Asymptotics of σ0(i)
Asymptotics of (i)
σ
0
The following (modified) example is given in the documentation for (i).
σ
0
Plot the running average of the number of divisors with its asymptotic value:
In[]:=
ShowDiscretePlotDivisorSigma[0,i],{n,100},Plot[Log[n]+2EulerGamma-1,{n,1,100}]
1
n
n
∑
i
Out[]=
However, GeneratingFunction does not work:
In[]:=
GeneratingFunction[DivisorSigma[0,n],n,x]
Out[]=
GeneratingFunction[DivisorSigma[0,n],n,x]
Nevertheless, the following sum is computed in closed-form:
In[]:=
gf=DivisorSigma[0,k]
∞
∑
k=1
k
x
Out[]=
Log[1-x]+QPolyGamma[0,1,x]
Log[x]
Multiplying by computes the cumulative sums:
-1
(1-x)
In[]:=
sum=gf+
1
1-x
11
O[x]
Out[]=
x+3+5+8+10+14+16+20+23+27+
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
10
x
11
O[x]
Check:
In[]:=
TableDivisorSigma[0,i],{n,10}
n
∑
i
Out[]=
{1,3,5,8,10,14,16,20,23,27}
Integrating the series of cumulative sums generates the average of the number of divisors:
In[]:=
x
sum
x
Out[]=
x+++2+2++++++
3
2
x
2
5
3
x
3
4
x
5
x
7
6
x
3
16
7
x
7
5
8
x
2
23
9
x
9
27
10
x
10
11
O[x]
Here are the first 100 terms:
In[]:=
ListPlotCoefficientListgf+x,x,FillingAxis
1
1-x
199
O[x]
Out[]=