An expansion of the Riemann Zeta function on the critical line

We give an expansion of the Riemann Zeta function on the critical line as a converging series $\sum_{m\geq 0}a_m q_m(\frac12+it) $ in the space $L^2(\mathbb{R},\frac{dt}{\text{cosh}(\pi t) })$, where the functions $q_m$ are related to Meixner polynomials of the first kind and the coefficients $a_m$ are linear combinations of the Euler constant $\gamma$ and the values $\zeta(2),\zeta(3),\dots,\zeta(m+1).$

Bernard Candelpergher (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

We give an expansion of the Riemann zeta function on the critical line as a converging series m≥0 a m q m (1 2 + it) in the space L 2 (R, dt cosh(πt)), where the functions q m are related to Meixner polynomials of the first kind and the coefficients a m are linear combinations of the Euler constant γ and the values ζ(2), ζ(3),. .. , ζ(m + 1).

Domaines: Mathématiques [math]

Bernard Candelpergher. An expansion of the Riemann Zeta function on the critical line. 2021. ⟨hal-03265364⟩