Proof that the zeta function converges for Re(s)>1

Question

It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that

$\sum^{\infty}_{n=1}\frac{1}{n^{s}}= \prod_{p \, \mathrm{prime}} \frac{1}{1-p^{-s}}$ converges for Re(s)>1 and so, proving that $p^{-s}$ converges for all $s$. How may I formulate the epsilon delta arguments for this?

Thanks a lot

It is more challenging to show that the series in the OP diverges for $\text{Re}(s)\le 1$. See THIS ANSWER and THIS ONE for an approach. — Mark Viola, Nov 17 '16 at 05:00

score 3 · Answer 1 · answered Apr 30 '13 at 20:33

3

HINT: Cauchy condensation test.

answered Apr 30 '13 at 20:33

Hagen von Eitzen

374,180

score 0 · Answer 2 · answered Apr 30 '13 at 20:32

0

Hint:

Prove the Zeta function converges absolutely (p-series test)
Absolute convergence implies convergence

answered Apr 30 '13 at 20:32

Long

1,630

Proof that the zeta function converges for Re(s)>1

2 Answers2

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