A Short Proof Of The Riemann Hypothesis

A short proof of the Riemann hypothesis Werner Raab We consider the meromorphic function u(s) =

1 (s − 1)ζ(s)

(<s > 0) and the Mellin transform u(3/2 − s) v(s) = π = sin(πs)

Z

∞

ts−1 w(t) dt

0

of the function w(t) =

=

∞ X

1 2πi

Z

1/2+i∞

t−s v(s) ds =

1/2−i∞

∞ X

Ress=−k t−s v(s)

k=0

u(3/2 + k)(−t)k =

k=0

∞ 1 X m t m ∆ u(3/2)( ) 1 + t m=0 1+t

with the differences m

∆ u(3/2) =

m µ ¶ X m k=0

k

(−1)k u(3/2 + k)

(|t| < 1 or −1/2 respectively). For positive real values of t we have Z ∞ 1 1 tiy w(t) = √ = dy → u(3/2) t 0 cosh(πy)y ζ(1 + iy) when t → 0 and 1 1 1 w( ) = √ t t t

Z 0

∞

1 t−iy = dy → u(1/2) cosh(πy)y ζ(1 + iy)

when t → 0. The properties w(t) = O(1) when t → 0 and w(t) = O(1/t) when t → ∞ of the Mellin inverse w(t) imply that the Mellin transform v(s) is holomorphic within the complex strip: 0 < <s < 1, as Riemann conjectured.

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