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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 .
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 ...
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.
The year in art included plenty of snark over new depictions of the British royal family that were perceived as too aggressive or too generic, as well as an infamous banana that returned to ...
As modular forms also satisfy a certain kind of functional equation with respect to the group action of the modular group, this Fourier series may be expressed in terms of =. So if f {\displaystyle f} is a modular form, then there are coefficients c ( n ) {\displaystyle c(n)} such that f ( z ) = ∑ n ∈ N c ( n ) q n {\displaystyle f(z)=\sum ...
Teri Hatcher has a delightfully delicious — and architecturally impressive — holiday tradition.. The actress tells PEOPLE that she looks forward to crafting over-the-top gingerbread creations ...