# On the nearest integer to $\zeta(1-1/B),B \ge 2$

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 $$B \ge 2$$?

• $\zeta(1+x)=\frac{1}{x}+\gamma+o(1)$. Commented Sep 25, 2021 at 13:04
• @Wojowu Many thanks, I suppose this would require some additional effort to get rid of $+\gamma$.
– joro
Commented Sep 25, 2021 at 13:13
• You don't want to get rid of it - when the $o(1)$ term gets small enough, this implies $\frac{1}{x}<\zeta(1+x)<\frac{1}{x}+1$. To see that it is good enough for $B\geq 2$ would require an explicit estimate on $o(1)$ which I will leave to someone else. Commented Sep 25, 2021 at 13:43
• @Wojowu I suspect the nearest integer might be outside the bounds.
– joro
Commented Sep 25, 2021 at 17:03

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.
• Fractional parts seem to decrease, this might suggest a proof working for these cases too...$$\begin{array}{rl}1 & -0.5 \\ 2 & -1.46035 \\ 3 & -2.44758 \\ 4 & -3.44129 \\ 5 & -4.43754 \\ 6 & -5.43505 \\ 7 & -6.43328 \\ 8 & -7.43196 \\ 9 & -8.43093 \\ 10 & -9.43011 \\ 11 & -10.4294 \\ 12 & -11.4289 \\ 13 & -12.4284 \\ 14 & -13.428 \\ 15 & -14.4277 \\ 16 & -15.4274 \\ 17 & -16.4271 \\ 18 & -17.4268 \\ 19 & -18.4266 \\ 20 & -19.4264 \\ 21 & -20.4263 \\ 22 & -21.4261 \\ 23 & -22.426 \\ 24 & -23.4258 \\ 25 & -24.4257 \\ 26 & -25.4256\end{array}$$ Commented Sep 26, 2021 at 5:37
• (In fact these fractional parts seem to monotonously converge to $1-\gamma$) Commented Sep 26, 2021 at 7:50