Is this :$\sum_{n=1}^{\infty }\zeta(2n)-\zeta(2n+1)=\frac{1}{2}$?

Question

I would like to know the behavior of the Riemann zeta function values at even and odd integers for studying irrationality between those values. I have tried using wolfram alpha to check the value of this sum: $$ \sum_{n = 1}^{\infty}\left[\zeta\left(2n\right)-\zeta\left(2n + 1\right)\right]. $$ It tells me it equals $\frac{1}{2}$ .

Note: I don't have any method to show if the titled sum is true . Maybe I find who is help me here for evaluating the titled sum.

Thanks for any help.

Answer 1

Note that , due to the absolute convergence, we have$$\sum_{n\geq1}\left(\zeta\left(2n\right)-\zeta\left(2n+1\right)\right)=\sum_{n\geq1}\left(\sum_{k\geq1}\frac{1}{k^{2n}}-\sum_{k\geq1}\frac{1}{k^{2n+1}}\right) $$ $$=\sum_{n\geq1}\sum_{k\geq1}\frac{k-1}{k^{2n+1}}=\sum_{k\geq2}\sum_{n\geq1}\frac{k-1}{k^{2n+1}} $$ $$=\sum_{k\geq2}\frac{1}{k\left(k+1\right)}=\sum_{k\geq2}\left(\frac{1}{k}-\frac{1}{k+1}\right)=\color{red}{\frac{1}{2}}.$$

Answer 2

The computation is not hard:

\begin{align*} \sum_{n=1}^{\infty}[\zeta(2n) - \zeta(2n+1)] &= \sum_{n=1}^{\infty}(-1)^{n+1}[\zeta(n+1) - 1] \\ &= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}\sum_{k=2}^{\infty} \frac{n!}{k^{n+1}} \\ &= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}\sum_{k=2}^{\infty} \int_{0}^{\infty} x^n e^{-kx} \, dx \\ &\stackrel{(1)}{=} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}\int_{0}^{\infty} \frac{x^n e^{-2x}}{1 - e^{-x}} \, dx \\ &\stackrel{(2)}{=} \int_{0}^{\infty} \bigg( \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!} x^n \bigg)\frac{e^{-2x}}{1 - e^{-x}} \, dx \\ &= \int_{0}^{\infty} e^{-2x} \, dx = \frac{1}{2}. \end{align*}

For $\text{(1)}$, we utilized Tonelli's theorem to interchange the order of integral and summation. For $\text{(2)}$, we utilized Fubini's theorem along with the estimate:

$$ \int_{0}^{\infty} \bigg( \sum_{n=1}^{\infty}\frac{x^n }{n!} \bigg)\frac{e^{-2x}}{1 - e^{-x}} \, dx = \int_{0}^{\infty} e^{-x} \, dx < \infty. $$

Answer 3

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\sum_{n = 1}^{\infty}\bracks{\zeta\pars{2n}-\zeta\pars{2n + 1}} = \overbrace{% \sum_{n = 1}^{\infty}\bracks{\zeta\pars{2n} - 1}}^{\ds{3 \over 4}}\ -\ \overbrace{% \sum_{n = 1}^{\infty}\bracks{\zeta\pars{2n + 1} - 1}}^{\ds{1 \over 4}} = \bbx{1 \over 2} \end{align}

The 'even' and 'odd' above sums are well known identities.