I have been playing around with slices like this of $1 + 2 + 3 + 4 + ...$ e.g. $1 + 3 + 5 + 7 + ...$, or $2 + 5 + 8 + 11 + ...$ using Dirichlet regularization in Mathematica and it seems the result is always an integer multiple of $1/12$.
Does anyone here have a hint why this would be the case?
We have $$ \sum_{n=1}^{+\infty} 1 \underset{\zeta}{=}\zeta(0)=-\frac{1}{2}\qquad \sum_{n=1}^{+\infty}n \underset{\zeta}{=} \zeta(-1)=-\frac{1}{12}$$ hence $$ \sum_{n=1}^{+\infty}(kn+i) \underset{\zeta}{=} \color{red}{-\frac{k+6i}{12}}.$$