Arithmetic Randomness



The aim of the Austrian-French joint project ArithRand ("Arithmetic Randomness", 2021-2025) is to make progress on various questions that interrelate and link the notion of randomness (or pseudo-randomness) with the notion of determinism in the context of analytic number theory, combinatorics on words, automata theory, quasi-random sequences and nearby disciplines. It is financially supported by the FWF (Fonds zur Förderung der wissenschaftlichen Forschung, I 4945-N) and the ANR (Agence Nationale de la Recherche, ANR-20-CE91-0006).

Research topics

Many numbertheoretic sequences are per se deterministic but resemble much of the overall behaviour of a random sequence. One important example is the Möbius function. In this context we are interested in studying the independence between the Möbius function and various deterministic functions such as functions that are produced by a dynamical system of zero entropy or by a simple algorithm based on the binary digital representation. Informally, the difficulty of this independence problem reflects the difficulty of the transition from the digital representation of an integer to its multiplicative representation as a product of prime factors. This field of questions is the source of many important open problems in mathematics and computer science. In particular, we mention the construction of normal numbers, the analysis of pseudo-random and complexity measures, and the search for optimal discrepancy estimates for digitally based sequences or other quasi-random sequences which are some of the major lines of research in this joint project.


The Austrian-French Consortium consists of 16 researchers and is based in Graz, Vienna, Bordeaux, Calais, Marseille, Nancy and St Etienne.

Postdoctoral researchers:

Doctoral students:

  • Vincent Gozé (Calais-Littoral)


  • Talks by visitors (or online)
    Arithrand-related talks at Ernest seminar (Luminy, Marseille, FR)
  • 2021-12-07: Joni Teräväinen, On a hybrid of the Hardy-Littlewood and Chowla conjectures
  • 2021-06-08: Gregory Debruyne, Malliavin-type remainders for Beurling generalized prime number systems
  • 2021-05-11: Johann Verwee, Théorèmes d'Erdős-Wintner effectifs
  • 2021-04-13: Pierre-Adrien Tahay, Corrélations discrètes d’ordre 2 de suites généralisées de Rudin-Shapiro par une approche combinatoire
  • 2020-12-08: Youness Lamzouri, La répartition du maximum des sommes partielles de sommes d'exponentielles

    Seminars with links to ArithRand ("Metz-Nancy Number Theory Seminar" at IECL, Nancy)
  • 9/12/2021, Journée Scientifique FCH "Pseudorandomness, cryptography and number theory" (IECL/LORIA)
  • 2/12/2021 Benli Kübra (IECL), Changes in digits of primes
  • 18/11/2021 Sébastien Darses (Aix-Marseille Université), On probabilistic generalizations of the Nyman-Beurling criterion for the Zeta function
  • 21/10/2021 Manfred Madritsch (IECL), Construction d'un nombre normal tel que son inverse soit également normal
  • 30/09/2021 Yann Bugeaud, Approximation rationnelle des nombres sturmiens
  • 20/05/2021 (online) Kübra Benli, Small prime power residues modulo p
  • 15/04/2021 (online) Gérald Tenenbaum (IECL), Répartition des fonctions multiplicatives dans les progressions arithmétiques de grands modules et applications
  • 08/04/2021, (online) Paul Pollack (University of Georgia), Multiplicative orders mod p
  • 01/04/2021 (online) Sarah Peluse (IAS/Princeton), Modular zeros in the character table of the symmetric group
  • 11/03/2021 (online) Alessandro Languasco, On computing L'/L(1,chi) and related problems
  • 18/02/2021 (online) Brad Rodgers, The distribution of random polynomials with multiplicative coefficients
  • 07/01/2021 (online) Marc Munsch, Comportement aléatoire local des suites réelles: résultats métriques, énergie additive et inégalités diophantiennes
  • 10/12/2020 (online) Johann Verwee, Théorèmes d'Erdös-Wintner effectifs
  • 19/11/2020 (online) Amri Myriam, Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidus
  • 12/11/2020 (online) Jakub Konieczny, Automatic multiplicative sequences
  • 15/10/2020 Shuo Li, On the classification of completely multiplicative automatic sequences

Guests (selection)

  • to come

For your upcoming or to be published papers, please include : "This paper was supported by the joint FWF-ANR project Arithrand: FWF: I 4945-N and ANR-20-CE91-0006."

