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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 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 (1916) observed, but did not prove, ... Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)
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 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".
For example, the Ramanujan tau function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a 1 = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell ...
Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.