Original file (SVG file, nominally 600 × 480 pixels, file size: 103 KB)
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
The green lines above and below provide a limit of . 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
Created by User:Linas, Linas Vepstas, 3 July 2006
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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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|current||17:48, 3 July 2006||600 × 480 (103 KB)||Linas|
|16:42, 3 July 2006||600 × 480 (15 KB)||Linas||== 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(\|