Timeline for Asymptotic behavior of weighted sums involving the fractional part function

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Nov 15 at 8:46	comment	added	Babar		The fact that $\theta$ is the same in my previous comment seems to be due to the conjectured relation $$S_{m+1}(n)-S_{m}(n)=\eta\left(m\right)n^{m}+O\left(1\right)$$ for $m\geq0$.
Nov 15 at 8:24	comment	added	Babar		Let $$S_m(n) = \sum_{k=1}^{n-1} (-1)^{k-1} \left(1 - \frac{k}{n}\right)^{-m} \left\{ \frac{n}{k} \right\}$$ then I claim that:$$S_0(n) = O\left(n^\theta\right)$$, where $\theta = \frac{1}{4} + \varepsilon$ is the conjectured optimal value (similar to the Dirichlet divisor problem). $$ S_1(n) = O\left(n^\theta\right)$$ And in general for $m\geq2$ $$S_m(n) = \sum_{k=1}^{m-1} \eta(k)n^k + O\left(n^\theta\right)$$ where $ \eta(s) = \sum_{n \geq 1} \frac{(-1)^{n-1}}{n^s}$. For instance $$S_4(n) = \frac{3}{4}\zeta(3)n^3 + \frac{1}{2}\zeta(2)n^2 + \log(2)n + O\left(n^\theta\right)$$.
Nov 14 at 20:37	comment	added	Babar		Very nice! I encountered binomials in another conjecture that is probably more accessible. This is probably known but I haven't found any references. If for any integer value $\lambda \geq 0$ we have: $$C_{\lambda}(j) := \sum_{i \geq 0} f(i)i^{\lambda+j}$$ converges, then through experimentation I discovered that for any $n_0 \geq 1$ the following asymptotic formula holds: $$\sum_{k=1}^{n-1} f(n-k) \left\{\frac{n}{k}\right\}^{\lambda}= \sum_{j=0}^{n_0} \frac{1}{n^{\lambda+j}} \binom{\lambda-1+j}{\lambda-1} C_{\lambda}(j) + O(n^{-n_0-1})$$
Nov 13 at 22:21	history	answered	Henri Cohen	CC BY-SA 4.0