Number Theory – Do These Series Converge to the von Mangoldt Function?

number theorysequences-and-series

Jeffrey Shallit formulated this recurrence for me:
$\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$

which is the lower triangular array equal to 1 if k divides n, 0 otherwise.

By changing the recurrence so that it takes values from either the vertical or horizontal direction:

$\displaystyle T(n,1)=1, T(1,k)=1, n>=k: -\sum\limits_{i=1}^{k-1} T(n-i,k), n<k: -\sum\limits_{i=1}^{n-1} T(k-i,n)$

we get this array starting:

$$\displaystyle T = \left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \\ \vdots&&&&&&&\ddots \end{array} \right)
$$

Do the series $\displaystyle \sum\limits_{k=1}^{\infty}\frac{T(n,k)}{k}$ $\;$ converge to the Mangoldt function $\Lambda(n)$?

http://mathworld.wolfram.com/MangoldtFunction.html

Edit 14.7.2011, added Mathematica program:

Clear[t]; 
nn = 100; 
mm = 15; 
t[n_, 1] = 1; 
t[1, k_] = 1; 
t[n_, k_] := 
t[n, k] = 
If[n < k, 
If[And[n > 1, k > 1], Sum[-t[k - i, n], {i, 1, n - 1}], 0], 
If[And[n > 1, k > 1], Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; 
a = Table[Table[t[n, k], {k, 1, mm}], {n, 1, nn}]; 
b = Range[1, nn]; 
c = a/b; 
MatrixForm[c]; 
d = N[Table[Total[c[[All, i]]], {i, 1, mm}]] 
d[[1]] = 0; 
mangoldt = Exp[d] 
mangoldtexponentiated = Round[Exp[d]]

that outputs the sequence: $1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1…$
which is the Mangoldt function exponentiated.

Edit 9.2.2014:

Just for memory:

$$\varphi (n) =
n\lim\limits_ {s \rightarrow 1}\zeta (s)\sum\limits_ {d | n}\mu (d) (e^{1/d})^{(s – 1)}$$

$$a(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \mu(d)(e^{d})^{(s-1)}$$

$$\Lambda(n) =\sum\limits_{k=1}^{\infty} \frac{a(GCD(n,k))}{k}$$

$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$

$$\text{Fourier Transform of } \Lambda(n) \sim \sum\limits_{n=1}^{n=k} \frac{1}{n} \zeta(1/2+i \cdot t)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot t-1)}}$$

Edit 3.3.2014:

Just for memory:
Mathematica:

nn = 12;
mm = nn;
MatrixForm[
 Chop[N[Total[
    Transpose[
     Table[Table[
       If[Mod[n1, k1] == 0, 
        Table[(Table[
             Sum[Exp[-a*b/n*2*Pi*I], {b, 1, n}], {a, 1, mm}]), {n, 1, 
            nn}][[n1/k1]]*MoebiusMu[n1/k1], 0], {k1, 1, nn}], {n1, 1, 
       nn}]]]]]]

Edit 14.9.2014:

Just for memory:
Conjectured formula from Dirichlet characters:

nn = 12;
b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn}];
j = 1;
MatrixForm[Table[Table[Product[(b[[n]][[m]] * DirichletCharacter[b[[n]][[m]], j, k] - (b[[n]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]]
(* Conjectured expression as Dirichlet characters. Mats Granvik, Nov 23 2013 *)

Just for memory (18.1.2015) :

A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, 12}], {n, 1, 12}];
B = Table[
   Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {k, 1, 12}], {n, 1, 
    12}];
MatrixForm[A.B]

Just for memory (20.1.2015):

nn = 42
Z = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}];
A = Table[Table[If[Mod[n, k] == 0, k, 0], {k, 1, nn}], {n, 1, nn}];
B = Table[
   Table[If[Mod[k, n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1, nn}];
MatrixForm[T = Z.A.B];
T[[All, 1]] = 0;
a = Table[Total[Total[T[[1 ;; n, 1 ;; n]]]], {n, 1, nn}]
a = Table[Total[Total[T[[1 ;; n, 1 ;; n]]]]/n, {n, 1, nn}]
g1 = ListLinePlot[a];
b = Accumulate[MangoldtLambda[Range[nn]]];
g2 = ListLinePlot[b];
Show[g1, g2]

