Christian Mauduit, Joël Rivat. Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, (2010) 171(3):1591. DOI:10.4007/annals.2010.171.1591
Tuesday, September 14, 2010
Many arithmetical problems involve prime numbers and remain unresolved even after centuries. For example, the sequence of prime numbers is infinite, but it is still not known if an infinity of prime numbers p exists such that p+2 is also a prime number (the problem of twin prime numbers). One hypothesis about prime numbers, first formulated in 1968 by Alexandre Gelfond, has recently been proven by Christian Mauduit and Joel Rivat from the Institut de Mathématiques de Luminy. It states that on average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd. In order to arrive at this result, the researchers employed highly groundbreaking methods derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis. This proof should pave the way for the resolution of other difficult questions concerning the representation of certain sequences of integers. Apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography.