New 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 ...
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}]
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]
...
5
In short NSum cannot handle this sort of sequence. Indeed, strictly the series is not convergent, and some notion of summability/regularization needs to be chosen.
Given the nature of MoebiusMu, "Dirichlet" seems appropriate:
Sum[MoebiusMu[k], {k, 1, \[Infinity]}, Regularization -> "Dirichlet"]
(* -2 *)
Here's how one can see NSum is not working ...
4
Graham, Knuth, and Patashnik in their book Concrete Mathematics
(pages 118 and 150) discuss the Farey series. Their recurrence
does not require finding Subsets, computing the elements in
order starting with $0/1$ and $1/n$. Although very fast, Subsets
can use too much memory when very large $n$ are required, as for
some PE problems.
...
5
Here's a functional way to use the property (the property, which has been removed from the original question, was $N'/D' = N/D + 1/D'D$ or equivalently $N'D-D'N=1$):
farey1[n_] :=
NestWhileList[
With[{num0 = Numerator[#], den0 = Denominator[#]},
First @ Minimize[{num/den,
num den0 - num0 den == 1 && 1 <= den <= n && 1 ...
5
Here's a way to exploit the mediant property of the Farey series. To calculate the mediant:
med[{a_, b_}] := (Numerator[a] + Numerator[b])/(Denominator[a] + Denominator[b]);
Then the Farey series is:
farey[n_] := farey[n] = DeleteCases[ Riffle[
farey[n - 1], med /@ Partition[farey[n - 1], 2, 1]], _?(Denominator[#] > n &)];
with initial ...
10
This is caused by a bug in RootReduce for Root objects representing last coordinates of solutions of triangular systems. The bug affects cases where the last coordinate of the solution is real, but some of the other coordinates are not real. Thanks for pointing it out.
The problem can be fixed with the following patch (you can put it in your init.m file).
...
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