Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence: $$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$ And let $\left\{a_{n}\right\}_{n\in\mathbb{N}}$ be a sequence, whose ordinary generating function is $g(x)$: $$g(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\;\;\;\;\;\ |x|<x_{0}\leq 1$$ We want to write the related Dirichlet series: $$\Phi (z)=\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$ in terms of $\Theta(s)$ and $g(x)$. A practical example of such a construction is the everywhere-convergent series for the Riemann zeta function: $$\zeta(z)-\frac{1}{z-1}=\sum_{n=0}^{\infty}\left | G_{n+1} \right |\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{1}{(k+1)^{z}}$$ $\left | G_{n+1} \right |$ being the absolute Gregory coefficients.
My attempt :
We use the formula (due to Apostol) : $$\theta(k+1)=\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\Theta(\sigma+it)(k+1)^{\sigma+it}dt\;\;\;\;\;\;\sigma>\beta$$ Ignoring issues of convergence, we have: $$\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$$$=\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\Theta(\sigma+it)\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(k+1)^{\sigma+it-z}dt$$ Now, we have that : $$\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(k+1)^{-\mu}=\frac{1}{\Gamma(\mu)}\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\int_{0}^{\infty}x^{\mu-1}e^{-(k+1)x}dx$$ $$=\frac{1}{\Gamma(\mu)}\sum_{n=0}^{\infty}a_{n}\int_{0}^{\infty}x^{\mu-1}e^{-x}\left(1-e^{-x}\right)^{n}dx=\frac{1}{\Gamma(\mu)}\int_{0}^{\infty}x^{\mu-1}e^{-x}g(1-e^{-x})dx\;\;\;\Re(\mu)>0$$ Assuming this last integral admits analytic continuation, and denoting it by $\Omega (\mu)$, we have: $$\sum_{n=0}^{\infty}a_{n}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}=\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\Theta(\sigma+it)\Omega(\sigma+it-z)dt$$ But i doubt that this kind of integrals could ever be evaluated, even using the residue theorem, due to the limit, and the $T^{-1}$ factor. Is there an alternative to my approach ?
