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George Andrews [14] showed that several of Ramanujan's fifth order mock theta functions are equal to quotients Θ(𝜏) / θ(𝜏) where θ(𝜏) is a modular form of weight 1 / 2 and Θ(𝜏) is a theta function of an indefinite binary quadratic form, and Dean Hickerson [15] proved similar results for seventh order mock theta ...
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".
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.
Srinivasa Ramanujan is credited with discovering that the partition function has nontrivial patterns in modular arithmetic. For instance the number of partitions is divisible by five whenever the decimal representation of n {\displaystyle n} ends in the digit 4 or 9, as expressed by the congruence [ 7 ] p ( 5 k + 4 ) ≡ 0 ( mod 5 ...
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. These results also follow from the Weil conjectures, except for the case k = 1, where it is a result of Deligne ...
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An elliptic function is a modular function if this function in dependence on the elliptic nome as an internal variable function results in a function, which also results as an algebraic combination of Legendre's elliptic modulus and its complete elliptic integrals of the first kind in the K and K' form.
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.