MANGOLDT's FUNCTION
Hans Carl Friedrich von Mangoldt
\[\Lambda (n)\]
$$\Lambda ~~~~ Lambda $$
\[\Lambda(n) = \left\{\begin{matrix} ln(p)\ if\ n\ = \ p^{k}\\ 0\ otherwise \end{matrix}\right. \\ \\ with \ p \ prime \ and \ k \geqslant \ 1\]
\[\Lambda (n)= \begin{cases} & log_{e}(p)~~~~if~n~\forall~A000961~Powers~of~primes.\\ & 0~~~~~~~~~~if~n~\forall~A024619~Not~Powers~of~primes. \end{cases}\]
A024619 Not Powers of primes.
6,10,12,14,15,18,20,21,22,24,26,28,30,33,34,35,36,38,39,40,42,44
A000961 Powers of primes.
1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49
\[\Lambda (n)= ln(A014963(n))\]
\[A014963(n)=e^{\Lambda (n)} \\ =\frac{lcm(1,2,...,n)}{lcm(1,2,...,n-1)} \\ = \frac{A003418(n)}{A003418(n-1)}\]
n=6 # Example with n=6
lcm(range(1, n+1)) # A003418(n)
lcm(range(1, n+1))/lcm(range(1, n)) # A003418(n)/A003418(n-1)
numerical_approx(log(lcm(range(1, n+1))/lcm(range(1, n)),exp(1))) # Mangoldt's Function
table([(n,numerical_approx(log(lcm(range(1, n+1))/lcm(range(1, n)),exp(1))) ) for n in [1..50]], header_row=['n', 'MangoldtFunction(n)'])
Mangoldt = [numerical_approx(log(lcm(range(1, n+1))/lcm(range(1, n)),exp(1))) for n in range(1, 50)]
graphique = list_plot(Mangoldt, color='red', pointsize=10, title='Mangoldt function', aspect_ratio='automatic')
graphique.axes_labels(['$x$', '$y$'])
graphique.axes_labels_size(1)
graphique.set_axes_range(0, 50, 0, 4)
graphique.axes_width(0.5)
graphique
n MangoldtFunction(n)
+----+---------------------+
1 0.000000000000000
2 0.693147180559945
3 1.09861228866811
4 0.693147180559945
5 1.60943791243410
6 0.000000000000000
7 1.94591014905531
8 0.693147180559945
9 1.09861228866811
10 0.000000000000000
11 2.39789527279837
12 0.000000000000000
13 2.56494935746154
14 0.000000000000000
15 0.000000000000000
16 0.693147180559945
17 2.83321334405622
18 0.000000000000000
19 2.94443897916644
20 0.000000000000000
21 0.000000000000000
22 0.000000000000000
23 3.13549421592915
24 0.000000000000000
25 1.60943791243410
26 0.000000000000000
27 1.09861228866811
28 0.000000000000000
29 3.36729582998647
30 0.000000000000000
31 3.43398720448515
32 0.693147180559945
33 0.000000000000000
34 0.000000000000000
35 0.000000000000000
36 0.000000000000000
37 3.61091791264422
38 0.000000000000000
39 0.000000000000000
40 0.000000000000000
41 3.71357206670431
42 0.000000000000000
43 3.76120011569356
44 0.000000000000000
45 0.000000000000000
46 0.000000000000000
47 3.85014760171006
48 0.000000000000000
49 1.94591014905531
50 0.000000000000000
\[log_{e}(n)=ln(n)=\sum_{d|n} \Lambda (d)\]
numerical_approx(log(n,exp(1))) == sum([numerical_approx(log(lcm(range(1, d+1))/lcm(range(1, d)),exp(1))) for d in divisors(n)])
\[\Lambda (n)=\sum_{d|n} \mu (\frac{n}{d})ln(d)\]
numerical_approx(log(lcm(range(1, n+1))/lcm(range(1, n)),exp(1))) == sum([moebius(n/d)*numerical_approx(log(d,exp(1))) for d in divisors(n)])
A014963 Exponential of Mangoldt's function.
1,2,3,2,5,1,7,2,3,1,11,1,13,1,1,2,17,1,19,1,1,1,23,1,5,1,3,1
# Exponential of Von Mangoldt's function
psi=4
exp(sum([ log(prod([1 - exp( 2*pi*I*k/(i+2) ) for k in range(i+3) if gcd(i+2,k) == 1])) for i in range(psi-1)])).numerical_approx();
exp(sum([ log(prod([1 - exp( 2*pi*I*k/(i+2) ) for k in range(i+3) if gcd(i+2,k) == 1])) for i in range(psi-1)]))
A003418 Least common multiple.
1,1,2,6,12,60,60,420,840,2520,2520,27720,27720,360360,360360,360360
[lcm(range(1, n)) for n in range(1, 10)]
[1, 1, 2, 6, 12, 60, 60, 420, 840]