Tao’s resolution of the Erdős discrepancy problem
Author:
K. Soundararajan
Journal:
Bull. Amer. Math. Soc. 55 (2018), 81-92
MSC (2010):
Primary 14K38, 11B75, 11N64, 11Z05
DOI:
https://doi.org/10.1090/bull/1598
Published electronically:
September 11, 2017
MathSciNet review:
3737211
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: This article gives a simplified account of some of the ideas behind Tao’s resolution of the Erdős discrepancy problem.
- Peter Borwein, Stephen K. K. Choi, and Michael Coons, Completely multiplicative functions taking values in $\{-1,1\}$, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6279–6291. MR 2678974, DOI https://doi.org/10.1090/S0002-9947-2010-05235-3
- P. D. T. A. Elliott, On the correlation of multiplicative functions, Notas Soc. Mat. Chile 11 (1992), no. 1, 1–11. MR 1292619
- Paul Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300. MR 98702
- Paul Erdős, Some of my favourite unsolved problems, A tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, 1990, pp. 467–478. MR 1117038
- P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de “L’Enseignement Mathématique” [Monographs of L’Enseignement Mathématique], vol. 28, Université de Genève, L’Enseignement Mathématique, Geneva, 1980. MR 592420
- N. Frantzikinakis and B. Host. The logarithmic Sarnak conjecture for ergodic weights. arXiv:1708.00677, 2017.
- W. Timothy Gowers, Erdős and arithmetic progressions, Erdös centennial, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc., Budapest, 2013, pp. 265–287. MR 3203599, DOI https://doi.org/10.1007/978-3-642-39286-3_8
- Andrew Granville and Kannan Soundararajan, Pretentious multiplicative functions and an inequality for the zeta-function, Anatomy of integers, CRM Proc. Lecture Notes, vol. 46, Amer. Math. Soc., Providence, RI, 2008, pp. 191–197. MR 2437976, DOI https://doi.org/10.1090/crmp/046/15
- G. Halász, On the distribution of additive and the mean values of multiplicative arithmetic functions, Studia Sci. Math. Hungar. 6 (1971), 211–233. MR 319930
- O. Klurman. Correlations of multiplicative functions and applications. Compositio Math., 153 (2017), 1622–1657.
- O. Klurman and A. Mangerel. Rigidity theorems for multiplicative functions. arXiv:1707.07817, 2017.
- Boris Konev and Alexei Lisitsa, A SAT attack on the Erdős discrepancy conjecture, Theory and applications of satisfiability testing—SAT 2014, Lecture Notes in Comput. Sci., vol. 8561, Springer, Cham, 2014, pp. 219–226. MR 3252248, DOI https://doi.org/10.1007/978-3-319-09284-3_17
- A. R. D. Mathias, On a conjecture of Erdős and Čudakov, Combinatorics, geometry and probability (Cambridge, 1993) Cambridge Univ. Press, Cambridge, 1997, pp. 487–488. MR 1476466
- Kaisa Matomäki and Maksym Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056. MR 3488742, DOI https://doi.org/10.4007/annals.2016.183.3.6
- Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory 9 (2015), no. 9, 2167–2196. MR 3435814, DOI https://doi.org/10.2140/ant.2015.9.2167
- Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum Math. Sigma 4 (2016), Paper No. e14, 44. MR 3513734, DOI https://doi.org/10.1017/fms.2016.6
- Jiří Matoušek and Joel Spencer, Discrepancy in arithmetic progressions, J. Amer. Math. Soc. 9 (1996), no. 1, 195–204. MR 1311824, DOI https://doi.org/10.1090/S0894-0347-96-00175-0
- A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. Reine Angew. Math. 357 (1985), 138–160. MR 783538, DOI https://doi.org/10.1515/crll.1985.357.138
- Polymath5. The Erdős discrepancy problem. Website at http://michaelnielsen.org/polymath1, 2012.
- K. F. Roth, Remark concerning integer sequences, Acta Arith. 9 (1964), 257–260. MR 168545, DOI https://doi.org/10.4064/aa-9-3-257-260
- P. Sarnak, Three lectures on the Möbius function: randomness and dynamics, https://www.math.ias.edu/files/wam/2011/PSMobius.pdf, 2011.
- Peter Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc. 43 (2012), no. 2, 89–97. MR 3014544
- K. Soundararajan, The Liouville function in short intervals [after Matomäki and Radziwiłł]. Seminaire Bourbaki, arXiv:16016.08021, 2016.
- Terence Tao, The Erdős discrepancy problem, Discrete Anal. , posted on (2016), Paper No. 1, 29. MR 3533300, DOI https://doi.org/10.19086/da.609
- Terence Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi 4 (2016), e8, 36. MR 3569059, DOI https://doi.org/10.1017/fmp.2016.6
- T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, arXiv:1708.02610, 2017.
- Aurel Wintner, The Theory of Measure in Arithmetical Semi-Groups, publisher unknown, Baltimore, Md., 1944. MR 0015083
- E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen. II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467 (German). MR 223318, DOI https://doi.org/10.1007/BF02280301
Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 14K38, 11B75, 11N64, 11Z05
Retrieve articles in all journals with MSC (2010): 14K38, 11B75, 11N64, 11Z05
Additional Information
K. Soundararajan
Affiliation:
Department of Mathematics , Stanford University , 450 Serra Mall, Bldg. 380, Stanford, California 94305-2125
MR Author ID:
319775
Email:
ksound@math.stanford.edu
Received by editor(s):
June 15, 2017
Published electronically:
September 11, 2017
Additional Notes:
The author is partially supported by the NSF, and a Simons Investigator grant from the Simons Foundation.
Article copyright:
© Copyright 2017
American Mathematical Society
