# nt.number theory – Continuing an analytic continuation of the Dirichlet \$eta\$-function?

The Dirichlet $$eta$$-function is defined as:
$$eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} qquad Re(s) > 0$$

and has the full analytical continuation:

$$eta(s) = sum_{n=1}^N frac{(-1)^{n+1}}{n^s} + frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dx qquad s in mathbb{C} tag{1}$$

valid for all integers $$N ge 0$$.

Wondered what would happen for negative $$N$$ and found numerically that:

$$eta(s) = sum_{n=1}^{-N-1} frac{(-1)^{n+1}}{n^s} + (-1)^{s+1},frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dxqquad s in mathbb{Z} tag{2}$$

valid for all integers $$N < 0$$.

Note: assume the sums to be zero when their end values are $$< 1$$.

Question:

Is there a way to also expand equation (2) to $$s in mathbb{C}$$? If possible, I believe this would require some smart continuation of the $$(-1)^{s+1}$$ factor. Experimented with functions like $$cosleft(pi(s+1)right)$$, but no success yet.