Joseph Dillon

Curriculum Vitae

Research Interests

Spent a decade as a theoretical chemist developing methods to model energy and charge transfer processes in molecular systems. For the past few years I’ve been investigating connections between this work and research in other areas of physics and mathematics.

Fig. 4 - Graph of |ζ|²(1/2,t) / |ζ’|²(1/2,t)[Green], 16/j(t)²[Blue] and the
difference between 16/j(t)² and 4/Log²(t/2π)[Dashed Red] where
j(t)=2 log(π)-ψ(1/4-it/2)-ψ(1/4+it/2), ψ(s) denotes the digamma
function and j(t)~-2 Log(t/2π) for sufficiently large t.

Fig. 1 - Graph of non-adiabatic potential energy function for (CH)₂N₃.

Fig. 2 - Graph of non-adiabatic potential energy function for H₃O.

Fig. 3 - Transformation groups of graphs of a few special transcendental complex functions f(z), Re(t,x), Im(t,x) and the set of zeros and poles R ∪ γ, where z=t+ix.

Fig. 5 - Graph of |Ξ|²(σ,0)/a(σ)=|ζ|²(1/2,σ) on the critical line along with expansions for zeros ρ< 10 ⁵ at precision of ±2.5 x 10^-31, primes p < 10² and integers j < 10², for 0 < σ < 2.5 x 10² and 10⁵ < σ < 10⁵ + 10².

Recent Conference Presentations

INTERNATIONAL CONGRESS OF MATHEMATICIANS 2026 - Philadelphia, USA

Symmetry and the Riemann zeta function. Poster. (2026).

On Riemann's symmetric functional equation. ICM2026. p 1-27. (2026).

IRISH MATHEMATICAL SOCIETY CONFERENCE 2025 - Maynooth, IE

Properties of the square of the modulus of the Riemann Xi function. Poster. (2025).

IRISH MATHEMATICAL SOCIETY CONFERENCE 2024 - Belfast, UK

Symmetry and the Riemann zeta function. Poster. (2024).

Envelopes of the square of the modulus of the zeta function on the critical line. IMS2024. p 1-10. (2024).

Below are links to a few articles on the Riemann zeta function…

On the convexity of the square of the modulus of the zeta function in the critical strip. p 1-13. (2023).

On the oscillations of the square of the modulus of the Xi function along the real line. p 1-10. (2019).

Algorithms to locate zeros of the zeta function. p 1-5. (2016).

On the oscillations of the square of the modulus of the zeta function along the critical line. p 1-6. (2018).

Poloynomials of a quaternion variable with real coefficients. p 1. (2025/2017).

Complex vector functions: Polynomials, affine varieties and the FTOA. p 1-20. (2025/2015).

Complex vector functions: Rational functions, affine varieties and the icosahedron equation. p 1-17. (2025/2015).

Complex vector functions: Transcendental functions, varieties and the zeta and Xi functions. p 1-19. (2025/2015).

Below are links to a poster on quaternions and physics and a few related articles on complex functions…