I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at least positive real input, and likely real analytic.
\begin{equation} f(z)=\sum_{n=0}^{\infty} \sum_{m=1}^{n+1} \text{sinc}(\pi (n^2 + n - mz)) \end{equation}
Why I am interested in this specific series:
One can prove (this hinges on a combination of the Chinese Remainder Theorem and Hensel's Lemma) that when $z$ is a positive integer,
\begin{equation} 2^{\omega(z)}=\sum_{n=0}^{\infty} \sum_{m=1}^{n+1} \text{sinc}(\pi (n^2 + n - mz)) \end{equation}
where $\omega(z)$ gives the number of distinct prime factors of $z$. This is a modification of a previous, related result that is not everywhere analytic.
An interesting side note (albeit probably not useful) is that the Riemann Hypothesis implies the equation below. (See "Ueber die zahlentheoretische Funktion $\omega(n)$" by Dieter Wolke.)
\begin{equation} \sum_{k \leq x} \log_2 (\sum_{n=0}^{\infty} \sum_{m=1}^{n+1} \text{sinc}(\pi (n^2 + n - mk))) = x \ln \ln x + Bx -x \int_1^{\sqrt{x}} \frac{\{ t \} }{t^2 (\ln x - \ln t)} dt + \mathcal{O}(x^{\frac{2}{3}} (\ln x)^{\frac{1}{3}}) \end{equation}
