Evaluating shintani cone zeta functions mathoverflow. Multidimensional shintani zeta function is introduced and its definable probability distributions on rd are studied. The special case when k 1 is the barnes zeta function. 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. Shintanis prehomogeneous zeta functions and multiple sine. A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. The purpose of this paper is to determine the principal part of the adjusted zeta function. When the group is gl2, shintani determined the poles and the residues of the zeta functions for the spaces of binary forms of degree 2 and 3. On zeta elements and refined abelian stark conjectures livingstone boomla, alice, tokyo journal of mathematics, 2017. Their connection will be discussed at the end and thus the full importance of montgomerys conjecture is established. We show that shintanis work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. Multidimensional shintani zeta function is introduced and its definable probability.
High moments of the riemann zeta function conrey, j. These explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed. We develop the general theory of shintani data, and show that the order of vanishing part of grosss conjec. Multidimensional shintani zeta functions and zeta distributions on rd. On the shintani zeta function for the space of pairs of. Just like shintani zeta functions, witten zeta functions are defined by polynomials which are products of linear forms with nonnegative coefficients. This class contains some fundamental probability distributions such as.
Here the authors aim is to generalize shintanis approach from the viewpoint of geometric invariant theory, and in some special cases he also determines not only the pole structure but also the principal part of the zeta function. Shintani zeta functions and grossstark units for totally real fields, duke mathematical journal, 143. The brumerstark conjecture itself was already proven in the function. This is amongst the first books on the theory of prehomogeneous vector spaces, and represents the authors deep study of the subject. The basic example is the wellknown factorization of the riemann zeta function. The main objective of this paper is to discuss these zeta functions from an analytic point of view. High moments of the riemann zetafunction conrey, j.
We reprove many of shintani s formulas and prove several new ones. Definition of zeta functions and statement of the main results. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using. The subject begins with the pioneering work of shintani 27, who proved that these zeta functions enjoy an analytic continuation and a functional equation. Shintani showed that his zeta function admits a meromorphic scontinuation with the same pole locations as the function. Included are techniques of analytic continuation and relationships to special functions. Yukieon the shintani zeta function for the space of binary trihermitian forms. On shintani zeta functions for gl2 internet archive. Some generalizations of the riemann zeta function are outlined, as well as the calculation of zeta constants and the development of some identities. On the shintani zeta function for the space of binary trihermitian forms. In mathematics, a shintani zeta function or shintani lfunction is a generalization of the riemann zeta function. Multidimensional shintani zeta functions and zeta distributions on r d. We use shintanis domains and his formulas for the special values in order to prove. Here the authors aim is to generalize shintani s approach from the viewpoint of geometric invariant theory, and in some special cases he also determines not only the pole structure but also the principal part of the zeta function.
These explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. First, we obtain a functional equation satisfied by the shintani double zeta functions in addition to shintanis functional equations. Analytic properties of shintani zeta title functions. Shintanion zeta functions associated with vector spaces of quadratic forms. Analytic properties of shintani zeta functions university of south. Shintanis prehomogeneous zeta functions and multiple sine functions. On the shintani zeta function for the space of binary tri. A remark on zeta functions of algebraic number fields. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. Therefore, from an analytic perspective, the shintani zeta functions might be regarded as \black sheep in the family of zeta functions, which motivated us to further study their analytic properties. Cameron franc special values of riemanns zeta function. However, the formula 2 cannot be applied anymore if the real part. We introduce the notion of shintani data, which axiomatizes algebraic aspects of shintani zeta functions. Shintani 12 introduced zeta functions enumerating the class numbers hpmq.
Satoshintani showed that there exists a zeta function associated to g, v. We show that shintani s work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. Zeta functions in algebraic geometry mircea mustat. Title on shintanis zeta functions of ternary zero forms. Shintani zeta functions and grossstark units for totally real fields. Let f be a totally real number field and let p be a finite prime of f, such that p splits completely in the finite abelian extension h of f.
Among the latter is a generalization to the shintanibarnes gamma functions of raabes 1843 formula. In this note we show that the similar result holds for arbitrary not necessarily totally real algebraic number field. However, formatting rules can vary widely between applications and fields of interest or study. On the shintani zeta function for the space of binary quadratic forms. This is the first book on this topic, and represents the authors deep study of prehomogeneous vector spaces. Citeseerx document details isaac councill, lee giles, pradeep teregowda. We also relate barnes triple gamma function to the elliptic gamma.
