Proof Of The Riemann Hypothesis

Proof of the Riemann hypothesis Werner Raab The Mellin transform v(s) =

π = sin(πs)(1/2 − s)ζ(3/2 − s)

of the function 1 w(t) = 2πi

Z

1/2+i∞

Z

∞

ts−1 w(t) dt

0

t−s v(s) ds

1/2−i∞

is holomorphic within the complex strip: 0 < <s < 1, as Riemannn conjectured, since r ∞ 2 X µ(ν) t w(t) = √ arctan = O(1) when t → 0 ν t ν=1 ν and

∞

2 X µ(ν) w(t) = − √ arctan t ν=1 ν

r

ν = O(1/t) when t

t → ∞.

These formulae follow from the fact that the Dirichlet series ∞

X µ(ν) 1 = ζ(s) ν=1 ν s with the M¨obius values µ(ν) converges in the complex half plane: <s > 1 and from the transformation r Z ∞ t ν s−1/2 s−3/2 t arctan dt = π ν sin(πs)(1 − 2s) 0 (cf. W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Funnctions of Mathematical Physics, Third Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1966, p. 454), which is valid within the strip: 0 < <s < 1/2. Finally, one has to observe that r r π t ν arctan + arctan = ν t 2 and that

∞ X µ(ν) ν=1

ν

1

= 0.