Some mathematics programs

The compiled programs

The uncompiled programs + compiler

Number theory

PrimeDecomp: The prime factors and their multiplicity of a natural number.

Sylvester: Sylvester's formula for the number of ways into which an even number can be written as a sum of two primes.

Fourier: The Fourier-Mellin transform of sin(x) and sin(x) recreated from this function.

Moebius: Sinus curve recreated as a sum of saw-tooth functions being smaller and smaller.

Merten: The sum ∑_{n = 1, ..., N-1}μ(n).

NumbPrimes: The number of primes less than N: 1. the exact number, 2. Gram's formula, 3. the Li series, 4. N/log(N).

Riemann's zeta function: ζ(z) = ∑_{n = 1, ..., ∞}1/n^z:

   ZetaFunc: The graph of Riemann's zeta function along a line parallel with the y-axis.

   ZeroFunc: The graph of the function y → │ζ(0.5 + iy)│ which determines the non-trivial zeros of Riemann's zeta function.

   Newton: Newton-iteration for Riemann's zeta function.

   Riemann: The graph of Riemann's formula for "the number of primes less than a given number" in which all the trivial and the first n non-trivial zeros are included.

   Zerolines: The zerolines for the real and imaginary part of Riemann's zeta function.

   MacZeta: The value of Riemann's zeta function calculated by the Euler-Maclaurin formula.

PrimeZeta: The zeta function Z(z) = ∑_{p prime} 1/p^z. The graph and the integral formula (containing Z(z)) for the number of primes less than N.

Vinogradov: Vinogradov's formula for the ∑_{p odd prime < N}z^p (N large). The value for z being a rational point on the unit circle.

Miscellaneous:

   Recursion: The Bernoulli numbers calculated by recursion.

   PrimeSums: The integral formula for the average of the sums of the prime factors for the first N natural numbers.

The units used by the programs:

   compnumb: the four basic arithmetical operations for complex numbers plus arg(z), x^z, log(z), sin(z) and cos(z).

   functions: calculation of LiC(x) and LiR(x^z) (integral logarithm), θ(x) and H(x) (Riemann), prime(n), μ(n) (Möbius) and φ(n).

   bernou: the first 16 Bernoulli numbers.

   gammazeta: calculation of γ(z), γ'(z), ζ(z) and ζ'(z).

   zeropoint: the ordinates of the first 79 zeros for ζ(z) on the line Re(z) = 0.5.

Tomography (the roentgen scanning principle)

   Scan: Scans a picture in BMP format called "pict.bmp". Enter the number of angles (e.g. 360) and the number of displacements (e.g. 400). The result is a picture called "trans_pict.bmp" of width = "number of angles" and height = "number of displacements".

   Recreate: Recreates "pict.bmp" from "trans_pict.bmp".

Planet movement round a double star

   PlanetCases: Three cases.

   Planet: The movement for a given start position.

Music

   Noergaard: The infinity series of Per Nørgård.

   PN_generel: A generalization of the Nørgård construction.

   Roel: the Roel series (wavelength parts of the Nørgård series).

   KAaR: the Karl-Aage Rasmussen series.

March 2011

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