Values Of Zeta Functions Research Articles

Higher Kronecker “limit” formulas for real quadratic fields

Abstract For every integer k ≧ 2 we introduce an analytic function of a positive real variable and give a universal formula expressing the values ζ(ℬ, k) of the zeta functions of narrow ideal classes in real quadratic fields in terms of this function and its derivatives up to order k − 1 evaluated at reduced real quadratic irrationalities associated to ℬ. We show that our functions satisfy functional equations and use these to deduce explicit formulas for the rational numbers ζ(ℬ, 1 − k). We also give an interpretation of our formula for ζ(ℬ, k) in terms of cohomology groups of SL(2, ℤ) with analytic coefficients and describe a “twisted” extension of the main formula that allows one to treat zeta values of zeta functions of ray classes rather than just ideal classes. Finally, we use our formulas to compute some zeta-values numerically and test that they are expressible as combinations of higher polylogarithm functions evaluated at algebraic arguments.

Cohomological Hasse principle and motivic cohomology for arithmetic schemes

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.

Eulerian polynomials and B-splines

An interrelationship between Eulerian polynomials, Eulerian fractions and Euler–Frobe nius polynomials, Euler–Frobenius fractions, and B-splines is presented. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.

An explicit growth condition for middle-degree cuspidal cohomology of arithmetically defined quaternionic hyperbolic n-manifolds

Let \({G/\mathbb Q}\) be the simple algebraic group Sp(n, 1) and \({\Gamma=\Gamma(N)}\) a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group \({G(\mathbb R)}\) . Then a double quotient \({\Gamma\backslash G(\mathbb R)/K}\) is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology \({H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)}\) in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkhauser, Boston), pp. 1–48, 1984).

Diophantische Approximationen

Diophantische Approximationen

Volume of truncated fundamental domains

For T > 0, by applying Arthur's analytic truncation AT to the constant function 1 on G(F)\G(A)1, we obtain a characteristic function of a compact subset i(T) of G(F)\G(A)1. The aim of this paper is to give a general formula for the volume of a(T), which is an exponential polynomial function of T. Consequently, by letting T -* oo, we obtain a new way to evaluate the volume of fundamental domain for G(F)\G(A)1. While a(T) seems to be quite arbitrary, its volume admits beautiful structures: (i) Surprisingly enough, the coefficients of T are special values of zeta functions. (ii) The volume is an alternating sum associated to standard parabolic subgroups. ((ii) is suggested by geometry after all, a(T) is obtained from G(F)\G(A)1 by truncating off cuspidal regions which are known to be parametrized by standard parabolic subgroups.) Besides all basic properties of Arthur's analytic truncation, our method is based on a fundamental formula of Jacquet-Lapid-Rogawski on an integration of truncated Eisenstein series associated to cusp forms. Their formula, an advanced version of the Rankin-Selberg method, is obtained using the so-called Bernstein principle via regularized integration over cones.

Zeta functions and normalized multiple sine functions

By using normalized multiple sine functions we show expressions for special values of zeta functions and L-functions containing ζ(3), ζ(5), etc. Our result reveals the importance of division values of normalized multiple sine functions. Properties of multiple Hurwitz zeta functions are crucial for the proof.

A $q$-Mahler measure

We construct a $q$-analogue of the Mahler measure using a $q$-analogue of the logarithm. We present some basic examples, where we obtain $q$-analogues of special values of zeta functions. We calculate also the classical limit and the crystal limit.

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

The purpose of this paper is to derive a generalization of Kohnen-Zagier's results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces, and to refine our previous results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. Employing kernel functions, we construct a correspondence $\varPsi$ from modular forms of half integral weight $k+1/2$ belonging to Kohnen's spaces to modular forms of weight $2k$. We explicitly determine the Fourier coefficients of $\varPsi(f)$ in terms of those of $f$. Moreover, under certain assumptions about $f$ concerning the multiplicity one theorem with respect to Hecke operators, we establish an explicit connection between the square of Fourier coefficients of $f$ and the critical value of the zeta function associated with the image $\varPsi(f)$ of $f$ twisted with quadratic characters, which gives a further refinement of our results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces.

Hasse invariants for Hilbert modular varieties

Given a totally real fieldL of degreeg, we constructg Hasse invariants on Hilbert modular varieties in characteristicp and characterize their divisors. We show that these divisors give the type stratification defined by the action of\(\mathcal{O}_L \) on theαp-elementary subgroup. Under certain conditions, involving special values of zeta functions, the product of these Hasse invariants is the reduction of an Eisenstein series of weightp−1

Values of zeta functions at negative integers, Dedekind sums and toric geometry

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.

