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Furthermore, we can concretely calculate the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms). Deligne (1971) used the Eichler–Shimura isomorphism to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.
In mathematics, a Ramanujan–Sato series [1] [2] generalizes Ramanujan’s pi formulas such as, = = ()!! + to the form = = + by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (), and ,, employing modular forms of higher levels.
The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics. Vol. 102. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3368-1. Zbl 1119.11026. Ramanujan, S. (1919). "Some properties of p(n), the number of partitions of n".
Ramanujan (1916) observed, but did not prove, the following three properties of τ(n): τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicative function); τ(p r + 1) = τ(p)τ(p r) − p 11 τ(p r − 1) for p prime and r > 0.
A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in G, that is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C-vector spaces of modular and cusp forms of weight k are denoted M k (G) and S k (G), respectively.
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock 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 last 1920 letter to G. H. Hardy and in his lost notebook .
Ramanujan tau function; Ramanujan–Petersson conjecture; Rankin–Cohen bracket; Real analytic Eisenstein series; Ribet's theorem; Ring of modular forms; Rogers–Ramanujan continued fraction; Rogers–Ramanujan identities
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