File:Legendre's constant 10 000 000.svg
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Size of this PNG preview of this SVG file: 694 × 332 pixels. Other resolutions: 320 × 153 pixels | 640 × 306 pixels | 1,024 × 490 pixels | 1,280 × 612 pixels | 2,560 × 1,225 pixels.
Original file (SVG file, nominally 694 × 332 pixels, file size: 136 KB)
Summary
| Description |
English: The first 10, 000 000 elements of the sequence a_n = \ln(n) - \frac{n}{\pi(n)} (red line), where π is the prime-counting function. Adrien-Marie Legendre conjectured in 1808 that this sequence would converge to a value of approximately 1.08366 (blue line), which became known as Legendre's constant. The actual limit was later shown to be exactly 1. |
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Plotted in Maple and converted to SVG format in Inkscape by Sandrobt, using the following Maple code: > with(numtheory); > with(plots); > b := vector(10000, proc (i) options operator, arrow; [1000*i, 1.08366] end proc); > c := vector(10000, proc (i) options operator, arrow; [1000*i, 1] end proc); > a := vector(10000, proc (i) options operator, arrow; [1000*i, evalf(log(1000*i)-1000*i/pi(1000*i))] end proc); > display(plot(b, color = blue, thickness = 3), plot(a, color = red), plot(c, color = black, thickness = 3), view = [1 .. 10000000, .95 .. 1.1]); |
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| Author | Sandrobt | ||||
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 17:40, 2 September 2010 | | 694 × 332 (136 KB) | wikimediacommons>Sandrobt | {{Information |Description={{en|1=The first 10, 000 000 elements of the sequence a_n = \ln(n) - \frac{n}{\pi(n)} (red line), where π is the prime-counting function. Adrien-Marie Legendre conjectured in 1808 that this sequence would converge to a value of |
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