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Bubbles of Congruent Primes

Published online by Cambridge University Press:  04 November 2014

FRANK THORNE
FRANK THORNE*
Affiliation:
Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208U.S.A e-mail: [email protected]
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Abstract

In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu's theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 443 - 456
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]de Bruijn, N. G.On the number of positive integers ⩽ x and free of prime factors ⩾ y. Indag. Math. 13 (1951), 50–60.CrossRefGoogle Scholar
[2]Cojocaru, A. C. and Murty, M. R.An Introduction to Sieve Methods and Their Applications. (Cambridge University Press, Cambridge, 2005).CrossRefGoogle Scholar
[3]de Berg, M., van Kreveld, M., Overmars, M., and Schwarzkopf, O.Computational Geometry: Algorithms and Applications. (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
[4]Fogels, E.On the zeros of Hecke's L-functions I. Acta Arith. 7 (1961), 131–147.CrossRefGoogle Scholar
[5]Fogels, E.On the zeros of L-functions. Acta Arith. 11 (1965), 67–96; corrigendum, Acta Arith. 14 (1967/1968), 435.CrossRefGoogle Scholar
[6]Gallagher, P. X.A large sieve density estimate near σ = 1. Invent. Math. 11 (1970), 329–339.CrossRefGoogle Scholar
[7]Granville, A.Unexpected irregularities in the distribution of prime numbers. Proceedings of the International Congress of Mathematicians (Zürich, 1994), 388–399. (Birkhäuser, Basel, 1995).CrossRefGoogle Scholar
[8]Granville, A. and Soundararajan, K.An uncertainty principle for arithmetic sequences. Ann. of Math. 165 (2007), no. 2, 593–635.CrossRefGoogle Scholar
[9]Iwaniec, H. and Kowalski, E.Analytic number theory. Amer. Math. Soc. (Providence, 2005).CrossRefGoogle Scholar
[10]Krause, U.Abschätzungen für die Funktion ΨK(x, y) in algebraischen Zahlkörpern. Manuscripta Math. 69 (1990), 319–331.CrossRefGoogle Scholar
[11]Maier, H.Chains of large gaps between consecutive primes. Adv. in Math. 39 (1981), 257–269.CrossRefGoogle Scholar
[12]Maier, H.Primes in short intervals. Michigan Math. J. 32 (1985), 221–225.CrossRefGoogle Scholar
[13]Neukirch, J.Algebraic number theory. (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
[14]Rosen, M.A generalisation of Mertens' theorem. J. Ramanujan Math. Soc. 14 (1999), 1–19.Google Scholar
[15]Shiu, D. K. L.Strings of congruent primes. J. London Math. Soc. 61 (2000), 359–373.CrossRefGoogle Scholar
[16]Tenenbaum, G.Introduction to Analytic and Probabilistic Number Theory. (Cambridge University Press, Cambridge, 1995).Google Scholar