The " n "-th coefficient of the Dirichlet series is essentially the number of representations of " n " as a sum of two squares of nonnegative integers.
42.
There are now three possibilities regarding the convergence of a Dirichlet series, i . e . it may converge for all, for none or for some values of " s ".
43.
An elementary manipulation shows that the residue of the Dedekind zeta function at " s " = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function.
44.
After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.
45.
The function \ delta is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely in the half-plane \ mathrm { Re } ( s ), and there has Euler product
46.
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.
47.
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.
48.
Incidentally, this relation is interesting also because it actually exhibits as a Dirichlet series ( of ) which is non-absolutely ) in the larger half-plane ( not just ), up to an elementary factor.
49.
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:
50.
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.
