Maier’s Matrix Method and Irregularities in the Distribution of Prime Numbers

  • Andrei Raigorodskii
  • Michael Th. RassiasEmail author


This paper is devoted to irregularities in the distribution of prime numbers. We describe the development of this theory and the relation to Maier’s matrix method.



We would like to express our thanks to J. Friedlander and A. Granville for their useful comments.

A. Raigorodskii: I would like to acknowledge financial support from the grant NSh-6760.2018.1.

M. Th. Rassias: I would like to express my gratitude to the J. S. Latsis Foundation for their financial support provided under the auspices of my current “Latsis Foundation Senior Fellowship” position.


  1. 1.
    A. Balog, T.D. Wooley, Sums of two squares in short intervals. Can. J. Math. 52(4), 673–694 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2, 396–403 (1936)CrossRefGoogle Scholar
  3. 3.
    P.D.T.A. Elliott, H. Halberstam, A conjecture in prime number theory. Symp. Math. 4, 59–72 (1968)Google Scholar
  4. 4.
    P. Erdős, On the difference of consecutive primes. Q. J. Oxf. 6, 124–128 (1935)Google Scholar
  5. 5.
    J. Friedlander, A. Granville, Limitations to the equidistribution of primes I. Ann. Math. 129, 363–382 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Friedlander, A. Granville, Limitations to the equidistribution of primes IV. Proc. R. Soc. Lond. Ser. A 435, 197–204 (1991)CrossRefGoogle Scholar
  7. 7.
    J. Friedlander, A. Granville, A. Hildebrand, H. Maier, Oscillation theorems for primes in arithmetic progressions and for sifting functions. J. AMS 4(1), 25–86 (1991)MathSciNetzbMATHGoogle Scholar
  8. 8.
    P.X. Gallagher, A large sieve density estimate near σ = 1. Invent. Math. 11, 329–339 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Granville, Unexpected irregularities in the distribution of prime numbers, in Proceedings of the International Congress of Mathematicians, Zürich (1994), pp. 388–399Google Scholar
  10. 10.
    A. Granville, K. Soundararajan, An uncertainty principle for arithmetic sequences. Ann. Math. 165, 593–635 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Hildebrand, H. Maier, Irregularities in the distribution of primes in short intervals. J. Reine Angew. Math. 397, 162–193 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    H. Iwaniec, The sieve of Eratosthenes-Legendre. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4(4), 257–268 (1977)MathSciNetzbMATHGoogle Scholar
  13. 13.
    H. Maier, Chains of large gaps between consecutive primes. Adv. Math. 39, 257–269 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Maier, Primes in short intervals. Mich. Math. J. 32(2), 221–225 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Monks, S. Peluse, L. Ye, Strings of special primes in arithmetic progressions (English summary). Arch. Math. (Basel) 101(3), 219–234 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    H.L. Montgomery, Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, vol. 227 (Springer, New York, 1971)CrossRefGoogle Scholar
  17. 17.
    H.L. Montgomery, Problems concerning prime numbers. Proc. Symp. Pure Math. 28, 307–310 (1976)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Nair, A. Perelli, On the prime ideal theorem and irregularities in the distribution of primes. Duke Math. J. 77, 1–20 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    R.A. Rankin, The difference between consecutive prime numbers. J. Lond. Math. Soc. 13, 242–247 (1938)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Rosen, Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210 (Springer, New York, 2002)Google Scholar
  21. 21.
    D.K.L. Shiu, Strings of congruent primes. J. Lond. Math. Soc. 61(2), 359–373 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    F. Thorne, Irregularities in the distribution of primes in function fields. J. Number Theory 128(6), 1784–1794 (2008)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.Buryat State UniversityUlan-UdeRussia
  4. 4.Institute of MathematicsUniversity of ZürichZürichSwitzerland
  5. 5.Institute for Advanced Study, Program in Interdisciplinary StudiesPrincetonUSA

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