From this point, there are a few ways to prove that . } } One method, along the lines of Euler's reasoning, uses the relationship between the Riemann zeta function and the Dirichlet eta function " ? " ( " s " ).
32.
The usual denominator identities of semi-simple Lie algebras generalize as well; because the characters can be written as " deformations " or q-analogs of the highest weights, this led to many new combinatoric identities, include many previously unknown identities for the Dedekind eta function.
33.
Euler treated these two as special cases of for arbitrary " n ", a line of research extending his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
34.
Because of these functional equations the eta function is a modular form of weight 1 / 2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms.
35.
It expresses the value of the eta function as the limit of special Riemann sums associated to an integral known to be zero, using a relation between the partial sums of the Dirichlet series defining the eta and zeta functions for \ Re ( s ) > 1.
36.
While the Dirichlet series expansion for the eta function is convergent only for any complex number " s " with real part > 0, it is pole at " s " = 1, and perhaps poles at the other zeros of the factor 1-2 ^ { 1-s } ).
37.
Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
38.
Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
39.
Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines \ Re ( s ) = 1 / 2, \ Re ( s ) = 1, and on the perpendicular half line formed by the negative real axis.
40.
This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, " & zeta; " ( s ) & mdash; and for this reason the Dirichlet eta function is also known as the "'alternating zeta function "', also denoted " & zeta; " * ( s ).
