Sketch Of A Proof Of The Riemann Hypothesis
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Sketch of a proof of the Riemann hypothesis Werner Raab We consider the Euler series transformation ∞ X
u(s + k)(−t)k = f (s, t) =
k=0
∞ 1 X m t m ∆ u(s)( ) 1 + t m=0 1+t
(|t| < 1 or −1/2 respectively) with the finite differences m X m ∆m u(s) = (−1)k u(s + k) = ∆m u(s − 1) − ∆m+1 u(s − 1) k k=0
of the function u(s) =
1 (s − 1)ζ(s)
(for poisitive real values of s). It follows from the convergence of the series ∞ X u(1/2 + k)
2k
k=0
= f (1/2, −1/2) = 2
∞ X
∆m u(1/2)(−1)m
m=0
n
that limn→∞ ∆ u(1/2) = 0, which is why u(1/2) = u(1/2) − lim ∆n u(1/2) = lim n→∞
= lim
n→∞
n−1 X
m
∆ u(3/2) =
m=0
n→∞
∞ X
m
= lim
t→∞
m=0
∆m u(3/2)(
(∆m u(1/2) − ∆m+1 u(1/2))
m=0
∆ u(3/2) =
m=0 ∞ X
n−1 X
∞ X m=0
∆m u(3/2) lim ( t→∞
t m ) 1+t
t m ) = lim (f (3/2, t)t) , t→∞ 1+t
due to Abel’s theorem of continuity for power series. Thus we can conclude that Z ∞ π u(3/2 − s) = ts−1 w(t) dt v(s) = sin(πs) 0 is holomorphic in the complex strip: 0 < <s < 1, as Riemann conjectured, since Z 1/2+i∞ ∞ X 1 w(t) = t−s v(s) ds = Ress=−k t−s v(s) = f (3/2, t) 2πi 1/2−i∞ k=0
has the properties w(t) = O(1) when t → 0 and w(t) = O(1/t) when t → ∞. 1
u(s + k)(−t)k = f (s, t) =
k=0
∞ 1 X m t m ∆ u(s)( ) 1 + t m=0 1+t
(|t| < 1 or
of the function u(s) =
1 (s − 1)ζ(s)
(for poisitive real values of s). It follows from the convergence of the series ∞ X u(1/2 + k)
2k
k=0
= f (1/2, −1/2) = 2
∞ X
∆m u(1/2)(−1)m
m=0
n
that limn→∞ ∆ u(1/2) = 0, which is why u(1/2) = u(1/2) − lim ∆n u(1/2) = lim n→∞
= lim
n→∞
n−1 X
m
∆ u(3/2) =
m=0
n→∞
∞ X
m
= lim
t→∞
m=0
∆m u(3/2)(
(∆m u(1/2) − ∆m+1 u(1/2))
m=0
∆ u(3/2) =
m=0 ∞ X
n−1 X
∞ X m=0
∆m u(3/2) lim ( t→∞
t m ) 1+t
t m ) = lim (f (3/2, t)t) , t→∞ 1+t
due to Abel’s theorem of continuity for power series. Thus we can conclude that Z ∞ π u(3/2 − s) = ts−1 w(t) dt v(s) = sin(πs) 0 is holomorphic in the complex strip: 0 < <s < 1, as Riemann conjectured, since Z 1/2+i∞ ∞ X 1 w(t) = t−s v(s) ds = Ress=−k t−s v(s) = f (3/2, t) 2πi 1/2−i∞ k=0
has the properties w(t) = O(1) when t → 0 and w(t) = O(1/t) when t → ∞. 1
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