×

Prime numbers in two bases. (English) Zbl 1484.11016

For two coprime integer bases \(q_1, q_2 \ge 2\), let \(f\) be a proper strongly \(q_1\)-multiplicative function and \(g\) a strongly \(q_2\)-multiplicative function, for example exponentiated sum-of-digits functions in bases \(q_1\) and \(q_2\). The main result states that \[ \Big|\sum_{n\le x} \Lambda(n) f(n) g(n) \exp(2\pi i\vartheta n)\Big| \ll x \exp(-c \log x/\log \log x) \] uniformly for \(\vartheta \in \mathbb{R}\), where \(\Lambda\) denotes the von Mangoldt function. Here, the positive constant \(c\) and the implicit constant depend on \(f\) and \(g\). The same result holds for the Möbius function instead of the von Mangoldt function. In particular, \(f(n)g(n)\) is orthogonal to the Möbius function. Since these sequences are produced by zero-entropy dynamical systems, they provide a new class satisfying the Sarnak conjecture. For certain multiplicative functions, more explicit upper bounds and a prime number theorem are given. The proofs combine Fourier analysis, Diophantine approximation, and combinatorial arguments.

MSC:

11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11N60 Distribution functions associated with additive and positive multiplicative functions
11L20 Sums over primes
11N05 Distribution of primes
11L03 Trigonometric and exponential sums (general theory)
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] N. L. Bassily and I. Kátai, Distribution of the values of \(q\)-additive functions on polynomial sequences, Acta Math. Hungar. 68 (1995), no. 4, 353-361. · Zbl 0832.11035
[2] R. Bellman and H. N. Shapiro, On a problem in additive number theory, Ann. of Math. 49 (1948), no. 2, 333-340. · Zbl 0031.25401 · doi:10.2307/1969281
[3] J.-P. Bertrandias, Suites pseudo-aléatoires et critères d’équirépartition modulo un, Compos. Math. 16 (1964), 23-28. · Zbl 0207.05801
[4] J.-P. Bertrandias, Espaces de fonctions bornées et continues en moyenne asymptotique d’ordre \(p\), Bull. Soc. Math. France 5, Soc. Math. France, Paris, 1966. (1966), 106. · Zbl 0148.11701
[5] J. Bésineau, Indépendance statistique d’ensembles liés à la fonction “somme des chiffres,” Acta Arith. 20 (1972), 401-416. · Zbl 0252.10052
[6] J. Bourgain, On the Fourier-Walsh spectrum of the Moebius function, Israel J. Math. 197 (2013), no. 1, 215-235. · Zbl 1336.11038 · doi:10.1007/s11856-013-0002-2
[7] J. Bourgain, Prescribing the binary digits of primes, Israel J. Math. 194 (2013), no. 2, 935-955. · Zbl 1309.11065 · doi:10.1007/s11856-012-0104-2
[8] J. Bourgain, Monotone Boolean functions capture their primes, J. Anal. Math. 124 (2014), no. 1, 297-307. · Zbl 1370.11113 · doi:10.1007/s11854-014-0033-6
[9] J. Bourgain, Prescribing the binary digits of primes, II, Israel J. Math. 206 (2015), no. 1, 165-182. · Zbl 1310.11097 · doi:10.1007/s11856-014-1129-5
[10] J. Bourgain, On the Fourier-Walsh spectrum of the Moebius function, II, J. Anal. Math. 128 (2016), no. 1, 355-367. · Zbl 1417.11046 · doi:10.1007/s11854-016-0012-1
[11] M. Drmota, The joint distribution of \(q\)-additive functions, Acta Arith. 100 (2001), no. 1, 17-39. · Zbl 1057.11006
[12] M. Drmota, C. Mauduit, and J. Rivat, Normality along squares, J. Eur. Math. Soc. (JEMS) 21 (2) (2019), no. 2, 507-548. · Zbl 1430.11010 · doi:10.4171/JEMS/843
[13] J.-H. Evertse, “The subspace theorem of W. M. Schmidt” in Diophantine Approximation and Abelian Varieties (Soesterberg, 1992), Lecture Notes in Math. 1566, Springer, Berlin, 1993, 31-50. · Zbl 0812.11039
[14] S. Ferenczi, J. Kułaga Przymus, and M. Lemańczyk, “Sarnak’s conjecture: What’s new” in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math. 2213, Springer, Cham, 2018, 163-235. · Zbl 1421.37003
[15] E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith. 77 (1996), no. 4, 339-351. · Zbl 0869.11073 · doi:10.4064/aa-77-4-339-351
