This document provides a proof of the Riemann Hypothesis by investigating the characteristics of the Dirichlet eta function, which has the same zeros as the Riemann zeta function. It shows that the derivative of the implicit function for the real component of the eta function when the real and imaginary components are equal is always non-zero. This means there can be at most one zero for each value of the imaginary part of s. Combined with the fact that the zeros of the Riemann xi function are also zeros of the zeta function and xi(s)=xi(1-s), this leads to the conclusion that all non-trivial zeros must lie on the critical line with real part of 1/2, proving

1. Why the Dirichlet eta function and
so the Riemann zeta Function have
only zeros with real part = ½.
A Proof of the Riemann Hypothesis
A. A. Logan February 2019

2. Overview
• Abstract
• Zeta function observations (when Re(zeta) = Im(zeta)).
• Dirichlet eta function - Implicit function of Real
component = Imag component – derivative of value of
real component always nonzero so only one zero for
one value of imag part of s (using harmonic addition
theorem).
• Riemann xi function – implications.
• Nb xi(s) = xi(1-s).
• Conclusions, link to paper and contact details.

3. Abstract
• This paper investigates the characteristics of the zeros of the
Riemann zeta function (of s) in the critical strip by using the
Dirichlet eta function, which has the same zeros.
• The characteristics of the implicit functions for the real and
imaginary components when those components are equal are
investigated and it is shown that the function describing the value
of the real component when the real and imaginary components
are equal has a derivative that does not change sign along any of
its individual curves - meaning that each value of the imaginary
part of s produces at most one zero.
• Combined with the fact that the zeros of the Riemann xi function
are also the zeros of the zeta function and xi(s) = xi(1-s), this leads
to the conclusion that the Riemann Hypothesis is true.

4. Zeta function components

5. Zeta function components - detail

6. Zeta function observations
•
• Different symmetries of Re(zeta) and Im(zeta).
• Path of Re(zeta)=Im(zeta) example (slope).
• Value of Re(zeta) when Re(zeta)=Im(zeta).
• Apparent single valued curve.

7. Riemann Zeta Function and
Dirichlet Eta Function
Note Zeta converges absolutely for s>1
Eta converges uniformly (not absolutely) for s>0
Eta zeroes in critical strip same as Zeta zeroes.

8. Dirichlet eta function
Extracting real and imaginary parts for one term:
Real part:
Imag part:

9. Implicit function Real(eta)=Imag(eta)

10. Differentiating Implicit Function(1)
Implicit function:
Differentiating:
Substituting into exp for real component:

11. Differentiating Implicit Function(2)
Similarly:
And:
Nb this describes the same function as the last
slide.

12. Harmonic addition Theorem(1)
Using partial sums and substituting in (exp 1):
And:(exp 2)
Also (exp3):
And: (exp4)

13. Harmonic addition Theorem(2)
Beta does not equal zero
Csc has no zeros
Exp2 is the derivative of the real part of eta
(when re(eta)=imag(eta)) and is always positive
(or always negative) for all a, sigma (except
where undefined).
The same holds for Exp4 (undefined at different
points).
This means only one zero for each curve
segment (or max one zero for each value of a)

14. Riemann xi Function
• xi function - Power Series
• xi function zeros
• See above slide for only one root per value of
a - so sigma = ½.

15. Conclusions
• Eta(s) has same roots as zeta(s)
• Eta(s) real component value when Re(eta) =
Im(eta) – shows always non-zero derivative.
• eta(s) only 1 zero per fixed imag part of s.
• xi(s) = xi(1-s) means that root must be at real
part of s = ½: Riemann Hypothesis confirmed.
• http://vixra.org/abs/1802.0124
• andrewalogan@gmail.com