**Some mathematics 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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