# File:Chebyshev-big.svg

Original file(SVG file, nominally 600 × 480 pixels, file size: 103 KB)

## 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.

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


## File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:48, 3 July 2006600 × 480 (103 KB)Linas (talk | contribs)
16:42, 3 July 2006600 × 480 (15 KB)Linas (talk | contribs)== 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 :[itex]\psi(x)-x+\log(\
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