On the race between primes with an odd versus an even sum of the last $k$ binary digits

Youness Lamzouri (1) , Bruno Martin (2, 3)

1 IECL - Institut Élie Cartan de Lorraine
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
3 CRM - Centre de recherches mathématiques (IRL3457)

Motivated by Newman's phenomenon for the Thue-Morse sequence (−1)s(n), where s(n) is the sum of the binary digits of n, we investigate a similar problem for prime numbers. More specifically, for an integer k≥2, we explore the signs of Sk(x)=∑p≤x(−1)sk(p), where sk(n) is the sum of the last k binary digits of n, and p runs over the primes. We prove that Sk(x) changes signs for infinitely many integers x, assuming that all Dirichlet L-functions attached to primitive characters modulo 2k do not vanish on (0,1). Our result is unconditional for k≤18. Furthermore, under stronger assumptions on the zeros of Dirichlet L-functions, we show that for k≥4, the sets {x>2:Sk(x)>0} and {x>2:Sk(x)<0} both have logarithmic density 1/2.

Mots clés: prime number races; sum of digits; Newman's phenomenon

Domaines: Théorie des nombres [math.NT]

Youness Lamzouri, Bruno Martin. On the race between primes with an odd versus an even sum of the last $k$ binary digits. Functiones et Approximatio Commentarii Mathematici, 2019, 61 (1), pp.7-25. ⟨10.7169/facm/1687⟩. ⟨hal-02938375⟩