# Question on convergence of a formula for the Dirichlet eta function $\eta(s)=(1-2^{1-s})\,\zeta(s)$

Question: Is it true that formula (1) below for $$\eta(s)$$ converges for $$\Re(s)>-m$$ as $$N\to\infty$$ and if so, is formula (1) somehow related to the derivation of formula (2) below for $$\eta(s)$$ which is globally convergent?

(1) $$\quad\eta_{\,m,N}(s)=\sum\limits_{n=1}^N\frac{(-1)^{n-1}}{n^s}+\frac{1}{2^m}\sum\limits_{n=1}^m\left(\sum\limits_{i=1}^{m-n+1}\binom{m}{m-n+1-i}\right)\frac{(-1)^{N+n-1}}{(N+n)^s}\,,\\$$ $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad Re(s)>-m?\land N\to\infty$$

(2) $$\quad\eta(s)=\sum\limits_{n=0}^\infty\frac{1}{2^{n+1}}\sum\limits_{k=0}^n (-1)^k\,\binom{n}{k}\frac{1}{(k+1)^s}\quad$$(see Dirichlet eta function: Numerical Algorithms)

The following figures illustrate formula (1) for $$\eta(s)$$ above seems to converge for $$\Re(s)>-m$$ as $$N\to\infty$$ when evaluated with $$m\in\{1,2,3,4\}$$. In the following figures formula (1) is evaluated with upper limits $$N=100$$, $$N=101$$, $$N=1000$$, and $$N=1001$$ in orange, green, red, and purple respectively overlaid on the blue reference function.

The red discrete dot at $$s=-m$$ visible in most of the figures below illustrates $$\frac{1}{2}\left(\eta_{\,m,100}(-m)+\eta_{\,m,101}(-m)\right)$$. I'll note that I believe $$\frac{1}{2}\left(\eta_{\,m,N}(-m)+\eta_{\,m,N+1}(-m)\right)$$ evaluates exactly to $$\eta(-m)$$ for all values of $$N\ge 0$$, and $$\eta_{\,m,N}(-j)$$ evaluates exactly to $$\eta(-j)$$ where $$j\in \mathbb{Z}\land 0\le j for all values of $$N\ge 0$$.

This leads to the following representations for $$\eta(-j)$$ which I believe are valid for $$j\in\mathbb{Z}\land j\geq 0$$ where $$\binom{a}{b}$$ is the binomial coefficient and $$(a)_k$$ is the Pochhammer symbol.

(3) $$\quad\eta(-j)=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\sum\limits_{i=1}^{j+2-n}\binom{j+1}{j+2-n-i}\\$$ $$\qquad\qquad\quad\,\,=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\binom{j+1}{j+1-n}\,_2F_1(1,n-j-1;n+1;-1)\\$$ $$\qquad\qquad\quad\,\,=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\binom{j+1}{j+1-n}\sum\limits_{k=0}^{\infty}\frac{(-1)^k(n-j-1)_k}{(n+1)_k}$$

Figure (1): Illustration of Formula (1) for $$\eta(s)$$ with $$m=1$$

Figure (2): Illustration of Formula (1) for $$\eta(s)$$ with $$m=2$$

Figure (3): Illustration of Formula (1) for $$\eta(s)$$ with $$m=3$$

Figure (4): Illustration of Formula (1) for $$\eta(s)$$ with $$m=4$$

The following table illustrates $$f(m,n)=\sum\limits_{i=1}^{m-n+1}\binom{m}{m-n+1-i}$$ for $$m=1..n$$ which seems to correlate to https://oeis.org/A104709.

$$\begin{array}{cc} \text{n} & \text{f(m,n)} \\ 1 & \{1\} \\ 2 & \{3,1\} \\ 3 & \{7,4,1\} \\ 4 & \{15,11,5,1\} \\ 5 & \{31,26,16,6,1\} \\ 6 & \{63,57,42,22,7,1\} \\ 7 & \{127,120,99,64,29,8,1\} \\ 8 & \{255,247,219,163,93,37,9,1\} \\ 9 & \{511,502,466,382,256,130,46,10,1\} \\ 10 & \{1023,1013,968,848,638,386,176,56,11,1\} \\ \end{array}$$

• Forjmula (2) comes from MSE question 3182855 "Question on convergence of formula for Dirichlet eta function $\eta(s)$". Jun 7, 2019 at 1:15