File:Chebyshev-big.svg

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 $$\pm\frac12\sqrt{x}$$. 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.

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