# On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

## TL;DR

This paper investigates the integer part of the reciprocal of the tail of the Riemann zeta function at specific rational points within the critical strip, providing explicit descriptions for these values.

## Contribution

It establishes explicit formulas for the integer part of the reciprocal of the zeta tail at certain rational numbers, utilizing finiteness results on integral points of algebraic curves.

## Key findings

- Explicit formulas for the integer part of 1/ζ(s) tail at s=1/p and s=2/p
- Application of finiteness of integral points on algebraic curves
- Advances understanding of zeta function behavior at rational points

## Abstract

We prove that the integer part of the reciprocal of the tail of $\zeta(s)$ at a rational number $s=\frac{1}{p}$ for any integer with $p \geq 5$ or $s=\frac{2}{p}$ for any odd integer with $p \geq 5$ can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p},$ we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$.