Journal of Advances in Applied & Computational Mathematics, 2015, 2, 01-04 1 E-ISSN: 2409-5761/14 © 2014 Avanti Publishers Detection and Discrimination of the Periodicity of Prime Numbers by Discrete Fourier Transform – Symphony of Primes– L. Csoka * University of West Hungary, Institute of Wood Based Products and Technologies, 9400 Sopron, Hungary Abstract: A novel representation of a quasi-periodic modified von Mangoldt function L(n) on prime numbers and its decomposition into Fourier series has been investigated. We focus on some particular quantities characterizing the modified von Mangoldt function. The results indicate that prime number progression can be decomposed into periodic sequences. The main approach is to decompose it into sin or cosine function. Basically, it is applied to extract hidden periodicities in seemingly quasi periodic prime function. Numerical evidences were provided to confirm the periodic distribution of primes. Keywords: Von Mangoldt function, discrete fourier transform, prime numbers. 1. THE VON MANGOLDT FUNCTION The von Mangoldt arithmetic function is defined by !(n) = log p if n = p k for p a prime and integar k " 1 0 otherwise # $ % & % (1) This function carries the information about the properties of the primes and approximate their weighted characteristic function. The discrete Fourier transform of the von Mangoldt function is defined by the Riemann-Weil explicit formula h(! ) ! " = h i 2 # $ % & ’ ( + h ) i 2 # $ % & ’ ( ) g 0 () log * + 1 2* h(r ) + , , )- - . 1 4 + 1 2 ir # $ % & ’ ( dr ) 2 /(n) n n =1 - " g(log n) (2) and gives a spectrum with spikes at ordinates equal to the imaginary part of the Riemann zeta function zeros [1]. 2. THE MODIFIED VON MANGOLDT FUNCTION Starting with the function above, let us consider a special, simple case of von Mangoldt function at for k=1, L(n) ! ln (e) if n = p for p a prime 0 otherwise " # $ (3) The function L(n) is a representative way to introduce the set of primes with weight of unity attached to the *Address correspondence to these authors at the Inst of Wood Based Products and Technologies, University of West Hungary, Hungary; Tel: +36-99518-305; Fax: +36-99-518-302; E-mail: levente.csoka@skk.nyme.hu location of a prime. From the summation of that function we can see that primes contribute equally and that representation exist. The sum of the modified von Mangoldt function L(n) is same as the Riemann hypothesis [2, 3] with a slight error: ! (n) = 1 In t 2 pn " dt = n + 0 n In x ( ) 2 ( ) # L(n) # pn n$% & (4) but the L(n) function predict the number of primes with no error, e.g. the primes are not randomly distributed (large numbers’ law). Therefore, this function is oscillatory but not diverging. The modified von Mangoldt function L(n) is related to Dirichlet seriesor the Riemann zeta function by ! (a) = 1 na = L(n) n a n=1 " # n=1 " # $(a) > 1. (5) The discrete Fourier transform of the modified von Mangoldt function L(n) gives a spectrum with periodic, real parts as spikes at the frequency axis ordinates. Hence the modified von Mangoldt function L(n) and the prime series can be approximated by periodic sin or cos functions. Let us consider that natural numbers form a discrete function g(n) ! g(n k ) and, g k ! g(n k ) by uniform sampling at every sampling point n k ! k " , where Δ is the sampling period and k=2, 3,.., L−1. If the subset of discrete prime numbers is generated by the uniform sampling of a wider discrete set, it is possible to obtain the Discrete Fourier Transform (DFT) of L(n) . The sequence of L(n) . prime function is transformed into another sequence of N complex numbers according to the following equation: FLn () { } = X ! ( ) (6) where L(n) is the modified von Mangoldt function. The the operator F is defined as: