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

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

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 any 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.

See also

Licensing

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

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. GFDL GNU Free Documentation License https://en.wikipedia.org/wiki/Text_of_the_GNU_Free_Documentation_License

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. This licensing tag was added to this file as part of the GFDL licensing update. Cc-by-sa-3.0Attribution-ShareAlike 3.0https://creativecommons.org/licenses/by-sa/3.0/

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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