Domain Of Riemann Zeta Function

Riemann s zeta function is defined by the dirichlet series begin equation label sum zeta s sum n 1 infty frac 1 n s quad s sigma it end equation which converges absolutely and uniformly in any bounded domain of the complex s plane for which sigma geq1 delta delta 0.

Domain of riemann zeta function. The riemann zeta function plays a pivotal role in analytic number theory and has applications. The riemann zeta functional leads to a relationship between the zeros of the riemann zeta function on either sides of the critical line. The riemann zeta function for s c s in mathbb c s c with re s 1 operatorname re s 1 r e s 1 is defined as ζ s n 1 1 n s.

The riemann zeta function or euler riemann zeta function ζ s is a function of a complex variable s that analytically continues the sum of the dirichlet series which converges when the real part of s is greater than 1. In mathematics the riemann hypothesis is a conjecture that the riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2 many consider it to be the most important unsolved problem in pure mathematics bombieri 2000 it is of great interest in number theory because it implies results about the distribution of prime numbers. The series defines a function f on its region of convergence which is the set of complex numbers s σ i t where the real part σ 1.

3 the riemann zeta function in terms of prime numbers. Phic functions is by expansion into power series. I ll call this restricted function f to distinguish it from the full zeta function ζ.

It is then defined by analytical continuation to a meromorphic function on the whole c mathbb c c by a functional equation. Such quantum riemann zeta function analogies have led to the bender brody muller bbm conjecture which involves a non hermitian hamiltonian that maps to the zeros of the riemann zeta function. I e f z is equal to the power series in the disc fjz aj rgas in fig 3.

ζ s n 1 n s 1. More general representations of ζ s for all s are given below. How was the domain of the riemann zeta function extended to the negative real numbers.