A Sharp Estimate for Divisors of Bernoulli Sums

Let $S_n={\text{e}}_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\text{e}}_n$, where ${\text{e}}_i$ are i.i.d. Bernoulli r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d -r_d(n)$and$\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar r_d(n)^2/2 n}}]$\mathrm{.}Weshowthat$$\sup_{2\le d\le n} \big|\P\big{d|S_n\big}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over n^{3/2}}\big), $$where$E(n,d) $verifies$c_1\b(n,d)\le E(n,d)\le c_2\b(n,d) $and$c_1,c_2 $ are numerical constants.