arXiv subjects

Lukas Turčinskas

Publications and source records attributed to Lukas Turčinskas.

2 recordsLinked to original sources

arXiv2026

Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function

We prove four explicit continued fraction representations for the Lerch transcendent $Φ(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $ζ(2, M+1)$ and $ζ(3, M+1)$.

math.NT↗

arXiv2025

Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line

In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $ξ$-function in the region $1/2+1/\sqrt{\log t}<σ<1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}\sum_ρ1/(s-ρ)$, where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the log-derivative of the Riemann $ξ$-function in the provided region, assuming that there are non-trivial zeta-zeros off the critical line.

math.NT↗