An extended pair-correlation conjecture and primes in short intervals
HTML articles powered by AMS MathViewer
- by A. Languasco, A. Perelli and A. Zaccagnini PDF
- Trans. Amer. Math. Soc. 369 (2017), 4235-4250 Request permission
Abstract:
In this paper we extend the well-known investigations of Montgomery (1973) and Goldston and Montgomery (1987), concerning the pair-correlation function and its relations with the distribution of primes in short intervals, to a more general version of the pair-correlation function.References
- Daniel A. Goldston and Hugh L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 183–203. MR 1018376
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- D. R. Heath-Brown and D. A. Goldston, A note on the differences between consecutive primes, Math. Ann. 266 (1984), no. 3, 317–320. MR 730173, DOI 10.1007/BF01475582
- Serge Lang, Complex analysis, 4th ed., Graduate Texts in Mathematics, vol. 103, Springer-Verlag, New York, 1999. MR 1659317, DOI 10.1007/978-1-4757-3083-8
- Alessandro Languasco, Alberto Perelli, and Alessandro Zaccagnini, Explicit relations between pair correlation of zeros and primes in short intervals, J. Math. Anal. Appl. 394 (2012), no. 2, 761–771. MR 2927496, DOI 10.1016/j.jmaa.2012.04.058
- Alessandro Languasco, Alberto Perelli, and Alessandro Zaccagnini, An extension of the pair-correlation conjecture and applications, Math. Res. Lett. 23 (2016), no. 1, 201–220. MR 3512883, DOI 10.4310/MRL.2016.v23.n1.a10
- Helmut Maier, Primes in short intervals, Michigan Math. J. 32 (1985), no. 2, 221–225. MR 783576, DOI 10.1307/mmj/1029003189
- Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR 0337847, DOI 10.1007/BFb0060851
- H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
- H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82. MR 337775, DOI 10.1112/jlms/s2-8.1.73
- B. Saffari and R. C. Vaughan, On the fractional parts of $x/n$ and related sequences. II, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 2, v, 1–30 (English, with French summary). MR 480388
Additional Information
- A. Languasco
- Affiliation: Dipartimento di Matematica Tullio Levi-Civita, Università di Padova, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 354780
- ORCID: 0000-0003-2723-554X
- Email: languasco@math.unipd.it
- A. Perelli
- Affiliation: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
- MR Author ID: 137910
- Email: perelli@dima.unige.it
- A. Zaccagnini
- Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, 43124 Parma, Italy
- Email: alessandro.zaccagnini@unipr.it
- Received by editor(s): March 30, 2014
- Received by editor(s) in revised form: January 24, 2015, and June 18, 2015
- Published electronically: December 27, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4235-4250
- MSC (2010): Primary 11M26, 11N05
- DOI: https://doi.org/10.1090/tran/6835
- MathSciNet review: 3624407