations.

Eisenstein and Modular SeriesformulaSource catalogue

Open Formula

Ramanujan Expansions formula

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

A central formula or notation associated with Ramanujan Expansions.

Ramanujan SumsmethodSource catalogue

Open Formula

Ramanujan Primes formula

π (x) - π (x /2) \geq n

The term Ramanujan prime was introduced later and is based on Ramanujan's theorem.

Prime-Number TheoryconceptSource catalogue

Open Formula

Ramanujan Summation formula

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

This is not an ordinary sum; it belongs only in regularisation or summation-method contexts.

Ramanujan Summation of Divergent SeriesmethodSource catalogue

Open Formula

Ramanujan Sums formula

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

A central formula or notation associated with Ramanujan Sums.

Ramanujan SumsformulaSource catalogue

Open Formula

Ramanujan-Nagell Equation formula

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

Ramanujan conjectured the equation's solutions; Trygve Nagell later proved the result.

Diophantine EquationsequationSource catalogue

Open Formula

Ramanujan-Petersson Conjecture formula

∣ τ (p) ∣ \leq 2 p^{11/2}

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

Ramanujan's Tau FunctionconjectureSource catalogue

Open Formula

Ramanujan's 1 psi 1 Summation formula

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

A central formula or notation associated with Ramanujan's 1 psi 1 Summation.

q-Series and Basic Hypergeometric SeriestheoremSource catalogue

Open Formula

Ramanujan's 1/pi Series formula

\frac{1}{π} = \frac{2 2}{9801} n = 0 \sum \infty \frac{( 4 n )! ( 1103 + 26390 n )}{( n ! ) ^{4} 39 6 ^{4 n}}

A central formula or notation associated with Ramanujan's 1/pi Series.

Formulas for piseriesPublished Papers

Open Formula

Ramanujan's Cubic Continued Fraction formula

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

A central formula or notation associated with Ramanujan's Cubic Continued Fraction.

Other Continued Fractionscontinued-fractionSource catalogue

Open Formula

Ramanujan's Factorial Approximation formula

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

A central formula or notation associated with Ramanujan's Factorial Approximation.

Ramanujan's Factorial ApproximationapproximationSource catalogue

Open Formula

Ramanujan's General Theta Function formula

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

A central formula or notation associated with Ramanujan's General Theta Function.

Ramanujan Theta FunctionsfunctionSource catalogue

Open Formula

Ramanujan's Magic Square formula

22 + 12 + 18 + 87 = 139

A central formula or notation associated with Ramanujan's Magic Square.

Magic SquaresconceptSource catalogue

Open Formula

Ramanujan's Master Theorem formula

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

A central formula or notation associated with Ramanujan's Master Theorem.

Ramanujan's Master TheoremtheoremSource catalogue

Open Formula

Ramanujan's Modular Differential System formula

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

A central formula or notation associated with Ramanujan's Modular Differential System.

Eisenstein and Modular SeriesformulaSource catalogue

Open Formula

Ramanujan's Partition Congruences formula

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

Ramanujan discovered striking congruence properties of partition numbers.

Partition TheorytheoremPublished Papers

Open Formula

Ramanujan's Tau Function formula

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

A central formula or notation associated with Ramanujan's Tau Function.

Ramanujan's Tau FunctionfunctionPublished Papers

Open Formula

Representations by Squares formula

r_{2} (n) = 4 d ∣ n \sum χ (d)

A central formula or notation associated with Representations by Squares.

Representation of Integers by Quadratic FormsformulaSource catalogue

Open Formula

Rogers-Ramanujan Continued Fraction formula

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

The continued fraction has historical connections to Rogers and Schur as well as Ramanujan.

Rogers-Ramanujan Continued Fractioncontinued-fractionSource catalogue

Open Formula

Second Rogers-Ramanujan Identity formula

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

Rogers discovered the identities earlier; Ramanujan independently rediscovered them.

Rogers-Ramanujan IdentitiesidentitySource catalogue

Open Formula

Sigma-Function Identities formula

σ_{k} (n) = d ∣ n \sum d^{k}

A central formula or notation associated with Sigma-Function Identities.

Divisor FunctionsidentitySource catalogue

Open Formula

Tau Congruence Modulo 691 formula

τ (n) \equiv σ_{11} (n) (mod 691)

A central formula or notation associated with Tau Congruence Modulo 691.

Ramanujan's Tau FunctionformulaSource catalogue

Open Formula

Tau Multiplicativity formula

τ (mn) = τ (m) τ (n) ((m, n) = 1)

A central formula or notation associated with Tau Multiplicativity.

Ramanujan's Tau FunctionconjectureSource catalogue

Open Formula

Tau Prime-Power Recurrence formula

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

A central formula or notation associated with Tau Prime-Power Recurrence.

Ramanujan's Tau FunctionformulaSource catalogue

Open Formula

Tau-Function Congruences formula

τ (n) \equiv σ_{11} (n) (mod 691)

A central formula or notation associated with Tau-Function Congruences.

Ramanujan's Tau FunctionformulaSource catalogue

Open Formula

Taxicab Number Context formula

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

A central formula or notation associated with Taxicab Number Context.

