Variation of Archimedean Zeta Function and $n\mathrm{/}d$-Conjecture for Generic Multiplicities

For $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_r\in \mathbb{C}[z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n]\setminus \mathbb{C}$, we introduce the variation of archimedean zeta function. As an application, we show that the $n\mathrm{/}d$-conjecture, proposed by Budur, Mustaţă, and Teitler, holds for generic multiplicities. Consequently, strong monodromy conjecture holds for hyperplane arrangements with generic multiplicities as well.