Number Theory


Riemann Zeta Function Constants, Approximations, and Some Related Functions

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: ζ(z)=∑_(k=1)^∞ k^(-z) 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 ζ(z) 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 new propositions to advance in the knowledge of the Riemann Zeta function. a) A constant C that can be used to express ζ(2n+1)≡a/b*C^(2n+1) b) An approximation to the values of ζ(s) in R given by ζ(s)=1/(1-π^(-s)-2^(-s)) c) A theorem that states that the infinite sums ∑_(j=1)^∞[ζ(u*k±n)-ζ(v*k±m)] converge to a value in the interval (-1,1) for all u≥1,v≥1,n,m ∈N such that (u*k±n)>1 and (v*k±m)>1 for all j∈N d) A new set of constants CZ_(u,n,v,m)calculated from infinite sums involving ζ(z) e) A function in C2(x,a,b)= 2*x^(-a)*(∑_(j=1)^(x-1) [j^(-a)*cos(b*(ln(x/j)))]) in R with zeros in(a,b) with a=1/2 and b=Im(z*), with z*=non-trivial zero of ζ(z). f) A C-transformation that allows for a decomposition of ζ(z) that can be used to study the Riemann Hypothesis. g) Linearization of the Harmonic function using Non-Trivial zeros of ζ(z). h) An expression that links any two Non-Trivial zeros of ζ(z).

Comments: 21 Pages.

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[v1] 2018-03-10 16:37:39

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