Named Constants Connected with RamanujanconceptSource catalogue

Open Formula

Theta Product Representations formula

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

A central formula or notation associated with Theta Product Representations.

Ramanujan Theta FunctionsidentitySource catalogue

Open Formula

Third-Order Mock Theta Functions formula

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

A central formula or notation associated with Third-Order Mock Theta Functions.

Mock Theta FunctionsfunctionSource catalogue

Open Formula

The Formula Archive

palm Bernoulli-Number Identities formula

palm Birthday Magic Square Construction formula

palm Divisor Functions formula

palm Eisenstein Series E2 formula

palm Eisenstein Series E4 formula

palm Eisenstein Series E6 formula

palm Eta Quotients formula

palm f(-q) formula

palm First Rogers-Ramanujan Identity formula

palm Formula for zeta(3) formula

palm Gamma-Function Identities formula

palm Generating Function for tau(n) formula

palm Golden-Ratio Connections formula

palm Hardy-Ramanujan Asymptotic Formula formula

palm Hardy-Ramanujan Number 1729 formula

palm Higher-Power Partition Congruences formula

palm Highly Composite Numbers formula

palm Infinite Nested Radical formula

palm Infinite-Product Transformations formula

palm Jacobi Triple Product Context formula

palm Landau-Ramanujan Constant formula

palm Mellin Transforms formula

palm Mock Theta Functions formula

palm Modular Transformations formula

palm Near-Integer e^(pi sqrt 163) formula

palm Partition Congruences Modulo Prime Powers formula

palm Partition Generating Function formula

palm phi(q) formula

palm Prime-Counting Approximations formula

palm Product Form of the Continued Fraction formula

palm psi(q) formula

palm q-Binomial Identities formula

palm q-Pochhammer Product Formulas formula

palm Ramanujan Differential Equations formula

palm Ramanujan Expansions formula

palm Ramanujan Primes formula

palm Ramanujan Summation formula

palm Ramanujan Sums formula

palm Ramanujan-Nagell Equation formula

palm Ramanujan-Petersson Conjecture formula

palm Ramanujan's 1 psi 1 Summation formula

palm Ramanujan's 1/pi Series formula

palm Ramanujan's Cubic Continued Fraction formula

palm Ramanujan's Factorial Approximation formula

palm Ramanujan's General Theta Function formula

palm Ramanujan's Magic Square formula

palm Ramanujan's Master Theorem formula

palm Ramanujan's Modular Differential System formula

palm Ramanujan's Partition Congruences formula

palm Ramanujan's Tau Function formula

palm Representations by Squares formula

palm Rogers-Ramanujan Continued Fraction formula

palm Second Rogers-Ramanujan Identity formula

palm Sigma-Function Identities formula

palm Tau Congruence Modulo 691 formula

palm Tau Multiplicativity formula

palm Tau Prime-Power Recurrence formula

palm Tau-Function Congruences formula

palm Taxicab Number Context formula

palm Theta Product Representations formula

palm Third-Order Mock Theta Functions formula

Bernoulli-Number Identities formula

Birthday Magic Square Construction formula

Divisor Functions formula

Eisenstein Series E2 formula

Eisenstein Series E4 formula

Eisenstein Series E6 formula

Eta Quotients formula

f(-q) formula

First Rogers-Ramanujan Identity formula

Formula for zeta(3) formula

Gamma-Function Identities formula

Generating Function for tau(n) formula

Golden-Ratio Connections formula

Hardy-Ramanujan Asymptotic Formula formula

Hardy-Ramanujan Number 1729 formula

Higher-Power Partition Congruences formula

Highly Composite Numbers formula

Infinite Nested Radical formula

Infinite-Product Transformations formula

Jacobi Triple Product Context formula

Landau-Ramanujan Constant formula

Mellin Transforms formula

Mock Theta Functions formula

Modular Transformations formula

Near-Integer e^(pi sqrt 163) formula

Partition Congruences Modulo Prime Powers formula

Partition Generating Function formula

phi(q) formula

Prime-Counting Approximations formula

Product Form of the Continued Fraction formula

psi(q) formula

q-Binomial Identities formula

q-Pochhammer Product Formulas formula

Ramanujan Differential Equations formula

Ramanujan Expansions formula

Ramanujan Primes formula

Ramanujan Summation formula

Ramanujan Sums formula

Ramanujan-Nagell Equation formula

Ramanujan-Petersson Conjecture formula

Ramanujan's 1 psi 1 Summation formula

Ramanujan's 1/pi Series formula

Ramanujan's Cubic Continued Fraction formula

Ramanujan's Factorial Approximation formula

Ramanujan's General Theta Function formula

Ramanujan's Magic Square formula

Ramanujan's Master Theorem formula

Ramanujan's Modular Differential System formula

Ramanujan's Partition Congruences formula

Ramanujan's Tau Function formula

Representations by Squares formula

Rogers-Ramanujan Continued Fraction formula

Second Rogers-Ramanujan Identity formula

Sigma-Function Identities formula

Tau Congruence Modulo 691 formula

Tau Multiplicativity formula

Tau Prime-Power Recurrence formula

Tau-Function Congruences formula

Taxicab Number Context formula

Theta Product Representations formula

Third-Order Mock Theta Functions formula