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Aim A particular instance of a rational zeries that has as of yet not been evaluated is: \begin{align} Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1} \end{align} This sum ...

Background Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...

In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is ...

The Ramanujan's $\tau$-function is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$. Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....

I have been playing with the following function: $$ f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k} $$ It is hard to get correct numerical values. I'll start with ...

start at the origin, first step number is 1. turn $\pi/n$ move $1/n$ units forward Angles are cumulative, so this procedure is equivalent (finitely) to $$ u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...