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On the race between primes with an odd versus an even sum of the last \(k\) binary digits. (English) Zbl 1469.11371

Summary: 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 \(1/2\).

MSC:

11N13 Primes in congruence classes
11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11N37 Asymptotic results on arithmetic functions
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