Publications-Preprints - in preparation (explicitely mentioning ArithRand)

  • C. Aistleitner, B. Borda: Quantum invariants of hyperbolic knots and extreme values of trigonometric products. Preprint, arXiv:2006.08578
  • C. Aistleitner, D. El-Baz, M. Munsch: A pair correlation problem, and counting lattice points with the zeta function. GAFA 31 (2021), no. 3, 483--512
  • C. Aistleitner, S. Baker, N. Technau, N. Yesha: Gap statistics and higher correlations for geometric progressions modulo one. Preprint, arxiv:2010.10355
  • C. Aistleitner, N. Gantert, Z. Kabluchko, J. Prochno, K. Ramanan: Large Deviation Principles for Lacunary Sums. Preprint, axriv:2012.05281
  • C. Aistleitner, B. Borda: Maximizing Sudler products via Ostrowski expansions and cotangent sums. Preprint, arXiv:2104.01379
  • C. Aistleitner, D. El-Baz, M. Munsch: Difference sets and the metric theory of small gaps. Preprint, arXiv:2108.02227
  • C. Aistleitner, B. Borda: A conjecture of Zagier and the value distribution of quantum modular forms. Preprint, arXiv:2110.07407
  • Bettin, S.; Drappeau, S. Two arithmetic applications of perturbations of composition operators. J. Anal. Math., to appear. arXiv:2002.11417. hal-02493843v1.
  • Bettin, S.; Drappeau, S. Effective estimation of some oscillatory integrals related to infinitely divisible distributions. Ramanujan J., to appear. arXiv: 2010.15494. HAL:hal-02998544v1.
  • Bettin, S.; Drappeau, S. Two arithmetic applications of perturbations of composition operators. J. Anal. Math., to appear. arXiv:2002.11417. hal-02493843v1.
  • Bettin, S.; Drappeau, S. Effective estimation of some oscillatory integrals related to infinitely divisible distributions. Ramanujan J., to appear. arXiv: 2010.15494. HAL:hal-02998544v1.
  • Bettin, S.; Drappeau, S. Modularity and value distribution of quantum invariants of hyperbolic knots. Math. Ann., to appear. arXiv:1905.02045. HAL: hal-02998581v1.
  • Bettin, S.; Drappeau, S. Limit laws for rational continued fractions and value distribution of quantum modular forms. Submitted. arXiv:1903.00457. HAL: hal-02998562v1.
  • La Bretèche, R.; Dress, F.; Tenenbaum, G., Remarques sur une somme liée à la fonction de Möbius, Mathematika 66, no. 2 (2020), 416--421.
  • La Bretèche, R.; Munsch, M.; Tenenbaum, G., Small Gál sums and applications, J. London Math. Soc. 103}(2021), 336--352.
  • La Bretèche, R. Tenenbaum, G., Remarks on the Selberg--Delange method, Acta Arith. 200.4 (2021), 349-369.
  • La Bretèche, R. Tenenbaum, G., On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution, Lith. Math. J. 61, n°3 (2021), 301-311.
  • La Bretèche, R. Tenenbaum, G., On the gap distribution of prime factors, prépublication.
  • Fernando Chamizo, Bruno Martin,  The convergence of certain Diophantine series. Journal of Number Theory, to appear.  DOI: 10.1016/j.jnt.2021.04.022
  • Dartyge, C; Feutrie, D.;  Tenenbaum, G., Entiers ultrafriables en progressions arithmétiques, Quart. J. Math. (Oxford) (2021), à paraître.
  • Drappeau, S.; Hanna, G. The Thue–Morse and Rudin–Shapiro sequences at primes in principal number fields. Acta Math. Hung. 162 (2020), 130--186. Zbl 1398.11121; MR4169024. DOI: 10.1007/s10474-020-01030-9. arXiv:2001.07017. HAL:hal-02493862v1.
  • Drappeau, S.; Fiorilli, D. The first moment of primes in arithmetic progressions: beyond the Siegel-Walfisz range. Trans. London Math. Soc., to appear. arXiv: 2003.02201. HAL:hal-02567736v1.
  • Drappeau, S.; Pratt, K.; Radziwiłł, M. One-level density estimates for Dirichlet L-functions with extended support. Submitted. arXiv:2002.11968. HAL: hal-02493847v1.
  • Debruyne, G.; Tenenbaum, G., The saddle point method for general partitions, Indag. Math. 31 (2020), 728-738.
  • M. Drmota, M. Lemańczyk, C. Müllner, J. Rivat; Some recent developments on the Sarnak conjecture, manuscript.
  • Michael Drmota, Christian Mauduit, Joël Rivat, Lukas Spiegelhofer: Möbius orthogonality of sequences with maximal entropy. Journal d'Analyse Mathématique, to appear.
  • Michael Drmota, Clemens Müllner, Lukas Spiegelhofer; Primes as sums of Fibonacci numbers; arXiv:2109.04068
  • Elsholtz, C.; Planitzer, S., Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709.
  • Elsholtz, C. Fermat's last theorem implies Euclid's infinitude of primes. Amer. Math. Monthly 128 (2021), no. 3, 250--257.
  • Fouvry, É.; Tenenbaum, G., Multiplicative functions in large arithmetic progressions and applications, Trans. Amer. Math. Soc., à paraître.
  • Jamet, D.; Popoli, P.; Stoll, T. Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences. Cryptography and Communications 13 (2021),  791--814.,
  • Kaneko, H.; Stoll T. Products of integers with few nonzero digits. Uniform Distribution Theory (2021), 17pp, to appear.
  • Elżbieta Krawczyk, Clemens Müllner, Automaticity of uniformly recurrent substitutive sequences,
  • Marcovici, I.; Tahay, P.-A.; Stoll, T. Discrete correlations of order 2 of generalised Golay-Shapiro sequences: a combinatorial approach. Integers 21 (2021), A45, 21pp.,
  • O. Ramare, Notes on restriction theory in the primes, arXiv:2109.10180
  • Tenenbaum, G., A note on the normal largest  gap between prime factors, J. Théor. Nombres Bordeaux 31, no. 3 (2019), 747-749.
  • Tenenbaum, G.;  Verwee, J., Effective Erd\H os--Wintner theorems, Proc. Steklov Inst. Math. 314 (2021), 275--289.
  • Diophantine Problems: Determinism, Randomness and Applications Editors: Dijana Kreso, Joel Rivat, Robert Tichy.


