Image:Chebyshev-big.svg

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Chebyshev-big.svg‎ (600 × 480 pixel, file size: 103 KB, MIME type: image/svg+xml)

1 Summary
2 See also
3 Licensing
4 Source code

[edit] 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

ψ(x) - x + log(π)

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.

[edit] See also

Chebyshev function to 10K

[edit] 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.

[edit] 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

File history

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(del) (cur) 17:48, 3 July 2006 . . Linas (Talk | contribs) . . 600×480 (105,926 bytes)
(del) (rev) 16:42, 3 July 2006 . . Linas (Talk | contribs) . . 600×480 (15,134 bytes) (== 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 :<math>\psi(x)-x+\log(\)