The Ramanujan tau function, studied by Ramanujan (1916), is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \rightarrow \mathbb {Z} } defined by...
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Tau function may refer to: Tau function (integrable systems), in integrable systems Ramanujan tau function, giving the Fourier coefficients of the Ramanujan...
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In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients...
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arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function). He proved many...
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{\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. ...
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\eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle \Delta } , see Ramanujan tau function. e 1 {\displaystyle e_{1}}...
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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...
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this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial...
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(−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...
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Dedekind eta function. The Fourier coefficients here are written τ ( n ) {\displaystyle \tau (n)} and called 'Ramanujan's tau function', with the normalization...
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) {\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 (...
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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...
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Mock modular form (redirect from Mock theta function)
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...
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function Multivariate gamma function p-adic gamma function Pochhammer k-symbol q-gamma function Ramanujan's master theorem Spouge's approximation Stirling's...
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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 )...
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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...
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Heegner number (redirect from Ramanujan constant)
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...
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In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: p ( 5 k + 4 ) ≡ 0 ( mod...
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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...
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the previous line τ ( 3 ) {\displaystyle \tau (3)} , where τ {\displaystyle \tau } is the Ramanujan tau function. σ 3 ( 6 ) {\displaystyle \sigma _{3}(6)}...
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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...
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inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm...
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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...
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functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel...
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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...
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J-invariant (redirect from Elliptic modular function)
)=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...
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Stirling's approximation (category Gamma and related functions)
} 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...
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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...
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Pi (redirect from Tau versus pi debate)
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 }}...
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up theta function in Wiktionary, the free dictionary. Theta functions ϑ ( z ; τ ) {\displaystyle \vartheta (z;\tau )} are special functions of several...
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