Relation to square roots:

nn = 32;
A = Table[
   Table[If[Mod[n, k] == 0, Sqrt[k], 0], {k, 1, nn}], {n, 1, nn}];
B = Table[
   Table[If[Mod[k, n] == 0, MoebiusMu[n]*Sqrt[n], 0], {k, 1, nn}], {n,
     1, nn}];
MatrixForm[A.B]

Best Answer

Yes, they do for $n\neq1$. This can be proved in two steps:

  1. The coefficients are given by $T(n,k)=\sum_{d\mid\gcd(n,k)}\mu(d)d$, where $\mu$ is the Möbius function.
  2. $\displaystyle \sum\limits_{k=1}^{\infty}\frac1k\sum_{d|\gcd(n,k)}\mu(d)d=\Lambda(n)$ for $n\neq1$, where $\Lambda$ is the von Mangoldt function.

We can prove 1. by induction over the recursion steps. The equation holds for the initial values: $T(n,1)=T(1,k)=\sum_{d\mid1}\mu(d)d=\mu(1)=1$. Assume that it holds before the recursion step for $T(n,k)$. Since the recursion is symmetric with respect to interchange of $k$ and $n$, we can assume $n\ge k$. Then we have

$$ \begin{eqnarray} \sum_{d\mid\gcd(n,k)}\mu(d)d-T(n,k) &=& \sum_{d\mid\gcd(n,k)}\mu(d)d+\sum_{i=1}^{k-1}T(n-i,k) \\ &=& \sum_{d\mid\gcd(n,k)}\mu(d)d+\sum_{i=1}^{k-1}\sum_{d\mid\gcd(n-i,k)}\mu(d)d \\ &=& \sum_{i=0}^{k-1}\sum_{d\mid\gcd(n-i,k)}\mu(d)d \;. \end{eqnarray} $$

Every square-free divisor $d$ of $k$ occurs $k/d$ times in this sum, so its contribution sums to $(k/d)\mu(d)d=\mu(d)k$, and the sum becomes

$$\sum_{d|k}\mu(d)k=k\sum_{d|k}\mu(d)=0\;,$$

since $k\neq1$ (see here). This proves the induction claim.

To prove 2., note that the series in question is the series

$$\sum_{k=1}^{\infty}\frac1k\sum_{d|\gcd(n,k)}\mu(d)dx^k$$

evaluated at $x=1$, and this is

$$ \begin{eqnarray} \sum_{k=1}^{\infty}\frac1k\sum_{d|\gcd(n,k)}\mu(d)dx^k &=& \sum_{d|n}\mu(d)d\sum_{k=1\atop d|k}^{\infty}\frac{x^k}k \\ &=& \sum_{d|n}\mu(d)d\sum_{l=1}^{\infty}\frac{x^{ld}}{ld} \\ &=& \sum_{d|n}\mu(d)\sum_{l=1}^{\infty}\frac{(x^d)^l}l \\ &=& -\sum_{d|n}\mu(d)\log\left(1-x^d\right)\;. \end{eqnarray} $$

Since $\sum_{d|n}\mu(d)=0$ for $n\neq1$ (see above), we can rewrite this as

$$-\sum_{d|n}\mu(d)\log\left(1-x^d\right)=-\sum_{d|n}\mu(d)\log\left(\frac{1-x^d}{1-x}\right)=-\sum_{d|n}\mu(d)\log\left(\sum_{j=0}^{d-1}x^j\right)\;,$$

which evaluates to

$$-\sum_{d|n}\mu(d)\log d=\Lambda(n)$$

at $x=1$ (see here).

To make this argument rigorous, we need to invoke Abel's theorem: All the power series occurring in the proof have radius of convergence $1$ and converge at $x=1$, and hence converge at $x=1$ to the function whose Taylor series they are.

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