/sci/ - Science & Math

https://arxiv.org/pdf/2005.04568.pdf
>The Chi Function, Tau Function, and Riemann Zeta Function
>Jianyun Zhang

>The fundamental relationship ζ(s)=χ(s)ζ(1−s) reveals that no special function is more suitable than the chi function χ(s) to study the Riemann zeta function ζ(s). When t is sufficiently large, the modulus and argument of χ(σ+it) are monotone about σ and t respectively, which accomodates the construction of a Riemann surface for the multivalued z=χ(s). The inverse of χ(s), which is branched and much similiar to the logarithm function, is introduced as tau function s=τ(z). Then ζ(s)=ζ∘τ(z) can be studied by its different branches, with a much simpler relationship in single-valued domains, which finally leads to the conclusion about the nontrivial zeros of ζ(s).