Trivial zeros in the Riemann Zeta function


by msariols
Tags: riemann, riemann zeta, zeta function
msariols
msariols is offline
#1
Oct8-11, 11:46 AM
P: 1
Hello, I have read in many articles that the trivial zeros of the Riemann zeta function are only the negative even integers (-2, -4, -6, -8, -10, ...).

The reason why these are the only ones is that when substituting them in the functional equation, the function is 0 because sin([itex]\frac{x\pi}{2}[/itex])=0.

My question is: why aren't positive even integers trivial zeros too?

The sinus of k[itex]\pi[/itex] =0 with either k[itex]\in[/itex]Z positive or negative.


Remember that the functional equation is:

[itex]\zeta[/itex](x)=[itex]\zeta[/itex](1-x)[itex]\Gamma[/itex] (1-x)2[itex]^{x}[/itex][itex]\pi[/itex][itex]^{x-1}[/itex]sin ([itex]\frac{x\pi}{2}[/itex])
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
jackmell
jackmell is offline
#2
Oct8-11, 01:06 PM
P: 1,666
At the even integers, the simple poles of [itex]\Gamma(1-z)[/itex] are canceled by the simple zeros of [itex]\sin(\pi z/2)[/itex] and since the poles and zeros are of the same order (simple), this cancelation is non-zero, that is, the singularity is a removable one. For example consider the limit:

[tex]\lim_{x\to 4} \; \Gamma(1-x) \sin(\pi x/2)=\frac{\pi}{12}[/tex]
camilus
camilus is offline
#3
Oct25-11, 04:34 PM
P: 150
also because at the positive even integers, the zeta function is defined the Dirichlet series 1+1/2^s+1/3^s+1/4^s+... which converges for all positive even numbers.


Register to reply

Related Discussions
Can someone explain zeros and zeta function for Riemann Hypothesis? (Yr13) Linear & Abstract Algebra 19
Trivial zeros of Zeta Riemann Function Linear & Abstract Algebra 1
trivial zeros of the Riemann zeta function Linear & Abstract Algebra 3
Riemann Zeta function zeros Calculus 1
Riemann Zeta zeros Linear & Abstract Algebra 31