## Number Theory   ## Riemann Zeta Function Nine Propositions

Authors: Pedro Caceres

The Riemann Zeta function or Euler–Riemann Zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series: () = ∑ ^(-z) from k=1,∞ The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of () in the critical line Re(z)=1/2. The Riemann Zeta function is one of the most studied and well known mathematical functions in history. In this paper, we will formulate nine new propositions to advance in the knowledge of the Riemann Zeta function