MMN-2877
Efficient estimate of the remainder for the Dirichlet function $\eta(p)$ for $p\in \mathbb{R}^+$
V. Lampret;Abstract
For any $k,n,q\in\N$, where $n\ge 2k+1\ge3$, and $p\in\R^+$ the
$n$th remainder $\rho(n,p)$ of the Dirichlet eta function
$\eta(p)$, known also as the alternating Riemann zeta function,
\[
\eta(p):=\sum_{i=1}^{\infty}(-1)^{i+1}\frac{1}{i^p},
\]
is given in the form
\[
\rho(n,p)=\Delta_q(n,p)+\delta_q(k,p),
\]
where $k$ and $q$ are parameters controlling the magnitude of the
error term $\delta_q(k,p)$. The function $\Delta_q(n,p)$ consists
of $\lfloor q/2\rfloor +1$ simple summands and $\delta_q(k,n,p)$
is estimated as
\begin{equation*}
\big|\delta_q(k,p)\big| <
\frac{2}{3}\cdot\frac{p^{(q-1)}}{\pi^{q-2}}\cdot
\frac{1}{(2k)^{p+q-1}} \,,
\end{equation*}
where $p^{(q-1)}$ is the rising Pochhammer product.
Vol. 21 (2020), No. 1, pp. 241-247