Functiones et Approximatio Commentarii Mathematici

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

Youness Lamzouri and Bruno Martin

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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 $k\ge 2$, we explore the signs of $ S_k(x)=\sum_{p \le x} (-1)^{s_k(p)}$, where $s_k(n)$ is the sum of the last $k$ binary digits of $n$, and $p$ runs over the primes. We prove that $S_k(x)$ changes signs for infinitely many integers $x$, assuming that all Dirichlet $L$-functions attached to primitive characters modulo $2^k$ do not vanish on $(0,1)$. Our result is unconditional for $k\leq 18$. Furthermore, under stronger assumptions on the zeros of Dirichlet $L$-functions, we show that for $k\geq 4$, the sets $\{x> 2: S_k(x)>0\}$ and $\{x> 2: S_k(x)<0\}$ both have logarithmic density $1/2$.

Article information

Funct. Approx. Comment. Math., Volume 61, Number 1 (2019), 7-25.

First available in Project Euclid: 25 September 2019

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Primary: 11N13: Primes in progressions [See also 11B25] 11A63: Radix representation; digital problems {For metric results, see 11K16} 11N37: Asymptotic results on arithmetic functions

prime number races sum of digits Newman's phenomenon


Lamzouri, Youness; Martin, Bruno. On the race between primes with an odd versus an even sum of the last $k$ binary digits. Funct. Approx. Comment. Math. 61 (2019), no. 1, 7--25. doi:10.7169/facm/1687.

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