Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Riemann zeta function

I'm reading something about the riemann zeta function and it says that zeta(0)=-1/2. How is that the case? Because if the riemann zeta function is zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s +....., surely zeta(0) = 1+ 1 + 1+1+1....=

Daniel Freedman

Wiki says:

"The Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:

\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

for values of s with real part greater than one "

Read the article, particularly the section about the functional equation. http://en.wikipedia.org/wiki/Riemann_zeta_function

OP

OK i've got that now. But now what I don't get is why all negative even numbers give zeros. Is this because

\eta (-2n)

=0 for all integers n>0?? If so, how come this is the case?

Daniel Freedman

lilman91

OK i've got that now. But now what I don't get is why all negative even numbers give zeros. Is this because

\eta (-2n)

=0 for all integers n>0?? If so, how come this is the case?

Whenever s is an even negative,

\sin{\left(\frac{\pi s}{2}\right)} = 0

OP

cheers got it now. sorry the book im reading hasn't got that much theory in it.

Daniel Freedman

lilman91

cheers got it now. sorry the book im reading hasn't got that much theory in it.

No problem. Wikipedia is often helpful.

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