On Bernoulli identities and applications

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.

"Special values of zeta functions of the simplest cubic fields and their applications"

In [2], Halbritter and Pohst computed the values of partial zeta functions of totally real cubic fields. In this paper, applying their results to the simplest cubic fields, we explicitly compute some special values of partial zetafunctions of these fields. And as applications, we give a necessary condition for class numbers of the simplest cubic fields to be 1 and construct the simplest cubic fields with class numbers divisible by a given rational integer n. First we restate the main theorem of [2]. (The meaning of notations such

Fourier coefficients of modular forms of half integral weight, periods of modular forms and the special values of zeta functions

Fourier coefficients of modular forms of half integral weight, periods of modular forms and the special values of zeta functions

Class Number 1 Criteria for Real Quadratic Fields of Richaud–Degert Type

In this paper we develop two ways of computing special values of zeta function attached to a real quadratic field. Comparing these values we obtain various class number 1 criteria for real quadratic fields of Richaud–Degert type

Euler-Poincare Characteristics for Motivic Cohomology and Arithmetic Groups

Euler-Poincare characteristics have a way of cropping up when one studies the values of zeta functions at integers. On the one hand, they arise in arithmetic versions of the Gauss-Bonnet theorem [On], [H], [S], [T2], and, on the other, in applications of etale cohomology and of Ktheory to varieties over finite fields [L1-4], [BN], [Sch], [M2-3]. Here we investigate some consequences of the Lichtenbaum-Milne theory and of Serre's theory of cohomology of S-arithmetic groups as they apply to elliptic surfaces over finite fields. Some of Serre's definitions are recalled briefly at the beginning of Section 1. This introduction is followed by a calculation of the EulerPoincare characteristic of a v-arithmetic of units in a maximal order of a algebra over a global field of positive characteristic. In Section 2 we show how to associate such a quaternion group to each irreducible fiber of an elliptic surface defined over a finite field and carrying a level two structure. The question then is how the Euler characteristics of these groups relate to the arithmetic of the surface. Their product diverges, but if each Euler characteristic is appropriately weighted, the product converges to the motivic Euler characteristic of the constant sheaf Z in the etale topology. (The statement of the theorem is not quite this clean as it is necessarily smudged by the presence of some global constants and by factors coming from the singular fibers. Two remarks following the theorem indicate how it might be polished.) The paper closes with an appendix that studies in detail the association of an algebraic family of algebras to the smooth fibers of an elliptic surface with level two structure. This construction may be of independent interest for the study of the class field theory of the surface.

A generating function for the Euler characteristic of out( F n)

Let A be a group satisfying suitable homological finiteness conditions, and A' C A any torsion-free subgroup of finite index. Then the rational Euler characteristic %(A) of A is defined as x(A')/[A : A ' ] 9 where #(/4') is the usual alternating sum of the ranks of the homology groups H^A'.J.) (cf. [10]). The rational Euler characteristic has good multiplicative properties which often make it easier to compute and more appropriate for groups with torsion than the usual Euler characteristic, In addition, for many groups there is a strong connection between X(A) and number theory. Harder [5] has shown that the rational Euler characteristics of arithmetic groups are closely related to values of zeta functions. In particular, the rational Euler characteristics of the Z-points of algebraic groups can often be given in terms of Bernoulli numbers. Harer and Zagier [7] have shown that the rational Euler characteristics of mapping class groups can likewise be given as simple formulas involving Bernoulli numbers. In this paper, we concentrate on the group Out(Fn) of outer automorphims of a free group on n generators. In [3] it is shown that Out(/^) has homological finiteness properties in common with arithmetic groups and mapping class groups; specifically, it is of type VFL (cf. [1]). In general, however, very little is known about the homology of Out(/?). In this paper, we study the rational Euler characteristic %(Out(/^)), and in particular, we give an effective method for computing it. In an interesting departure from the arithmetic case, we do not know whether there is a formula for #(Out(/^)) in terms of Bernoulli numbers, though the denominators share some divisibility properties with the denominators of Bernoulli numbers. The authors would like to thank Ken Brown for many helpful comments and suggestions.

Special values of zeta functions associated to cusp singularities

Special values of zeta functions associated to cusp singularities

Algebraic Relations Between Critical Values of Zeta Functions and Inner Products

Algebraic Relations Between Critical Values of Zeta Functions and Inner Products

The critical values of zeta functions associated to the symplectic group

The critical values of zeta functions associated to the symplectic group