Abstract
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.
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References
A. Balog, T.D. Wooley, Sums of two squares in short intervals. Can. J. Math. 52(4), 673–694 (2000)
H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2, 396–403 (1936)
P.D.T.A. Elliott, H. Halberstam, A conjecture in prime number theory. Symp. Math. 4, 59–72 (1968)
P. Erdős, On the difference of consecutive primes. Q. J. Oxf. 6, 124–128 (1935)
J. Friedlander, A. Granville, Limitations to the equidistribution of primes I. Ann. Math. 129, 363–382 (1989)
J. Friedlander, A. Granville, Limitations to the equidistribution of primes IV. Proc. R. Soc. Lond. Ser. A 435, 197–204 (1991)
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)
P.X. Gallagher, A large sieve density estimate near σ = 1. Invent. Math. 11, 329–339 (1970)
A. Granville, Unexpected irregularities in the distribution of prime numbers, in Proceedings of the International Congress of Mathematicians, Zürich (1994), pp. 388–399
A. Granville, K. Soundararajan, An uncertainty principle for arithmetic sequences. Ann. Math. 165, 593–635 (2007)
A. Hildebrand, H. Maier, Irregularities in the distribution of primes in short intervals. J. Reine Angew. Math. 397, 162–193 (1989)
H. Iwaniec, The sieve of Eratosthenes-Legendre. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4(4), 257–268 (1977)
H. Maier, Chains of large gaps between consecutive primes. Adv. Math. 39, 257–269 (1981)
H. Maier, Primes in short intervals. Mich. Math. J. 32(2), 221–225 (1985)
K. Monks, S. Peluse, L. Ye, Strings of special primes in arithmetic progressions (English summary). Arch. Math. (Basel) 101(3), 219–234 (2013)
H.L. Montgomery, Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, vol. 227 (Springer, New York, 1971)
H.L. Montgomery, Problems concerning prime numbers. Proc. Symp. Pure Math. 28, 307–310 (1976)
A. Nair, A. Perelli, On the prime ideal theorem and irregularities in the distribution of primes. Duke Math. J. 77, 1–20 (1995)
R.A. Rankin, The difference between consecutive prime numbers. J. Lond. Math. Soc. 13, 242–247 (1938)
M. Rosen, Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210 (Springer, New York, 2002)
D.K.L. Shiu, Strings of congruent primes. J. Lond. Math. Soc. 61(2), 359–373 (2000)
F. Thorne, Irregularities in the distribution of primes in function fields. J. Number Theory 128(6), 1784–1794 (2008)
Acknowledgements
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.
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Raigorodskii, A., Rassias, M.T. (2018). Maier’s Matrix Method and Irregularities in the Distribution of Prime Numbers. In: Pintz, J., Rassias, M. (eds) Irregularities in the Distribution of Prime Numbers. Springer, Cham. https://doi.org/10.1007/978-3-319-92777-0_9
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DOI: https://doi.org/10.1007/978-3-319-92777-0_9
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