TITLE:
How to Prove Riemann Conjecture by Riemann’s Four Theorems
AUTHORS:
Chuanmiao Chen
KEYWORDS:
Riemann Conjecture, Zeta-Function, Xi-Function, Functional Equation, Product Expression, Contradiction
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.8,
August
22,
2024
ABSTRACT: Riemann (1859) had proved four theorems: analytic continuation
ζ(
s
)
, functional equation
ξ(
z
)=G(
s
)ζ(
s
)
(
s=1/2
+iz
,
z=t−i(
σ−1/2
)
), product expression
ξ
1
(
z
)
and Riemann-Siegel formula
Z(
z
)
, and proposed Riemann conjecture (RC): All roots of
ξ(
z
)
are real. We have calculated
ξ
and
ζ
, and found that
ξ(
z
)
is alternative oscillation, which intuitively implies RC, and the property of
ζ(
s
)
is not good. Therefore Riemann’s direction is correct, but he used the same notation
ξ(
t
)=
ξ
1
(
t
)
to confuse two concepts. So the product expression only can be used in contraction. We find that if
ξ
has complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of
ζ
have real part 1/2. Of course, RH also holds, but can not be proved directly by
ζ(
s
)
.