## Duke Mathematical Journal

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

Anders Södergren

#### Abstract

We study the value distribution of the Epstein zeta function $E_{n}(L,s)$ for $0\textless s\textless \frac{n}{2}$ and a random lattice $L$ of large dimension $n$. For any fixed $c\in({1}/{4},{1}/{2})$ and $n\to\infty$, we prove that the random variable $V_{n}^{-2c}E_{n}(\cdot,cn)$ has a limit distribution, which we give explicitly (here $V_{n}$ is the volume of the $n$-dimensional unit ball). More generally, for any fixed $\varepsilon \textgreater 0$, we determine the limit distribution of the random function $c\mapstoV_{n}^{-2c}E_{n}(\cdot,cn)$, $c\in[{1}/{4}+\varepsilon ,{1}/{2}-\varepsilon ]$. After compensating for the pole at $c=1/2$, we even obtain a limit result on the whole interval $[1/4+\varepsilon ,1/2]$, and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function $h_{n}(L)$ of the flat torus ${\mathbb {R}}^{n}/L$: the random variable $n\{h_{n}(L)-(\log(4\pi)-\gamma+1)\}+\log n$ has a limit distribution as $n\to\infty$, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice $L\subset {\mathbb {R}}^{n}$ for which $E_{n}(L,s)$ has no zeros in $(0,\infty)$.

#### Article information

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

Dates
First available in Project Euclid: 14 January 2013

https://projecteuclid.org/euclid.dmj/1358172073

Digital Object Identifier
doi:10.1215/00127094-1903389

Mathematical Reviews number (MathSciNet)
MR3011871

Zentralblatt MATH identifier
1285.11065

#### 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

#### 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.