Sum of elements in finite Sidon sets. II

A set $S\subset\{1,2,\dots,n\}$ is called a Sidon set if all the sums $a+b$ are different for different unordered pairs $(a,b)\in S^2$. Let $S_n$ be the largest cardinality of a Sidon set in $\{1,2,\dots,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,\,|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon＞0$ is an arbitrarily small constant. In this note, we improve the error term by showing that $O\left(n^{11/8}\log n\right)$ is true for almost all integers $n$ in the above formula. Besides, we give some extensions of the former results. For any positive integers $\ell$ and $s$, we obtain the asymptotic formulae of the following summations $$\sum_{\substack{S=\{ a_1＜a_2＜\cdots ＜a_t\}\\S\subset[1,n]\operatorname{Sidon}}}a_i^\ell, \qquad \text{and}\qquad \sum_{\substack{S=\{a_1＜a_2＜\cdots ＜a_t\}\\S\subset[1,n]\operatorname{Sidon}}}i^sa_i^\ell,$$ when $t$ is near the magnitude $n^{1/2}$.