Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line
Theorem
The nontrivial zeroes of the Riemann $\zeta$ function are distributed symmetrically with respect to the critical line.
That is, suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\map \zeta s$.
Then there exists another nontrivial zero $s_2$ of $\map \zeta s$ such that:
- $s_2 = 1 - s_1$
Proof
From Functional Equation for Riemann Zeta Function, we have:
- $\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$
We suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\map \zeta s$.
Then we have:
\(\ds \map \zeta {s_1}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \pi^{-s_1/2 } \map \Gamma {\dfrac {s_1} 2} \map \zeta {s_1}\) | \(=\) | \(\ds 0\) | left hand side | |||||||||||
\(\ds \pi^{\paren {s_1/2 - 1/2 } } \map \Gamma {\dfrac {1 - s_1} 2} \map \zeta {1 - s_1}\) | \(=\) | \(\ds 0\) | right hand side |
It remains to be shown that of the three terms on the right hand side, $\map \zeta {1 - s_1}$ MUST equal zero.
We can rewrite the first term in terms of the exponential function
- $\ds \pi^{\paren {s_1 / 2 - 1 / 2} } = \map \exp {\map \ln {\pi^{\paren {s_1 / 2 - 1 / 2} } } }$
From the definition of the exponential function, we know $\map \exp {\map \ln {\pi^{\paren { {s_1}/2 - 1/2 } } }}$ never equals zero.
From Zeroes of Gamma Function, we know $\map \Gamma {\dfrac {1 - s_1} 2}$ never equals zero.
Therefore:
- $\map \zeta {1 - s_1} = 0$
$\blacksquare$
Also see
- Riemann Hypothesis: If this is true, all nontrivial zeroes are already on the critical line.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,5$