Thehodgerealisationfunctoronmtkgivesrisetoaregulatormap 1. The residue at s 1 of the riemann zeta function is 1. Dirichlet also showed that the l series can be written in a finite form, which gives a finite form for the class number. The resulting function s is called riemanns zeta function. Let kbe a number eld, n k its degree, and d k the absolute value of its discriminant. Special values of dedekind zeta functions and motivic cohomology for a totally real number eld f and an even integer n 2 the value f1 n of the dedekind zeta function fs at s 1 nis a nonzero rational number. Cameron franc special values of riemanns zeta function. Damerell 197071 studied special values of lfunctions in a special class including those we discussed above via the weierstra. In this paper, we establish the formulas expressing the special values of the spectral zeta function. A special case of this function is the hurwitz or generalized zeta function. Determinantal formula for the special values of the dedekind zeta function of the cyclotomic field fumio hazama department of natural sciences, college of science and engineering, tokyo denki university, hatoyama, hikigun, saitama 35003, japan communicated by k. Abstracta family of integer matrices, which generalize the demjanenko matrix and the matrix defined by silverman, is introduced and shown to compute the values at negative integers of the dedekind zeta function of thepth cyclotomic field for any primep.
We prove zagiers conjecture on the special value of the dedekind zeta function at s4, which tells that zeta f,4 can be expressed via classical tetralogarithms. This enables us to interpret geometrically the special values at negative integers of dedekind zeta function of maximal real subfields as well as of imaginary subfields of cyclotomic fields. Hyperbolic manifolds and special values of dedekind zeta functions 293. We discuss some aspects of indivisibility of the special values of dedekind zeta functions at negative odd integers associated to real quadratic fields. Sczech introduced a new type of dedekind sums and showed how they can be used to calculate derivatives of partial zeta functions. Determinantal formula for the special values of the dedekind zeta function of the cyclotomic field. Damerell 197071 studied special values of l functions in a special class including those we discussed above via the weierstra. Classical and elliptic polylogarithms and special values of l. In this paper, we are interested in the integral power sums of the coefficients of the dedekind zeta function of a nonnormal cubic extension k 3. We conclude with some numerical examples for elds of. Indivisibility of special values of dedekind zeta functions of real quadratic elds by dongho byeon seoul 1. Milne niranjan ramachandran november, 20 abstract beginning with the conjecture of artin and tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function.
While the dirichlet series expansion for the eta function is convergent only for any complex number s with real part 0, it is abel summable for any complex number. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial step in the proof of the prime number theorem. The bsd conjecture, regulators and special values of lfunctions frank gounelas contents 1. F s of an arbitrary algebraic number field f as the product of a simple factor involving the class number of the field with the determinant of a matrix whose entries are logarithms of units in the field.
Both dirichlet eta function and riemann zeta function are special cases of polylogarithm. Moreover, the meromorphic continuation and the same residue formula at s 0is valid for more general zeta functions, with r. Motivic complexes and special values of zeta functions. Values of dedekind sums for function fields hamahata, yoshinori, functiones et approximatio commentarii mathematici, 2015. The algebraic nature of special values of l functions is shrouded in mystery. Abstract we discuss some aspects of indivisibility of the special values of dedekind zeta functions at negative odd integers associated to real quadratic.
Was studied in depth by euler and others before riemann. 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. The dedekind zeta function and the dirichlet regulator 1. Dedekind zeta motives for totally real fields ihes. Maxplancklnstitut fiir mathematik, gottfriedclarenstral3e 26, d5300 bonn 3. A generalization of the secant zeta function as a lambert. On rapidly convergent series for dirichlet l function values via the modular relation springerlink. In this paper, we generalized secant zeta function as a lambert series and proved a result for the. Special values of various zeta and l functions in number theory and related areas at integral or almost integral arguments have been the major subject of research over the years. The first sums are written in terms of jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the barnes multiple zeta functions.
In 19, he obtained an expression for the values of dedekind zeta functions attached to real quadratic number. In the special case of a totally real number eldk, lichtenbaum conjectured that the values of the zeta function at negative integers are rational numbers which involve the rank of the algebraic or etale ktheory of k. We study relations among special values of zeta functions, invariants of toric varieties, and generalized dedekind sums. For k k a number field then all special values of the dedekind zeta function. Shouwu zhang, triple lseries and effective mordell conjecture some history is in. W e also show that the residue of the dedekind zeta function. The main tool is the eisenstein cocycle, a group cocycle for gl nz. Cluster polylogarithms and zagiers conjecture on zetaf,4 download report pdf we prove zagiers conjecture on the special value of the dedekind zeta function at s4, which tells that zetaf,4 can be expressed via classical tetralogarithms. The bsd conjecture, regulators and special values of l functions frank gounelas contents 1. The residue formula above is a special case of a result of bushnell and reiner 3. For ml, the function defined by 3 agrees with the function ax in theorem 1. Assume that the weaker schanuels conjecture is true. Motivic complexes and special values of zeta functions james s.
Mean values connected with the dedekind zetafunction of a. Explicit zerofree regions for dedekind zeta functions habiba kadiri abstract. The main directions of research conducted on the zeta function include. Values of the riemann zeta function at integers roman j. This formula is a special case of the analytic class number formula, and in. Hyperbolic manifolds and special values of dedekind zeta functions 287 the smallest natural number such that kce2in.
