Let $zeta(s)$ be the Riemann zeta function and $gamma$ be the Euler-Mascheroni constant. I observed the following result empirically. Looking for a proof or disproof.

$$

lim_{n to infty}sum_{k = 1}^n zetaBig(k – frac{1}{n}Big) = gamma

$$

Also, I searched for different summation formulas for the Euler-Mascheroni constant using the Riemann zeta function but could not find it any. Is there any reference to his sum in literature?