**Alessandro Zaccagnini**^{[1]}

*
The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function
*

**Pages:** 133-151

**Received:** 31 December 2015

**Accepted in revised form:** 1 March 2016

**Mathematics Subject Classification (2010):** 11M26, 11N05.

**Keywords:** Riemann zeta-function, Selberg integral, Montgomery's pair-correlation function.

**Author address:**

[1] : Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Parma, 43124 Italia

This research was partially supported by the grant PRIN2010-11, *Arithmetic Algebraic Geometry and Number Theory*.

**Abstract:**
The present paper is a report on joint work with Alessandro Languasco
and Alberto Perelli, collected in [10], [11] and [12], on our recent
investigations on the Selberg integral and its connections to
Montgomery's pair-correlation function.
We introduce a more general form of the Selberg integral and connect
it to a new pair-correlation function, emphasising its relations to
the distribution of prime numbers in short intervals.

**References**

[1]
J. Brüdern, R. J. Cook and A. Perelli ,
*The values of binary linear forms at prime arguments*,
Sieve methods, exponential sums and their application in number theory (Cardiff, 1995),
G. R. H. Greaves et al., ed., Cambridge Univ. Press, Cambridge 1997, 87-100.
MR1635730

[2]
T. H. Chan ,
*More precise pair correlation of zeros and primes in short
intervals*, J. London Math. Soc. (2) 68 (2003), no. 3, 579-598.
MR2009438

[3]
K. Ford and A. Zaharescu ,
*On the distribution of imaginary parts of zeros of the Riemann zeta function*,
J. Reine Angew. Math. 579 (2005), 145-158.
MR2124021

[4]
D. A. Goldston ,
*Notes on pair correlation of zeros and prime numbers*,
in "Recent perspectives in random matrix theory and number theory", F. Mezzadri
and N. C. Snaith, eds., London Math. Soc. Lecture Note Ser., 322 , Cambridge
University Press, Cambridge 2005, 79-110.
MR2166459

[5]
D. A. Goldston and D. R. Heath-Brown ,
*A note on the differences between consecutive primes*,
Math. Ann. 266 (1984), no. 3, 317-320.
MR0730173

[6]
D. A. Goldston and H. L. Montgomery ,
*Pair correlation of zeros and primes in short intervals*,
Analytic number theory and Diophantine problems (Stillwater, OK, 1984),
A. C. Adolphson et al., ed., Progr. Math., 70 , Birkhäuser Boston, Boston, MA 1987, 183-203.
MR1018376

[7]
S. M. Gonek ,
*An explicit formula of Landau and its applications to the theory of the zeta-function*,
Contemp. Math., 143 , Amer. Math. Soc., Providence, RI 1993, 395-413.
MR1210528

[8]
D. R. Heath-Brown ,
*Gaps between primes, and the pair correlation of zeros of the zeta-function*,
Acta Arith. 41 (1982), 85-99.
MR0667711

[9]
D. R. Heath-Brown ,
*The number of primes in a short interval*,
J. Reine Angew. Math. 389 (1988), 22-63.
MR0953665

[10]
A. Languasco, A. Perelli and A. Zaccagnini ,
*Explicit relations between pair correlation of zeros and primes in short intervals*,
J. Math. Anal. Appl. 394 (2012), 761-771.
MR2927496

[11]
A. Languasco, A. Perelli and A. Zaccagnini ,
*An extension of the pair-correlation conjecture and applications*,
Math. Res. Lett. 23 (2016), no. 1, 201-220.
URL

[12]
A. Languasco, A. Perelli and A. Zaccagnini ,
*An extended pair-correlation conjecture and primes in short intervals*, Trans. Amer. Math. Soc., to appear.

[13]
J. E. Littlewood ,
*Sur la distribution des nombres premiers*,
C. R. Math. Acad. Sci. Paris 158 (1914), 1869-1872.

[14]
H. Maier ,
*Primes in short intervals*,
Michigan Math. J. 32 (1985), 221-225.
MR0783576

[15]
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, 181-193.
MR0337821

[16]
H. L. Montgomery and K. Soundararajan ,
*Beyond pair correlation*,
Paul Erdos and his mathematics, I (Budapest, 1999), Bolyai Soc. Math.
Stud., 11 , János Bolyai Math. Soc., Budapest 2002, 507-514.
MR1954710

[17]
H. L. Montgomery and R. C. Vaughan ,
*Hilbert's inequality*, J. London Math. Soc. (2) 8 (1974), 73-82.
MR0337775

[18]
H. L. Montgomery and R. C. Vaughan ,
*Multiplicative number theory. I. Classical theory*,
Cambridge Univ. Press, Cambridge 2007.
MR2378655

[19]
M. R. Murty and A. Perelli ,
*The pair correlation of zeros of functions in the Selberg class*,
Internat. Math. Res. Notices 1999, no. 10, 531-545.
MR1692847

[20]
J. Pintz ,
*On the remainder term of the prime number formula and the zeros of the Riemann's zeta-function*,
Number Theory (Noordwijkerhout 1983), Lecture Notes in Math., 1068 , Springer, Berlin 1984, 186-197.
MR0756094

[21]
G. F. B. Riemann ,
*Über die Anzahl der Primzahlen unter einer gegebenen Grösse*,
Monatsber. Königl. Preuss. Akad. Wiss.
Berlin (1859), 671-680, in "Gesammelte Mathematische Werke" (ed. H. Weber),
reprint, Dover Publications, New York 1953.

[22]
B. Saffari and R. C. Vaughan ,
*On the fractional parts of \(x/n\) and related sequences. II*,
Ann. Inst. Fourier 27 (1977), 1-30.
MR0480388

[23]
A. Selberg ,
*On the normal density of primes in small intervals, and the difference between consecutive primes*,
Arch. Math. Naturvid. 47
(1943), 87-105.
MR0012624

[24]
A. Zaccagnini ,
*Primes in almost all short intervals*, Acta Arith.
84 (1998), no. 3, 225-244.
MR1617735

[25]
A. Zaccagnini ,
*A conditional density theorem for the zeros of the Riemann zeta-function*,
Acta Arith. 93 (2000), no. 3, 293-301.
MR1759919

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