[16] E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, Math. Ann. 305 (1996), no. 3, 571-599. · Zbl 0858.11050 · doi:10.1007/BF01444238
[17] A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1967/1968), no. 3, 259-265. · Zbl 0155.09003
[18] P. J. Grabner, P. Liardet, and R. F. Tichy, Spectral disjointness of dynamical systems related to some arithmetic functions, Publ. Math. Debrecen 66 (2005), no. 1-2, 213-243. · Zbl 1083.37008
[19] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004. · Zbl 1059.11001
[20] T. Kamae, “Mutual singularity of spectra of dynamical systems given by ‘sums of digits’ to different bases” in Dynamical Systems, Vol. I—Warsaw, Astérisque 49, Soc. Math. France, Paris, 1977, 109-114. · Zbl 0371.28018
[21] T. Kamae, Sum of digits to different bases and mutual singularity of their spectral measures, Osaka J. Math. 15 (1978), no. 3, 569-574. · Zbl 0391.10034
[22] T. Kamae, Cyclic extensions of odometer transformations and spectral disjointness, Israel J. Math. 59 (1987), no. 1, 41-63. · Zbl 0639.28010 · doi:10.1007/BF02779666
[23] I. Kátai, Distribution of digits of primes in \(q\)-ary canonical form, Acta Math. Hungar. 47 (1986), no. 3-4, 341-359. · Zbl 0603.10041
[24] D.-H. Kim, On the joint distribution of \(q\)-additive functions in residue classes, J. Number Theory 74 (1999), no. 2, 307-336. · Zbl 0920.11067 · doi:10.1006/jnth.1998.2327
[25] B. Martin, C. Mauduit, and J. Rivat, Théorème des nombres premiers pour les fonctions digitales, Acta Arith. 165 (2014), no. 1, 11-45. · Zbl 1395.11023 · doi:10.4064/aa165-1-2
[26] B. Martin, C. Mauduit, and J. Rivat, Fonctions digitales le long des nombres premiers, Acta Arith. 170 (2015), no. 2, 175-197. · Zbl 1395.11024 · doi:10.4064/aa170-2-5
[27] B. Martin, C. Mauduit, and J. Rivat, Nombres premiers avec contraintes digitales multiples, Bull. Soc. Math. France 147 (2019), no. 2, 259-287. · Zbl 1477.11014
[28] B. Martin, C. Mauduit, and J. Rivat, Propriétés locales des chiffres des nombres premiers, J. Inst. Math. Jussieu 18 (2019), no. 1, 189-224. · Zbl 1455.11117 · doi:10.1017/S1474748017000044
[29] C. Mauduit and B. Mossé, Suites de \(G_q\)-orbite finie, Acta Arith. 57 (1991), no. 1, 69-82. · Zbl 0726.11016
[30] C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math. 203 (2009), no. 1, 107-148. · Zbl 1278.11076 · doi:10.1007/s11511-009-0040-0
[31] C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2) 171 (2010), no. 3, 1591-1646. · Zbl 1213.11025 · doi:10.4007/annals.2010.171.1591
[32] C. Mauduit and J. Rivat, Prime numbers along Rudin-Shapiro sequences, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 10, 2595-2642. · Zbl 1398.11121 · doi:10.4171/JEMS/566
[33] J. Maynard, Primes with restricted digits, Invent. Math. 217 (2019), no. 1, 127-218. · Zbl 1477.11167 · doi:10.1007/s00222-019-00865-6
[34] M. Mendès France, Nombres normaux: Applications aux fonctions pseudo-aléatoires, J. Anal. Math. 20 (1967), no. 1, 1-56. · Zbl 0161.05002
[35] H. L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), no. 4, 547-567. · Zbl 0408.10033 · doi:10.1090/S0002-9904-1978-14497-8
[36] C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J. 166 (2017), no. 17, 3219-3290. · Zbl 1439.11089 · doi:10.1215/00127094-2017-0024
[37] M. Queffelec, Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme des chiffres, Israel J. Math. 34 (1979), no. 4, 337-342. · Zbl 0429.10035 · doi:10.1007/BF02760612
[38] P. Sarnak, Three lectures on the Mobius function randomness and dynamics, preprint, 2011, http://publications.ias.edu/sarnak/paper/512.
[39] J. D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 183-216. · Zbl 0575.42003 · doi:10.1090/S0273-0979-1985-15349-2
[40] M. · Zbl 0944.11024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.