Hot answers tagged number-theory
5
(This is a math question, not a Mathematica question.)
To add to Artes's answer, there is the well-known(!) identity
$$\zeta(-n)=\frac{(-1)^n}{n+1}B_{n+1}$$
so you might as well ask why
$$\begin{align*}
-\frac12\times B_2&=-\frac1{14}\times B_{14}\\
-\frac12\times \frac16&=-\frac1{14}\times \frac76
\end{align*}$$
A justification for the ...
5
In order to understand the issue, we should provide the underlying definitions.
Mathematica helps in verifying appropriate relations and definitions. The main functional equation relating Riemann's zeta function $\zeta\;$, to Euler's $\Gamma\;$, established in Riemann's famous paper Über die Anzahl der Primzahlen unter einer gegebener Grösse (1859, English ...
3
A function from the article that cormullion linked is shorter and faster than what I proposed below. Transcribed in terse style:
uf[m_, 1] := {{}}
uf[1, n_] := {{}}
uf[m_, n_?PrimeQ] := If[m < n, {}, {{n}}]
uf[m_, n_] := uf[m, n] =
Join @@ Table[Prepend[#, d] & /@ uf[d, n/d], {d, Select[Rest@Divisors@n, # <= m &]}]
uf[n_] := uf[n, n]
...
1
In fact there are lots of Zetas which produce the same value, e.g. try these:
$\{\zeta(-2), \zeta(-4), \zeta(-6),\ldots ,\zeta(-2n)\}$
{Zeta[-2], Zeta[-4], Zeta[-6]}
(* {0, 0, 0} *)
Using Plot may shed some light on this:
Plot[Zeta[x], {x, -15, 0.5}]
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