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gptkbp:instanceOf
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gptkb:modular_group
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gptkbp:alsoKnownAs
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gptkb:modular_discriminant
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gptkbp:appearsIn
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Ramanujan's 1916 paper on modular forms
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gptkbp:cusp_form
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true
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gptkbp:definedIn
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Δ(z) = q ∏_{n=1}^∞ (1 - q^n)^{24}, q = e^{2πiz}
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gptkbp:discoveredBy
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gptkb:Srinivasa_Ramanujan
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gptkbp:field
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modular forms
number theory
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gptkbp:firstPublished
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1916
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gptkbp:Fourier_coefficient
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gptkb:Ramanujan_tau_function
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gptkbp:Fourier_expansion
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Δ(z) = ∑_{n=1}^∞ τ(n)q^n
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gptkbp:growthForm
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bounded by O(n^{11/2}) for τ(n)
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gptkbp:Hecke_eigenform
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true
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gptkbp:holomorphic
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true
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gptkbp:L-function
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L(Δ,s) = ∑_{n=1}^∞ τ(n)n^{-s}
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gptkbp:level
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1
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gptkbp:modular_group_action
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Δ(γz) = (cz+d)^{12}Δ(z) for γ in SL(2,ℤ)
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gptkbp:multiplicative_property
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τ(mn) = τ(m)τ(n) for coprime m, n
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gptkbp:nonzero_elsewhere
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true
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gptkbp:relatedTo
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gptkb:modular_group_SL(2,ℤ)
gptkb:Eisenstein_series
gptkb:Ramanujan_tau_function
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gptkbp:satisfies_functional_equation
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true
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gptkbp:symbol
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Δ(z)
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gptkbp:vanishes_at_infinity
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true
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gptkbp:weight
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12
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gptkbp:zeroes
|
at the cusp only
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https://www.w3.org/2000/01/rdf-schema#label
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Ramanujan Delta function
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