Joseph Dillon

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. 3 - 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. 4 - Graph of |Υ|²(t,σ)=|ζ|²(1/2-t,σ) / |ζ|²(1/2+t,σ) for 7≤σ≤100 and
|t|< 1/2. Note that (σ,t,Y)∊ℝ³ such that |Υ|²(t,σ)=1 are marked
in Red and that t=0 corresponds with the critical line.

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

Fig. 5 - 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.

Below are links to posters I presented at IMS2024 and IMS2025, in addition to a supporting article…

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

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

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

Below are links to a few articles on properties of 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. (2017).

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. (2025/2018).

Foundations for a general theory of complex vector functions: Polynomial functions, affine varieties and the FTOA. p 1-20. (2025/2015).

Foundations for a general theory of complex vector functions: Rational functions, affine varieties and the icosahedron equation. p 1-17. (2026/2015).

Foundations for a general theory of complex vector functions: Transcendental functions, varieties and the zeta and Xi functions. p 1-19. (2026/2015).

On the method of separation of variables for complex partial differential equations. p 1-9. (2026/2015).

Research Interests

Below are links to posters I presented at IMS2024 and IMS2025, in addition to a supporting article…

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

Below is a link to a poster on quaternions and physics, in addition to a related article on complex functions that I’m working on…