They also gave modular transformation properties of the secant zeta function. On rapidly convergent series for dirichlet lfunction. Dedekind psi function in number theory, the dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes p dividing n by convention. Special values of multiple gamma functions 115 the existence and uniqueness of. The algebraic nature of special values of lfunctions is shrouded in mystery. Determinantal formula for the special values of the dedekind. Riemann zeta function an overview sciencedirect topics. Analogously to the riemann zeta function, the values of the dedekind zeta function at integers encode at least conjecturally important arithmetic data of the field k. Determinantal formula for the special values of the.
The chapter also provides an overview of the polylogarithm functions, which include the legendres chi function, clausens integral or clausens function, and the hurwitzlerch zeta function. Indivisibility of special values of zeta functions associated to real quadratic. Robert sczech rutgers university newark spoke on dedekind sums and derivatives of partial zeta functions in real quadratic. The l functions arise from various contexts like algebraic number theory riemann zeta function, dirichlet l functions, dedekind zeta functions, lseries associated with hecke grossencharacters, representation theory, and theory of automorphic forms. Siegel 1937 proved this conjecture, and klingen 19612 gave a simpler proof using hilbertblumenthal modular forms.
We discuss the relation between the numerator and denominator of this number and orders of motivic cohomology groups. The classical dedekind sums arise in formulas for special values of partial zeta values in real quadratic. Dedekind zeta functions the roman number theory association. We prove that, if d k is su ciently large, then the dedekind zeta function ks has no zeros in the re. Dedekind sums involving jacobi modular forms and special. Pdf determinantal formula for the special values of the. Transcendental nature of special values of lfunctions. Determinantal formula for the special values of the dedekind zeta function of the cyclotomic field article pdf available in journal of number theory 752. In section 6 we return to critical values to give one or two examples of special values of zeta functions at the symmetry point of their functional equation. Explicit formular for special value of dedekind zeta functions. Zeros of riemann zeta function university of chicago. Ks, is a generalization of the riemann zeta function which is obtained by specializing to the case where k is the rational numbers q.
In mathematics, the dedekind zeta function of an algebraic number field k, generally denoted. Then more generally, we consider the special values of artin lfunction motivated by the work of stark, we have the following result. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. We discuss computation of the special values of partial zeta functions associated to totally real number elds. Indivisibility of special values of zeta functions. Pdf transcendental nature of special values of lfunctions. Hence we obtain a polynomialtime algorithm for computing these special values. The lfunctions arise from various contexts like algebraic number theory riemann zeta function, dirichlet lfunctions, dedekind zeta functions, lseries associated with hecke grossencharacters, representation theory, and theory of automorphic forms. In such cases, we also study the algebraic nature of these l values. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in.
In mathematics, the study of special values of l functions is a subfield of number theory devoted to generalising formulae such as the leibniz formula for pi, namely. The dirichlet class number formula expresses the residue at s 1 of the dedekind zeta function. Sczech introduced a new type of dedekind sums and showed how they can be used to calculate derivatives of partial zeta functions at. Its covolume should be related to the values of the dedekind zeta function. The bsd conjecture, regulators and special values of contents. Indivisibility of special values of zeta functions associated to real. Class number formula project gutenberg selfpublishing. We also give a substantial evidence for the freeness conjecture in the weight 4, which describes periods of mixed tate motives. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial.
Eulers identity and other similar identities like can be viewed as the special value at, of the riemann zeta function this sum converges when and the basic observation due to riemann is that can be extended as a meromorphic function on on via analytic continuation. However, the formula 2 cannot be applied anymore if the real part. Hyperbolic manifolds and special values of dedekind zetafunctions. In mathematics, the dedekind eta function, named after richard dedekind, is a modular form of weight 12 and is a function defined on the upper halfplane of complex numbers, where the imaginary part is positive. After landaus famous work, many authors contributed to some mean values connected with the dedekind zetafunction.
Introduction much of what is written in this note follows the line of argument in the excellent survey paper by. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. The signs of the stieltjes constants associated with the dedekind zeta function saad eddin, sumaia, proceedings of the japan academy, series a, mathematical sciences, 2018. It is holomorphic everywhere except a simple pole at with residue. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. The theme of our present work is to investigate the nonvanishing of a varying class of l functions as well as their derivatives, mainly at the point s 1. Pdf evaluation of dedekind sums, eisenstein cocycles. Dwilewicz, j an min a c 1 introduction the riemann zeta function is one of the most leonhard euler important and fascinating functions in mathematics. Evaluation of dedekind sums, eisenstein cocycles, and special values of l functions article pdf available in duke mathematical journal 1182 october 1999 with 59 reads how we measure reads. In this special case, we can ignore the distinction between the. Hyperbolic manifolds and special values of dedekind zetafunctions 293. It is very natural as it deals with the series of powers. Many kinds of zeta dedekind zeta of an algebraic number field f such as qv2. The main tool is the eisenstein cocycle, a group cocycle for gl.
First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the eisenstein cocycle introduced by the second author. K s, is a generalization of the riemann zeta function which is obtained in the case where k is the rational numbers q. For a number eld k and a prime number p, we denote by hk the class number of kand by pk, pk the iwasawa, invariants of the cyclotomic zpextension of. Indivisibility of special values of zeta functions associated. Zeta function a zeta function is a function which is composed of an. Hyperbolic manifolds and special values of dedekind zeta. For example, the analytic class number formula relates the residue at s 1 to the class number h k of k, the regulator r k of k. Beilinsons conjectures on special values of l functions 10 1. Special values of zeta functions 3 the regulator is far from being controlled in relation to the discriminant, and there has been no essential progress to the proof of the conjecture. The analog for function fields is the weil zeta function, while the generalization to higher dimensional arithmetic geometry are the arithmetic zeta functions. Special values of dedekind zetafunctions and motivic cohomology. An analogue of wiltons formula and values of dedekind zeta functions.