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We prove that the integer part of the reciprocal of the tail of $\zeta(s)$ ata 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 describedessentially as the integer part of an explicit quantity corresponding to it. Todeal with the case when $s=\frac{2}{p},$ we use a result on the finiteness ofintegral points of certain curves over $\mathbb{Q}$.

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