Let $B \ge 2$ be integer and $[x]$ denote the nearest integer
to real $x$.


For $2 \le B \le 10^5$ computations with mpmath suggest:


$$ [\zeta(1-1/B)]=-B+1 \qquad (1)$$


Is (1) true for all $B \ge 2$?



On the nearest integer to $\zeta(1-1/B),B \ge 2$
We can make the error mentioned by Wojowu in his comment explicit by using some results on the Laurent coefficients of the zeta function. There are a few results on this, but I'll just use Theorem 2 of [this paper](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-2/issue-1/On-the-Hurwitz-zeta-function/10.1216/RMJ-1972-2-1-151.full) by Berndt (which gives a more general result for the Hurwitz zeta function). The relevant result is


**Theorem.** (Berndt) *Write*
$$
\zeta(s) = \frac{1}{s-1} + \sum\_{n=0}^\infty \gamma\_n (s-1)^n
$$
*for the Laurent series expansion of $\zeta$ around $s=1$. Then for $n\geq 1$, we have*
$$
\left|\gamma\_n\right| \leq \frac{4}{n\pi^n}.
$$
For $B\geq 2$ an integer, write
$$
\zeta\left(1-\frac{1}{B}\right) = -B + \gamma + R(B),
$$
where $\gamma=\gamma\_0\approx .577...$ is the usual Euler-Mascheroni constant and
$$
R(B) = \sum\_{n=1}^\infty \gamma\_n \left(-\frac{1}{B}\right)^n.
$$
Applying the triangle inequality and Berndt's result, we have
$$
\left|R(B)\right| \leq \sum\_{n=1}^\infty \frac{4}{n(B\pi)^n} \leq \frac{4}{B}\sum\_{n=1}^{\infty} \frac{1}{n\pi^n}.
$$
The sum on the right converges rapidly and is numerically $\approx 0.383...$. Thus
$$
\left| R(B) \right| \leq \frac{2}{B}.
$$
(By a more careful analysis, the constant 2 can be replaced with 1.) For $B \geq 26$, it follows that
$$
\frac{1}{2} < \gamma+ R(B) < 1.
$$
Therefore
$$
\left[ \zeta\left(1-\frac{1}{B} \right) \right] = -B+1
$$
for $B \geq 26$. The remaining cases can be verified via computer.


