Limit Theorems for the Riemann Zeta-Function
. Kluwer, Dordrecht, 1996.
Mitrovic, The signs of some constants associated with the Riemann zeta-function
, Michigan Math.
Ivic, Riemann zeta-function
: theory and applications, Oversea Publishing House, 2003.
where [zeta](s) = [[SIGMA].sup.[infinity].sub.k=1][k.sup.-s] (s > 1) is the Riemann zeta-function
. Concerning the computational aspects we shall derive an approximation to [gamma](a) in terms of the function [zeta](s, a), assumed to be numerically known.
 Ivic, A., 1985, The Riemann Zeta-Function
, Dover, New York, NY.
Clearly, [zeta](s, 1) = [zeta](s) and [zeta](s, 2) = ([2.sup.s] - 1) [zeta](s), where [zeta](s) is the Riemann zeta-function
. Therefore, in those cases, [zeta](s, [alpha]) is also universal, however, the approximated function f (s) must be non-vanishing on K.
The Riemann zeta-function
is a special case of the Hurwitz zeta-function:
Ivic, The Riemann Zeta-function
, John Wiley Sons, 1985.
The statistical behaviour and universality properties of the Riemann zeta-function
and other allied Dirichlet series.
Akatsuka, The Euler product for the Riemann zeta-function
in the critical strip.
In , Voronin discovered the universality property of the Riemann zeta-function
[zeta](s), s = [sigma] + it, on the approximation of analytic functions from a wide class by shifts [zeta](s + i[tau]), [tau] [member of] R.
Where [chi] denotes a Dirichlet character modulo d, L(n, [chi]) denotes the Dirichlet L-function corresponding to[chi], [empty set](d) and [zeta](s) are the Euler function and Riemann zeta-function