×

On the value distribution of the Epstein zeta function in the critical strip. (English) Zbl 1285.11065

The Epstein zeta function of a lattice \(L \subset \mathbb{R}^n\) of covolume \(1\) is defined as the meromorphic continuation to all \(s \in \mathbb{C}\) of \[ E_n(L, s) = \sum_{v \in L, v \neq 0} \|v\|^{-2s} \] which converges absolutely for \(\mathrm{Re}(s) > n/2\). The only pole of \(E_n(L, s)\) is at \(s=n/2\) (simple) with residue \(\pi^{n/2} \Gamma(n/2)^{-1}\). Moreover it satisfies a function equation under \(s \mapsto n/2 - s\). Call \(\{ s : 0 < \text{Re}(s) < n/2 \}\) the critical strip for \(E_n(L, s)\). The first goal of this paper is to study the value distribution of \(E_n(L, cn)\) as \(n \to \infty\), where \(0 < c < 1/2\); here \(L\) is viewed as a random lattice of covolume \(1\).
The first result (Theorem 1.1) of this paper describes the limit distribution of \(V_n^{-2c} E_n(\cdot, cn)\) as \(n \to \infty\) (regarded as a random variable), where \(c\) lies in a closed interval in \((1/4, 1/2)\) and \(V_n := \mathrm{vol}(\mathbb{S}^n)\): it equals \[ H(c) := \int_{V > 0} V^{-2c} \,dR(V) \] where \(R(V)\) is explicitly given in terms of a Poisson process on \(\mathbb{R}_{>0}\) with constant intensity \(1/2\). As a byproduct (Theorem 1.3), the author obtains \(R_n(V) \ll V^{1/2} (\log V)^{\varepsilon + 3/2}\) for almost every \(L\), where \(R_n\) is essentially the remainder term in the generalized circle problem.
By subtracting the singularities from both \(E_n\) and \(H\) at \(c=1/2\), the author extends the aforementioned result to intervals of the form \([1/4 + \varepsilon, 1/2]\) (Theorem 6.2). These results are closely related to P. Sarnak and A. Strömbergsson [Invent. Math. 165, No. 1, 115–151 (2006; Zbl 1145.11033)] on the heights \(h_n(L)\) of flat tori \(\mathbb{R}^n/L\) and the minima of \(E_n(\cdot, s)\). For example, the Theorem 1.4 in this paper sharpens the description of \(h_n(L)\) as \(n \to \infty\) in Theorem 3 of loc. cit. Such results are also of interest in arithmetic.

MSC:

11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11P21 Lattice points in specified regions
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 1145.11033
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] P. T. Bateman and E. Grosswald, On Epstein’s zeta function , Acta Arith. 9 (1964), 365-373. · Zbl 0128.27004
[2] V. Bentkus and F. Götze, On the lattice point problem for ellipsoids , Acta Arith. 80 (1997), 101-125. · Zbl 0871.11069
[3] V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms , Ann. of Math. (2) 150 (1999), 977-1027. · Zbl 0979.11048 · doi:10.2307/121060
[4] P. Billingsley, Probability and Measure , 3rd ed., Wiley Ser. Probab. Stat., Wiley, New York, 1995. · Zbl 0822.60002
[5] P. Billingsley, Convergence of Probability Measures , 2nd ed., Wiley Ser. Probab. Stat., Wiley, New York, 1999. · Zbl 0944.60003
[6] F. Götze, Lattice point problems and values of quadratic forms , Invent. Math. 157 (2004), 195-226. · Zbl 1090.11063 · doi:10.1007/s00222-003-0366-3
[7] P. Hartman and A. Wintner, On the law of the iterated logarithm , Amer. J. Math. 63 (1941), 169-176. · Zbl 0024.15802 · doi:10.2307/2371287
[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. · Zbl 1030.11053
[10] V. Jarník, Über Gitterpunkte in mehrdimensionalen Ellipsoiden , Math. Ann. 100 (1928), 699-721. · JFM 54.0203.01 · doi:10.1007/BF01448873
[11] J. F. C. Kingman, Poisson Processes , Oxford Stud. Probab. 3 , Oxford Univ. Press, New York, 1993. · Zbl 0771.60001
[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. · Zbl 0065.28201 · doi:10.1007/BF02392493
[14] C. A. Rogers, The number of lattice points in a set , Proc. Lond. Math. Soc. (3) 6 (1956), 305-320. · Zbl 0071.27403 · doi:10.1112/plms/s3-6.2.305
[15] W. Rudin, Real and Complex Analysis , 3rd ed., McGraw-Hill, New York, 1987. · Zbl 0925.00005
[16] G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance , Chapman & Hall, New York, 1994. · Zbl 0925.60027
[17] P. Sarnak and A. Strömbergsson, Minima of Epstein’s zeta function and heights of flat tori , Invent. Math. 165 (2006), 115-151. · Zbl 1145.11033 · doi:10.1007/s00222-005-0488-2
[18] C. L. Siegel, A mean value theorem in geometry of numbers , Ann. of Math. (2) 46 (1945), 340-347. · Zbl 0063.07011 · doi:10.2307/1969027
[19] A. Södergren, On the Poisson distribution of lengths of lattice vectors in a random lattice , Math. Z. 269 (2011), 945-954. · Zbl 1257.60002 · doi:10.1007/s00209-010-0772-8
[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. · Zbl 1228.11051 · doi:10.1016/j.jnt.2010.12.003
[21] H. M. Stark, On the zeros of Epstein’s zeta function , Mathematika 14 (1967), 47-55. · Zbl 0242.12010 · doi:10.1112/S0025579300008007
[22] A. Terras, Real zeroes of Epstein’s zeta function for ternary positive definite quadratic forms , Illinois J. Math. 23 (1979), 1-14. · Zbl 0392.10024
[23] A. Terras, Integral formulas and integral tests for series of positive matrices , Pacific J. Math. 89 (1980), 471-490. · Zbl 0397.10020 · doi:10.2140/pjm.1980.89.471
[24] A. Terras, The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions , J. Number Theory 12 (1980), 258-272. · Zbl 0432.10010 · doi:10.1016/0022-314X(80)90062-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.