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Tag Archives: Zeta
Sum of natural numbers equal 1/12 ?
$$\tag{1}1+2+3+4+\cdots =^?\frac{1}{12}$$ There are many problems with that equality. Can a summation of positive integers lead to a negative quantity ? The equality implies that $$\sum_{k=1}^\infty k = \frac{1}{12}$$ But we already know that $$S_N = \sum_{k=1}^Nk = \frac{N(N+1)}{2} \to … Continue reading
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Tagged 1/12, Divergent, natural numbers, negative, sum, value, Zeta
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Relation between Zeta and Dirichlet eta functions proof
$$\eta(s) = \left( 12^{1s} \right) \zeta(s) $$ $$\textit{proof}$$ We will start by the RHS $$\left( 12^{1s} \right) \zeta(s) = \zeta(s) – 2^{1s} \zeta(s)$$ Which can be written as sums of series $$\sum_{n=1}^\infty \frac{1}{n^s} – \frac{1}{2^{s1}}\sum_{n=1}^\infty \frac{1}{n^s}$$ $$\sum_{n=1}^\infty \frac{1}{n^s} – 2\sum_{n=1}^\infty … Continue reading
Integral representation of the zeta function proof
$$\zeta(s) = \frac{1}{\Gamma(s)}\int^\infty_0\frac{t^{s1}}{e^t1}dt$$ $$\textit{proof}$$ Start by the integral representation $$\int^\infty_0 \frac{e^{t}t^{s1}}{1e^{t}}\,dt$$ Using the power expansion $$\frac{1}{1e^{t}} = \sum_{n=0}^\infty e^{nt}$$ Hence we have $$\int^\infty_0\,e^{t}t^{s1}\left(\sum_{n=0}^\infty e^{nt}\right)\,dt$$ By swapping the series and integral $$\sum_{n=0}^\infty\int^\infty_0\,t^{s1}e^{(n+1)t}\,dt = \Gamma(s) \sum_{n=0}^\infty \frac{1}{(n+1)^s}=\Gamma(s)\zeta(s)\,$$
Relation between polygamma and Hurwitz zeta function proof
\( \forall \,\, n\geq 1 \) $$\psi_{n}(z) \, = \, (1)^{n+1}n!\,\zeta(n+1,z)$$ $$\textit{proof}$$ Use the series representation of the digamma $$\psi_{0}(z) = \gamma\frac{1}{z}+ \sum_{n=1}^\infty\frac{z}{n(n+z)}$$ This can be written as the following $$\psi_{0}(z) = \gamma + \sum_{k=0}^\infty\frac{1}{k+1}\frac{1}{k+z}$$ By differentiating with respect to … Continue reading
Zeta for Even integers proof (Bernoulli numbers)
$$\zeta(2k) \, = \, (1)^{k1} B_{2k} \frac{2^{2k1}}{(2k)!}{\pi}^{2k}$$ $$\textit{proof}$$ We start by the product formula of the sine function $$\frac{\sin(z)}{z} = \prod_{n=1}^\infty \left(1\frac{z^2}{n^2 \, \pi^2} \right)$$ Take the logarithm to both sides $$\log(\sin(z)) – \log(z) = \sum_{n=1}^\infty \log \left(1\frac{z^2}{n^2 \, \pi^2} … Continue reading