Graph of Chebyshev function, with the leading terms subtracted, for values of n from 1 to 10,000. Note the remarkably chaotic, unpredictable movement of this function.

More precisely, this is a graph of

${\displaystyle \psi (x)-x+\log(\pi )}$

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

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

and

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

infinitely often.

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

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GFDL GNU Free Documentation License //en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License true true

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

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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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