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The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form.
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.
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 .
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.
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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.
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