Hey, Gottfried very interesting.
I have too done indefinite summation excessively. I wrote a paper in my second year of undergrad. To summarize, I'll give the formula for indefinite summation as I wrote it.
If
\(
|f(s)| \le C e^{\tau|\Im(s)| + \rho |\Re(s)|}\,\,\text{for}\,\,0\le \tau < \pi/2\,\,\rho > 0\,\,C>0\\
f\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Then the indefinite sum,
\(
F(s) = \sum_{j=1}^s f(j)\\
F(s) + f(s+1) = F(s+1)\\
F(1) = f(1)\\
F\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Can be given by the formula, for Euler's Gamma function \( \Gamma(s) \),
\(
\vartheta(x) = \sum_{n=0}^\infty (\sum_{j=1}^{n+1} f(j))\frac{(-x)^n}{n!}\\
\Gamma(1-s)F(s) = \sum_{n=0}^\infty (\sum_{j=1}^{n+1} f(j))\frac{(-1)^n}{n!(n+1-s)} + \int_1^\infty \vartheta(x)x^{-s}\,dx\\
F(s) = \frac{d^{s-1}}{dx^{s-1}}|_{x=0} \vartheta(-x)\\
\)
If you're curious I can write a quick write-up. It's largely a simple consequence of Ramanujan's Master Theorem. I would link the original paper but, it has much to be desired from. It was one of the first papers I ever wrote so it's a tad hand-wavey.
This function will be unique, so if your bernoulli sum \( H \) satisfies,
\(
|H(s)| \le C e^{\tau|\Im(s)| + \rho |\Re(s)|}\,\,\text{for}\,\,0\le \tau < \pi/2\,\,\rho > 0\,\,C>0\\
H\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Then it is equivalent to \( F \) when taking \( f(s) = \log^a(s) \).
Not too sure if this helps at all; but exponentially bounded indefinite sums are very simple to construct (largely due to Ramanujan, I just made a few short-cuts in his construction).
I have too done indefinite summation excessively. I wrote a paper in my second year of undergrad. To summarize, I'll give the formula for indefinite summation as I wrote it.
If
\(
|f(s)| \le C e^{\tau|\Im(s)| + \rho |\Re(s)|}\,\,\text{for}\,\,0\le \tau < \pi/2\,\,\rho > 0\,\,C>0\\
f\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Then the indefinite sum,
\(
F(s) = \sum_{j=1}^s f(j)\\
F(s) + f(s+1) = F(s+1)\\
F(1) = f(1)\\
F\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Can be given by the formula, for Euler's Gamma function \( \Gamma(s) \),
\(
\vartheta(x) = \sum_{n=0}^\infty (\sum_{j=1}^{n+1} f(j))\frac{(-x)^n}{n!}\\
\Gamma(1-s)F(s) = \sum_{n=0}^\infty (\sum_{j=1}^{n+1} f(j))\frac{(-1)^n}{n!(n+1-s)} + \int_1^\infty \vartheta(x)x^{-s}\,dx\\
F(s) = \frac{d^{s-1}}{dx^{s-1}}|_{x=0} \vartheta(-x)\\
\)
If you're curious I can write a quick write-up. It's largely a simple consequence of Ramanujan's Master Theorem. I would link the original paper but, it has much to be desired from. It was one of the first papers I ever wrote so it's a tad hand-wavey.
This function will be unique, so if your bernoulli sum \( H \) satisfies,
\(
|H(s)| \le C e^{\tau|\Im(s)| + \rho |\Re(s)|}\,\,\text{for}\,\,0\le \tau < \pi/2\,\,\rho > 0\,\,C>0\\
H\,\,\text{is holomorphic for}\,\,\Re(s) > 0\\
\)
Then it is equivalent to \( F \) when taking \( f(s) = \log^a(s) \).
Not too sure if this helps at all; but exponentially bounded indefinite sums are very simple to construct (largely due to Ramanujan, I just made a few short-cuts in his construction).