Prime numbers with a positive proportion of preassigned digits

Abstract

Bourgain [Israel J. Math. 206 (2015) 165-182] estimated the number of prime numbers with a proportion c > 0 of preassigned digits in base 2 (c is an absolute constant not specified). We establish a generalization of this result in any base g 2 and we provide explicit admissible values for the proportion c depending on g. Our proof, which develops and enhances Bourgain's arguments, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet L-functions, notably a very strong zero-free region due to Iwaniec.By Lemma 5.28, since B 1 N κ 4(1+κ) and |A| cn, the double sum over q 2 and k 2 on the right-hand side of (84) is g,κ,c Ng −|A| (log N ) 2 . This completes the proof. Conclusion on the minor and major arcs under GRHProposition 15.8. Assume GRH. Let g 2 be an integer and c > 0 be a real number. Let n 100, A ⊂ {0, . . . , n − 1} and d = (d j ) j∈A ∈ {0, . . . , g − 1} A such that 0 ∈ A and (d 0 , g) = 1.

Refinements to the prime number theorem for arithmetic progressions

“…The arithmetic progression condition enables us to prescribe the A least significant digits, while the short interval condition enables us to prescribe the B most significant digits. Corollary 3.2 complements recent work of Swaenepoel [21] wherein a small proportion of the digits can be prescribed without any restriction on how significant the digits are. With the effective bounds in (1.7), Gallagher's work implies that there exist absolute and effectively computable constants c 12 , c 13 > 0 such that if…”

The Thue-Morse and Rudin-Shapiro sequences at primes in principal number fields

“…, where f is given as in Theorem A.3 of [65] for some small enough β > 0 and large enough α depending on B.…”

“…The recent years, a lot of progress has been made on our understanding of digits of multiplicatively constrained integers (e.g. primes): see [20,19,11,44,45,28,14] for sum of digits of primes in residue classes, [29,7,65] for primes with restricted digits, or [43,15,17,46,16] for digits of polynomials. Here we are interested in two particular digital functions (defined in terms of digit expansion), the sum-of-digits function…”

On the distribution of the Rudin-Shapiro function for finite fields

“…In recent years, many spectacular results have been obtained on important problems combining some arithmetic properties of the integers and some conditions on their digits in a given basis, see for example [1,2,8,[13][14][15]17,19,23]. In particular, Drmota, Mauduit and Rivat [8] and Müllner [17] showed that Thue-Morse sequence and Rudin-Shapiro sequence along squares are both normal.…”