    by Sary Drappeau
  • "Modularity of the q-Pochhammer symbol and application", 2020-11-27, CIRM Conference "Diophantine Problems, Determinism and Randomness".
  • "The distribution of the Estermann function and other quantum modular forms", 2021-03-01, Mittag-Leffler semester on Number Theory.
  • "Fractions continues et lois limites pour quelques valeurs de fonctions L", 2021-05-21, Séminaire de Dynamique de l'IMJ-PRG.
  • "Continued fractions and limit laws for values of L-functions", 2021-05-28, N-Cube days.
  • "Shifted convolutions, arithmetic progressions, exponential sums", 2021-06-28 to 2021-07-02, 4 talks at the Paris Summer School in Analytic Number

    by Michael Drmota
  • online, contributed, (Logarithmic) Densities for Automatic Sequences along Primes and Squares AG Diskrete Mathematik, TU Wien,  March 16, 2021
  • online, invited, (Logarithmic) Densities for Automatic Sequences along Primes and Squares One World Numeration Seminar, March 30, 2021
  • in person, contributed, Primes as Sums of Fibonacci Numbers AG Diskrete Mathematik, TU Wien,  October 12, 2021

    by Christian Elsholtz
  • Large sets in (Z_m)^n without progressions 7 Sep 2020 - 11 Sep 2020, Cirm (Luminy, online)
  • Improved cap constructions, and sets without arithmetic progressions, Conference: Diophantine Problems: Determinism, Randomness, Applications 23 Nov 2020 - 27 Nov 2020, CIRM Luminy (online)
  • Sums of unit fractions, March 18, 2021, Karl Franzens University Graz
  • Improved cap constructions, and sets without arithmetic progressions, March 25, 2021, Karl Franzens University Graz
  • Fermat's Last Theorem implies Euclid's infinitude of the primes, Combinatorial and additive number theory, CANT 2021 (New York, online), 24 Mai 2020 - 28 May 2021
  • Improved cap constructions, and sets without arithmetic progressions, June 3, 2021, Laboratoire Analyse, Géométrie et Applications, Paris, Frankreich
  • Arithmetic progressions in arithmetic sets, June 11, 2021, Salzburg-Debrecen-Graz seminar
  • Fermat's last theorem implies Euclid's infinitude of primes, Jun 24, 2021. Karl-Franzens-Universität Graz.
  • Improved cap constructions in affine and projective spaces, 29.9.2021, DMV-ÖMG Tagung, Passau (virtual)
  • Lower bound constructions for progression-free sets and caps in affine and projective spaces. Number theory seminar, Alfred Renyi-Institute, (virtual)
  • Lower bound constructions for progression-free sets and caps in affine and projective spaces. COmbinatorial Number Theory And Connected Topics (CONTACT - I), India, (virtual)

    by Bruno Martin
    invited online talk in the Conference "Diophantine Problems, Determinism and Randomness" in Cirm, 24th november 2020. Title " Some interactions between number theory and multifractal analysis".

    by Clemens Müllner
  • Multiplicative automatic sequences, Open University Dynamical Systems seminar of 2021, 13. January 2021 (invited)
  • Multiplicative automatic sequences, One World Numeration Seminar 9. February 2021 (contributed)

    by O. Ramare
  • O. Ramaré, 4 octobre 2021, Séminaire de Théorie des Nombres de Paris, Polynomes trigonom'etriques arithm'etiques aux points de petites hauteurs, in person
  • 6-17 December 2021, Density Estimates and Additive Combinatorics, brainstorming session with four participants, with the support of ArithRand

    by Andrei Shubin
    in person, contributed, Primes in subsets and exponential sums, AG Diskrete Mathematik, TU Wien,  Nov 9, 2021

  • Thomas Stoll
    On difference matrices and generalised Rudin-Shapiro sequences, Plenary talk, online, 7th International Conference on Uniform Distribution Theory (UDT2021), 24-02-2021.
    by G. Tenenbaum
  • Number Theory Web Seminar, 10 novembre 2020. Diophantine Problems, Determinism and Randomness, CIRM, Luminy, 23-27 novembre 2020.
  • Analytic & combinatorial number theory in honour of R. Balasubramanian, IMSc, Chennai 16 mars 2021.