Akihiko yukie the theory of prehomogeneous vector spaces is a relatively new subject although its origin can be traced back through the works of siegel to gauss. Shintani, on evaluation of zeta functions of totally real algebraic number fields at nonpositive integers, journal of the faculty of science, university of tokyo, vol. In this paper we consider an analogue of the zeta function for not necessarily prehomogeneous representations of gl2 and compute some of the poles. Shintanibarnes zeta and gamma functions request pdf. Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. The prime number theorem, hardys theorem on the zeros of. First, we give an explicit formula of the shintani double zeta functions.
This class contains some fundamental probability distributions such as binomial and poisson. Three applications of the explicit formula are given. Basic zeta functions and some applications in physics. Witten zeta functions are however not special cases of shintani zeta functions because in witten zeta. We show that the multidimensional zeta function due to shintani sh 1, which is a generalization of the multiple hurwitz.
They include hurwitz zeta functions, barnes zeta functions there is another type of zeta function attached to prehomogeneous vector spaces which is sometimes also called a shintani zeta function. In mathematics, a shintani zeta function or shintani lfunction is a generalization of the. Ander steele september 21, 2012 abstract the shintani cocycle on gl nq, as constructed by hill, gives a cohomological interpretation of special values of zeta functions for totally real elds of degree n. We introduce a zeta function attached to a representation of a group. Stark has proposed a conjecture stating the existence of a punit in h with absolute values at the places above p specified in terms of the values at zero of the partial zeta. A shintanitype formula for grossstark units over function fields, with alison miller, journal of mathematical sciences, the university of tokyo, 16 2009, no. Shintanibarnes zeta and gamma functions sciencedirect. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. His results, combined with the local theory of datskovskywright dawra, dawrb and datskovsky dats produced the zeta function theoretic proofs of davenpor.
These explicit formulas can be interpreted as vanishing period integral side. By a global zeta function we mean an object that takes into account all primes. In lemma 9 10, shintani proved that the one variable zeta function satisfies a. Let i be the correction of all fractional ideals of f. In this paper, we consider the zeta function defined for the space of pairs of binary hermitian forms. The zeta functions attached to prehomogeneous vector spaces. On the shintani zeta function for the space of pairs of binary hermitian forms. We reprove many of shintanis formulas and prove several new ones. The exposition is selfcontained, and required a preliminary knowledge of only the ele. This class contains some fundamental probability distributions such as binomial and poisson distributions. Next, as its applications, we derive new functional equatons, and prove an asymptotic formula of the weighted average of the central values of quadratic dirichlet lfunctions. We show that the multidimensional zeta function due to shintani sh 1, which is a generalization of the multiple hurwitz zeta function, can be obtained in this framework. Riemanns zeta function if s 1 is a real number, then the series s x n 1 1 ns converges.
Shintani zeta functions and grossstark units for totally. Both siegel and shintani considered equivalence classes of integral forms. Pdf multidimensional shintani zeta functions and zeta. Shintani zeta functions and grossstark units for totally real fields dasgupta, samit, duke mathematical journal, 2008. We show that the derivative at 0 of shintanis prehomogeneous zeta function for the space of symmetric matrices is expressed via special values of multiple sine functions. This thesis is an exposition of the riemann zeta function.
Shintani, on evaluation of zeta functions of totally. In this case one expects to have a product formula in terms of local factors. In one sense, our investigations were less successful than we hoped. Download freecourseweb lms 183 shintani zeta functions.
This is a joint work with henry kim and masao tsuzuki. Shintani zeta functions and grossstark units for totally real fields, duke mathematical journal, 143 2008, no. The riemann zeta function is defined as the analytic continuation of the function defined for. Shintani zetafunctions and grossstark units for totally. Equidistribution, shintani zeta function, maass form. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the casimir effect and boseeinstein condensation.
The next two sections develop the mathematical theory of and its zeros starting with basic results such as the intermediate value theorem from real analysis and the argument princi. Shintani zeta functions and a refinement of grosss leading term conjecture minoru hirose abstract. On zeta functions associated to symmetric matrices, i. Shintani zeta functions by akihiko yukie, 9780521448048, available at book depository with free delivery worldwide. In this paper, we give an explicit formula of the shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. For a totally real algebraic number field k, it is known that every partial zeta function of k is a finite sum of dirichlet series which are regarded as natural generalizations of the hurwits zeta function see 1 and 2. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function on c. The class of riemann zeta distribution is one of the classical classes of probability distributions on r. Riemann and his zeta function 5 the second form for this integral is called the mellin transform 70 of 1 ex. The integral 1 can also be viewed as a laplace transform.
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