Duke Mathematical Journal

On the value distribution of the Epstein zeta function in the critical strip

Anders Södergren

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the value distribution of the Epstein zeta function En(L,s) for 0<s<n2 and a random lattice L of large dimension n. For any fixed c(1/4,1/2) and n, we prove that the random variable Vn2cEn(,cn) has a limit distribution, which we give explicitly (here Vn is the volume of the n-dimensional unit ball). More generally, for any fixed ε>0, we determine the limit distribution of the random function cVn2cEn(,cn), c[1/4+ε,1/2ε]. After compensating for the pole at c=1/2, we even obtain a limit result on the whole interval [1/4+ε,1/2], and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function hn(L) of the flat torus Rn/L: the random variable n{hn(L)(log(4π)γ+1)}+logn has a limit distribution as n, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice LRn for which En(L,s) has no zeros in (0,).

Article information

Source
Duke Math. J., Volume 162, Number 1 (2013), 1-48.

Dates
First available in Project Euclid: 14 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1358172073

Digital Object Identifier
doi:10.1215/00127094-1903389

Mathematical Reviews number (MathSciNet)
MR3011871

Zentralblatt MATH identifier
1285.11065

Subjects
Primary: 11E45: Analytic theory (Epstein zeta functions; relations with automorphic
Secondary: 11P21: Lattice points in specified regions 60G55: Point processes

Citation

Södergren, Anders. On the value distribution of the Epstein zeta function in the critical strip. Duke Math. J. 162 (2013), no. 1, 1--48. doi:10.1215/00127094-1903389. https://projecteuclid.org/euclid.dmj/1358172073


Export citation

References

  • [1] P. T. Bateman and E. Grosswald, On Epstein’s zeta function, Acta Arith. 9 (1964), 365–373.
  • [2] V. Bentkus and F. Götze, On the lattice point problem for ellipsoids, Acta Arith. 80 (1997), 101–125.
  • [3] V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms, Ann. of Math. (2) 150 (1999), 977–1027.
  • [4] P. Billingsley, Probability and Measure, 3rd ed., Wiley Ser. Probab. Stat., Wiley, New York, 1995.
  • [5] P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley Ser. Probab. Stat., Wiley, New York, 1999.
  • [6] F. Götze, Lattice point problems and values of quadratic forms, Invent. Math. 157 (2004), 195–226.
  • [7] P. Hartman and A. Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176.
  • [8] E. Hecke, Mathematische Werke, 2nd ed., Vandenhoeck & Ruprecht, Göttingen, 1970.
  • [9] M. N. Huxley, “Integer points, exponential sums and the Riemann zeta function” in Number Theory for the Millennium, II (Urbana, Ill., 2000), A. K. Peters, Natick, Mass., 2002, 275–290.
  • [10] V. Jarník, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Ann. 100 (1928), 699–721.
  • [11] J. F. C. Kingman, Poisson Processes, Oxford Stud. Probab. 3, Oxford Univ. Press, New York, 1993.
  • [12] B. Riemann, “Über die Anzahl der Primzahlen unter einer gegebenen Grösse” (1859) in Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge, Springer, Berlin, 1990.
  • [13] C. A. Rogers, Mean values over the space of lattices, Acta Math. 94 (1955), 249–287.
  • [14] C. A. Rogers, The number of lattice points in a set, Proc. Lond. Math. Soc. (3) 6 (1956), 305–320.
  • [15] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • [16] G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.
  • [17] P. Sarnak and A. Strömbergsson, Minima of Epstein’s zeta function and heights of flat tori, Invent. Math. 165 (2006), 115–151.
  • [18] C. L. Siegel, A mean value theorem in geometry of numbers, Ann. of Math. (2) 46 (1945), 340–347.
  • [19] A. Södergren, On the Poisson distribution of lengths of lattice vectors in a random lattice, Math. Z. 269 (2011), 945–954.
  • [20] A. Södergren, On the value distribution and moments of the Epstein zeta function to the right of the critical strip, J. Number Theory 131 (2011), 1176–1208.
  • [21] H. M. Stark, On the zeros of Epstein’s zeta function, Mathematika 14 (1967), 47–55.
  • [22] A. Terras, Real zeroes of Epstein’s zeta function for ternary positive definite quadratic forms, Illinois J. Math. 23 (1979), 1–14.
  • [23] A. Terras, Integral formulas and integral tests for series of positive matrices, Pacific J. Math. 89 (1980), 471–490.
  • [24] A. Terras, The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions, J. Number Theory 12 (1980), 258–272.