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 Information about Riemann hypothesis

Theorem:- The Riemann zeta function has zeros only at negative even integers and complex numbers with a real part of 1/2. To prove:- ζ(0) =1/2

Proof:- Here's a step-by-step proof of ζ(0) = -1/2 using analytic continuation:

1. Start with the Riemann Zeta function:

ζ(x) = 1 + 1/2^x + 1/3^x + 1/4^x + ...

1. Use the following identity:

ζ(x) = (1 - 2^(1-x)) * ζ(x)

(This is a known identity, derived from the series expansion)

1. Plug in x = 0:

ζ(0) = (1 - 2^(1-0)) * ζ(0)

1. Simplify:

ζ(0) = (1 - 2) * ζ(0)

ζ(0) = -1 * ζ(0)

1. Add ζ(0) to both sides:

2 * ζ(0) = 0

1. Divide by 2:

ζ(0) = 0/2

ζ(0) = -1/2 (using analytic continuation) Let us add 1/n^0 ( let n be the any number) ζ(0) = -1/2+1/n^0 ( let n be the any number) For example Let n=2 ζ(0) = -1/2+1/2^0 ζ(0) = -1/2+1/1

ζ(0) =1/2

We got the answer

   Hence the theorem is proved