A Succinct Proof Of The Riemann Hypothesis

A succinct proof of the Riemann hypothesis Werner Raab The Mellin transform v(s) =

π = sin(πs)(1/2 − s)ζ(3/2 − s)

Z

∞

ts−1 w(t) dt

0

of the function w(t) =

=

=

1 2πi

Z

1/2+i∞

t−s v(s) ds =

1/2−i∞

∞ X

Ress=−k t−s v(s)

k=0

∞ ∞ X (−t)k X µ(ν) (−t)k = (1/2 + k)ζ(3/2 + k) 1/2 + k ν=1 ν 3/2+k k=0 k=0 ∞ X

µ ¶1/2+k ∞ ∞ (−t)k 2 X µ(ν) X (−1)k t √ = (1/2 + k)ν 1/2+k t ν=1 ν k=0 1 + 2k ν k=0

∞ ∞ X µ(ν) X ν=1

ν

with the M¨obius values µ(ν) is holomorphic in the complex strip: 0 < <s < 1, as Riemann conjectured, since r ∞ 2 X µ(ν) t w(t) = √ arctan = O(1) when t → 0 ν ν t ν=1 and

∞

2 X µ(ν) w(t) = − √ arctan t ν=1 ν

r

ν = O(1/t) when t

One has to observe that r arctan and that

t + arctan ν

∞ X µ(ν) ν=1

ν

1

r

= 0.

ν π = t 2

t → ∞.