Riesz mean

Summary

Graph of Chebyshev function, with the leading terms subtracted, for values of n from 1 to 10 million. Note the remarkably chaotic, unpredictable movement of this function.

More precisely, this is a graph of

$\psi (x)-x+\log (\pi )$

The green lines above and below provide a limit of $\pm \frac{1}{2}\sqrt{x}$. Note that the function occasionally exceeds this bound; a theorem stated by Erhard Schmidt in 1903 shows that, for any real, positive K, there are values of x such that

$\psi (x)-x<-K\sqrt{x}$

and

$\psi (x)-x>K\sqrt{x}$

infinitely often.

See also

File:Chebyshev.svg

Chebyshev function to 10K

Licensing

Created by User:Linas, Linas Vepstas, 3 July 2006

Source code

Created with gnuplot, with the following markup:

set term svg
set out 'chebyshev.svg'

set data style lines
unset zeroaxis
set xtics border
set ytics border

set bmargin 5
set lmargin 7

set title "Chebyshev (summatory von Mangoldt) function"
set xlabel "n" 1,0
set ylabel "psi(n)-n+log(pi)" 1, 0
plot "chebyshev.dat" using 1:2 title "" with lines linewidth 2

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