The sum of digits of Gaussian primes
Let \(b\geq 2\) be an integer. \textit{C. Mauduit} and \textit{J. Rivat} [Ann. Math. (2) 171, No. 3, 1591--1646 (2010; Zbl 1213.11025)] settled a long-standing problem of Gelfond (1967/68) about the distribution of the sum of digits function \(s_b\) of primes in the digital representation of integers in base \(b\). They showed that \((s_b(p))\), \(p\) running through the primes, is uniformly distributed in arithmetic progressions, and that \((\alpha s_b(p))\) is uniformly distributed modulo 1 if and only if \(\alpha\) is irrational. One has a similar notion of the sum of digits function in the Gaussian integers, where the base \(b\) is replaced by \(q=-a+i\in\mathbb{Z}[i]\) with \(a\geq 1\). The Mauduit-Rivat result has been extended to the Gaussian primes by \textit{M. Drmota, J. Rivat} and the reviewer [Monatsh. Math. 155, No. 3-4, 317--347 (2008; Zbl 1190.11053)] but there were rather severe restrictions on the base \(q\) (\(q\) prime and sufficiently large in modulus). In the paper under review, the author obtains the complete result, and he puts it in an even more general setting. He shows that the sequence \((\alpha s_{q}(p))\) running through Gaussian primes that lie in a circular sector is uniformly distributed modulo 1 if and only if \(\alpha\) is irrational. From this, he also deduces well-distribution in arithmetic progressions. The proof involves, besides other tools, the use of a Vaughan-type identity, a suitable Gaussian integer version of the van der Corput inequality, discrete Fourier analysis in \(\mathbb{Z}[i]\) and a careful analysis of the carry propagation rule in this numeration system. The paper is very well written.
scientific article; zbMATH DE number 6006054