Integral formulas and integral tests for series of positive matrices

Terras, Audrey

doi:10.2140/pjm.1980.89.471

Cited by 10 publications

(5 citation statements)

References 23 publications

(24 reference statements)

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“…. , c m ) and recall the definition of the random vectors E n (·, cn), E R n (δ) (·, cn), T (c) and T (c, δ) from (22), (47), (17) and (18). We call their distributions μ E n (·,cn) , μ E Rn (δ) (·,cn) , μ T (c) and μ T (c,δ) respectively.…”

Section: An Alternative Proof Of Theoremmentioning

confidence: 99%

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On the value distribution and moments of the Epstein zeta function to the right of the critical strip

Södergren¹

2011

Journal of Number Theory

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Section: An Alternative Proof Of Theoremmentioning

confidence: 99%

“…Note however that for all n ≥ 2 there exist lattices L ∈ X n for which the Riemann hypothesis for E n (L, s) is known to fail (cf. [17,Thm. 1]; see also [1], [14], [16] and [18]).…”

Section: Introductionmentioning

confidence: 99%

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

Södergren¹

2011

Journal of Number Theory

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“…Cf. [8], [1], [25], [31], [32], [33], [26]. We denote by N L (T ) the number of nontrivial zeros (counting multiplicity) of E n (L, s) with |ℑs| ≤ T .…”

Section: Introductionmentioning

confidence: 99%

On the location of the zero-free half-plane of a random Epstein zeta function

Strömbergsson¹,

Södergren²

2017

Math. Ann.

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Abstract. In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function En(L, s) and prove that this random variable scaled by n −1 has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function. Furthermore, we obtain a limit distribution for the frequency of zeros of En(L, s) in vertical strips contained in the half-plane ℜs > n 2 .

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“…They arise whenever there are several gamma factors in the functional equation of the Dirichlet series (see Goldfeld and Viola [222], Lavrik [399], and Terras [662,663] …”

Section: Theorem 362 (The Maass Correspondence Between Maass Wavefomentioning

confidence: 99%

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

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Integral formulas and integral tests for series of positive matrices

Cited by 10 publications

References 23 publications

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

On the location of the zero-free half-plane of a random Epstein zeta function

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

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