The Laplace transform " ( in which he gave a first solution to Landau's problem on the Dirichlet eta function ), " An introduction to transform theory ", and"
22.
In mathematics, in the area of analytic number theory, the "'Dirichlet eta function "'is defined by the following Dirichlet series, which converges for any complex number having real part > 0:
23.
The formulation of Euler's generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function.
24.
The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges.
25.
Which represents ( up to a normalizing constant ) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function.
26.
The functions " G " and " H " turn up in the Rogers Ramanujan identities, and the function " Q " is the Euler function, which is closely related to the Dedekind eta function.
27.
On the other hand, if one uses stronger methods of summability, then the Dirichlet series for ? defines a function on the whole complex plane the Dirichlet eta function and moreover, this function is analytic.
28.
This cusp form is the discriminant ? ( " q " ) whose Fourier coefficients are given by the Ramanujan-function and which is ( up to a constant multiplier ) the 24th power of the Dedekind eta function.
29.
The roots of the cubics can be exactly given by quotients of the Dedekind eta function " ? " ( " ? " ), a modular function involving a 24th root, and which explains the 24 in the approximation.
30.
This is known as the generating function for partitions, and is also written as " q " 1 / 24 times the weight " 1 / 2 modular form 1 / ? ( the Dedekind eta function ).
