International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume‐3, Issue‐10, October 2017                                ISSN: 2395‐3470  www.ijseas.com    87    Apéry’s Constant Calculation and Prime Numbers Distributions: A Matrix Approach F. Delplace ESI Group Scientific Committee, Paris, France fr.delplace@gmail.com Abstract. We built a matrix ܯmade of zeta function series terms in rows. It allows matrix ܯ .ܯ  and ܯ  ܯ.to be build giving interesting characteristics. Equalling sums of terms in rows and columns in ܯgave a general method for ߞሺ2 ൅ 1ሻ calculations. Some important results were deduced from the knowledge of all ߞሺݏሻ and related to matrix ܯ .ܯ  and Euler product formula. Finally, considering matrix ܯ  ܯ.characteristics, a method for prime numbers enrichment is proposed giving a promising tool for information technologies encoding. Key Words: Apéry – Zeta function – Odd Integers – Prime Numbers – Hilbert’s Matrix 1. INTRODUCTION. The Riemann zeta function or Euler-Riemann zeta function ߞሺݏሻ is a function of complex variable ݏgiven by: ߞሺݏሻൌ෍ 1  ௦ ାஶ ௞ୀଵ In 1735, Euler solved the famous Basel problem [1] and found ߞሺ2ሻ ൌ గ మ ଺ using the polynomial development of ௦௜௡௫ ௫ . He also brilliantly showed that ߞሺ4ሻ ൌ గ ర ଽ଴ and ߞሺ6ሻ ൌ గ ల ଽସହ . Finally, for even integers greater than 1, he established the general formula: ∀ ൐ 0 ߞሺ2ሻ ൌ∣ ܤ ଶ௞ ∣ ሺ2ߨሻ ଶ௞ 2ሺ2ሻ! Where ܤ ଶ௞ are the Bernoulli numbers ( ܤ ଶ ൌ ଵ ଶ ; ܤ ସ ൌ െ ଵ ଷ଴ ; …) [2,3]. But for odd integers, the problem appeared much more complicated [4-6] and Euler himself was not able to find an analytical result for ߞሺ3ሻ. The Indian mathematician Ramanujan [7] worked a lot on this problem without success. We had to wait for Apéry’s work in 1978, who demonstrated that ߞሺ3ሻ is an irrational number. This number was called Apéry’s constant in honour of this important result. But even now, there exist no analytical or closed form expression for ߞሺ3ሻ and for all others values of zeta function for odd integers i.e. ߞሺ2 ൅ 1ሻ  ൒ 1.