The Ramanujan tau function, studied by Ramanujan (1916), is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by the following...

Tau function may refer to: Tau function (integrable systems), in integrable systems Ramanujan tau function, giving the Fourier coefficients of the Ramanujan...

In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients...

Divisor function in number theory, also denoted d or σ0 Ramanujan tau function Golden ratio (1.618...), although φ (phi) is more common Kendall tau rank...

{\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),}     where τ(n) is Ramanujan's function.    ...

arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function). He proved many...

) {\displaystyle s(q)=s\left(e^{\pi i\tau }\right)=-R\left(-e^{-\pi i/(5\tau )}\right)} is the Rogers–Ramanujan continued fraction: s ( q ) = tan ⁡ (...

{\displaystyle \tau (n)} : the Ramanujan tau function All Dirichlet characters are completely multiplicative functions, for example ( n / p ) {\displaystyle...

\eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle \Delta } , see Ramanujan tau function. e 1 {\displaystyle e_{1}}...

Dedekind eta function. The Fourier coefficients here are written τ ( n ) {\displaystyle \tau (n)} and called 'Ramanujan's tau function', with the normalization...

mathematics, the tau conjecture may refer to one of Lehmer's conjecture on the non-vanishing of the Ramanujan tau function The Ramanujan–Petersson conjecture...

In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, 1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle...

special cusp form of Ramanujan, ahead of the general theory given by Hecke (1937a,1937b). Mordell proved that the Ramanujan tau function, expressing the coefficients...

the eta function is defined by, η ( τ ) = e π i τ 12 ∏ n = 1 ∞ ( 1 − e 2 n π i τ ) = q 1 24 ∏ n = 1 ∞ ( 1 − q n ) . {\displaystyle \eta (\tau )=e^{\frac...

the previous line τ ( 3 ) {\displaystyle \tau (3)} , where τ {\displaystyle \tau } is the Ramanujan tau function. σ 3 ( 6 ) {\displaystyle \sigma _{3}(6)}...

In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: p ( 5 k + 4 ) ≡ 0 ( mod...

employing the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} , define the Ramanujan G- and g-functions as 2 1 / 4 G n = q − 1 24 ∏ n > 0 ( 1 + q 2 n − 1 )...

Euler function is related to the Dedekind eta function as ϕ ( e 2 π i τ ) = e − π i τ / 12 η ( τ ) . {\displaystyle \phi (e^{2\pi i\tau })=e^{-\pi i\tau /12}\eta...

function Multivariate gamma function p-adic gamma function Pochhammer k-symbol q-gamma function Ramanujan's master theorem Spouge's approximation Stirling's...

theta function is essentially a mock modular form of weight ⁠1/2⁠. The first examples of mock theta functions were described by Srinivasa Ramanujan in his...

a function on the upper half-plane H = { τ ∈ C ∣ Im ⁡ ( τ ) > 0 } {\displaystyle {\mathcal {H}}=\{\tau \in \mathbb {C} \mid \operatorname {Im} (\tau )>0\}}...

1+\tau {}\alpha } ) to the term itself. Via substitution and arithmetic, the type expands to 1 + τ + τ 2 + τ 3 + ⋯ {\displaystyle 1+\tau +\tau ^{2}+\tau...

this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial...

functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel...

generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais. Ramanujan's constant...

1225 Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even...

} An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is Γ ( 1 + x ) ≈ π ( x e ) x ( 8 x...

[a;\sigma ,\tau ]={\frac {\theta _{1}(\pi \sigma a,e^{\pi i\tau })}{\theta _{1}(\pi \sigma ,e^{\pi i\tau })}}} where the Jacobi theta function is defined...

In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were...

of n. A000396 Ramanujan tau function 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ... Values of the Ramanujan tau function, τ(n) at n = 1...