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Power series expansion of the d.e. about the origin
In[1]:=Out[1]=
3 5 6 7 8 2 9 10 2 11
-s s s s s (-350 + 3 Pi ) s 491 s (1694 - Pi ) s
----- + ------ - ------- - -------- + -------- + ----------------- - ------------ + ----------------
12 Pi 360 Pi 2 20160 Pi 2 3 2 3
432 Pi 5400 Pi 5443200 Pi 63504000 Pi 239500800 Pi
Power series expansion of nn(s) about s=0
In[2]:=Out[2]=
2 2 4 4 6 6 6 7 6 5 7 8 9
2 Pi s 4 Pi s 2 Pi s 32 Pi s -8 Pi -4 Pi 2 Pi 8 2144 Pi s
-------- - -------- + -------- - --------- + (------ - 2 Pi (------ + -----)) s + -----------
3 45 315 2025 243 243 14175 496125
Numerical solution of the d.e., calculation
of the mean and comparison with power series
expansion
Here the power series solution at s=1.1 is used as the
initial condition
In[3]:=
Out[3]=
{{y -> InterpolatingFunction[{1.1, 16.}, <>]}}
In[4]:=
Here the numerical solution is integrated to
determine the mean of the distrubution
In[5]:=
Out[5]=
0.725227
Next the numerical solution is compared to the power series solution
In[6]:=
Out[7]=
-Graphics-
The function nn(s) can be tabulated
In[8]:=
In[9]:=In[10]:=
Comparison with data from the GUE
Code to generate GUE matrices and plot empirical value of nn(s) (output
suppressed)
In[11]:=
In[12]:=In[13]:=In[14]:=
Calculation of theoretical curve for nn(s) and comparison with numerical data
In[15]:=
In[16]:=In[17]:=
Out[18]=
-Graphics-
Comparison with data from zeros of Riemann
zeta function for 10^6 zeros about the 10^20
zero, 10^6 zero and first zero
Here we read in the data as a list. The data gives the number
of zeros in the intervals [j*.05, (j+1)*.05) (j=0,...40). The third,
fourth and fifth columns contain the data for zeros about
zero 1, zero 10^6 and zero 10^20 respectively. The theoretical
value of nn(s) is plotted from the Table t.