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

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...

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

{\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.    ...

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...

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

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

(−1)ω(n), where the additive function ω(n) is the number of distinct primes dividing n. τ(n): the Ramanujan tau function. All Dirichlet characters are...

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...

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...

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

) {\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 ⁡ (...

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 )...

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

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...

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

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

inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm...

2\pi } (6.283...). Kendall tau rank correlation coefficient, a measure of rank correlation in statistics Ramanujan's tau function in number theory shear stress...

of the j-function (as well as the well-known Dedekind eta function) uses q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} . However, Ramanujan, in his examples...

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...

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...

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...

theta function Ramanujan graph Ramanujan's tau function Ramanujan's ternary quadratic form Ramanujan prime Ramanujan's constant Ramanujan's lost notebook...

)=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )} , Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} , and modular invariants...

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

Jacobi's theta function θ ( τ ) = ∑ n = − ∞ ∞ e π i n 2 τ . {\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.} However, this...

theta function θ ( z , τ ) = ∑ n = − ∞ ∞ e 2 π i n z + i π n 2 τ {\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty }e^{2\pi inz+i\pi n^{2}\tau }}...