Ramanujan tau function information

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...

{\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...

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

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...

(−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...

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...

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...

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

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

) {\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 )...

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

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...

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...

Ono and others, Lehmer's question on whether the Ramanujan tau function τ ( n ) {\displaystyle \tau (n)} is ever zero for a positive integer n. As well...

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...

Rankin (1939), who used a similar idea with k = 2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of...

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

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...