## On a problem of Gelfond: the sum of digits of prime numbers. (Sur un problème de Gelfond: la somme des chiffres des nombres premiers.)(English)Zbl 1213.11025

Let $$q\geq 2$$ be an integer. Each integer $$n$$ can be uniquely written in base $$q$$ in the form $n=\sum_{k\geq 0}n_kq^k,\quad n_k\in\{0,\ldots,q-1\}.$ Let $$s_q(n)=\sum_{k\geq 0}n_k$$ be the sum of digits of $$n$$ in the base $$q$$. The authors prove:
Theorem 1. For $$q\geq 2$$ and $$\alpha$$ such that $$(q-1)\alpha\in{\mathbb R}\setminus{\mathbb Z}$$ there exists $$\sigma_q(\alpha)>0$$ such that $\sum_{n\leq x}\Lambda(n)e(\alpha s_q(n))=O_{q,\alpha}(x^{1-\sigma_q(\alpha)}),$ where $$\Lambda$$ is von Mangoldt’s function and $$e(x)=\exp(2\pi i x)$$.
Theorem 2. For $$q\geq 2$$ the sequence $$(\alpha s_q(p))_{p\in{\mathbb P}}$$ is equidistributed modulo $$1$$ if and only if $$\alpha\in{\mathbb R}\setminus{\mathbb Q}$$. ($${\mathbb P}$$ denotes the set of prime numbers.)
Theorem 3. For $$q,m\geq 2$$ there exists $$\sigma_{q,m}>0$$ such that for every $$a\in{\mathbb Z}$$ $\text{card}\{p\leq x,s_q(p)\equiv a\bmod m \}=\frac{(m,q-1)}{m}\pi(x;a,(m,q-1))+O_{q,m}(x^{1-\sigma_{q,m}}).$

### MSC:

 11A63 Radix representation; digital problems 11A41 Primes 11N37 Asymptotic results on arithmetic functions

### Keywords:

prime number; sum of digits
Full Text:

### References:

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