The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are evenly distributed modulo p for every p. This is the basis of a well-known heuristic, given by Siegel, for estimating the frequency of irregular primes. So far, analyses have shown that if Q(\sqrt{D}) is a real quadratic field, then the values of the zeta function ζ_{D}(1-2m)=ζ_{Q(\sqrt{D})}(1-2m) at negative odd integers are also distributed as expected modulo p for any p. However, it has proven to be very computationally intensive to calculate these numbers for large values of m. In this paper, we present the alternative of computing ζ_{D}(1-2m) for a fixed value of D and a large number of different m.
The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract incidence algebra of the division poset.
We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.
The aim of this paper is twofold: on one hand we study the invariants of traces of quadratic forms over a finite field of characteristic two. On the other hand, we give results about the zeta functions of certain curves studied by van der Geer and ven der Vlugt.
In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even a…
We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV's). In the limit q->1, it turns into Kawashima's relation for multiple zeta values. As a corollary we find that qMZV's satisfy the linear relation contained in Kawashima's relation. In the proof we make use of a q-analogue of Newton series and Bradley's duality formula for finite multiple harmonic q-series.
Let $K$ be a quadratic number field and $ζ_K(s)$ be the associated Dedekind zeta-function. We show that there are infinitely many normalized gaps between consecutive zeros of $ζ_K(s)$ on the critical line which are greater than $2.866$ times the average spacing.
We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the Lévy constant of a quadratic irrational appears naturally as the topological entropy of such maps.
Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In this paper, we develop the technique for obtaining more shuffle relations and produce many examples of shuffle products over an imaginary quadratic field. We also define the notion of self shuffle of a (multiple) Dedekind zeta value and use it at many instances. We define a refinement of the multiple Dedekind zeta values. Our key examples are self shuffles …
We compute the special values of partial zeta function at $s=0$ for family of real quadratic fields $K_n$ and ray class ideals $\fb_n$ such that $\fb_n^{-1} = [1,δ(n)]$ where the continued fraction expansion of $δ(n)$ is purely periodic and each terms are polynomial in $n$ of bounded degree $d$. With an additional assumptions, we prove that the special values of partial zeta function at $s=0$ behaves as quasi-polynomial. We apply this to obtain that the special values the Hecke's $L$-functions at $s=0$ for a family of for a Dirichlet character $χ$ behave as quasi-polynomial as well. We compute…
We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an a…
We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field applying an asymptotic version of Euler-Maclaurin formula to the lattice cone associated to the ideal considered. The Euler-Maclaurin formula involved is obtained by applying the Todd series of differential operators to an integral of a small perturbation of thecone. The additive property of Todd series w.r.t. the cone decomposition enables us to express the partial zeta values in terms of the continued fraction of the reduced element of the ideal. The expression obtained us…
New computations of the Riemann zeta function on the critical line
Heilbronn Institute for Mathematical Research, School of Mathematics, University of Bristol（布里斯托大学）；Department of Mathematics, The Ohio State University（俄亥俄州立大学）
A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of generating functions, and contains more data than $Z_f$. This $G_f$ resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitl…
We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…
We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.
We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip $ σ_1 < \Re s < σ_2 $, where $ 1/2 < σ_1 < σ_2 < 1 $. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.
Here we study problems related to the proportions of zeros, especially simple and distinct zeros on the critical line, of Dedekind zeta functions. We obtain new bounds on a counting function that measures the discrepancy of the zeta functions from having all zeros simple. In particular, for quadratic number fields, we deduce that more than 45% of the zeros are distinct. This extends work based on Montgomery's pair correlation approach for the Riemann zeta function. Our optimization problems can be interpreted as interpolants between the pair correlation bound for the Riemann zeta function and …
This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $ζ_{K,C}(2)ζ_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary q…
Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields
Institute of Basic Science, Korea University（韩国科学技术院基础科学研究所，韩国大学）；Department of Mathematics Education, Sungkyunkwan University（成均馆大学数学教育系）；American University of Beirut (AUB)（贝鲁特美国大学）
We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray cla…
The prime geodesic theorem for cycles in Bruhat-Tits buildings is applied to unit groups of division algebras to derive new asymptotic assertion on class numbers of orders in imaginary quadratic fields.
We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms for n=2. Our construction naturally yields a regularisation, which is necessary for the improvement of the properties of these zeta function, in particular for the analytic continuation if n>2. We further obtain upper and lower bounds on the mean value X^{-5/2}\sum_{E} \res_{s=1}ζ_E(s) as X->\infty, where E runs over totally real …
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are int…
We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the moments of the product over zeros we give a heuristic argument involving random matrix theory. The asymptotic for the first moment of the product over zeros is then proved for quadratic extensions. We are also able to reproduce our main conjecture in the quadratic case by using a modified version of the moments recipe. Finally, we generalise our methods to give a …
We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.
Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.
J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find some expressions for Dedekind zeta values attached to arbitrary real as well as quadratic number fields at any positive integer.
In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently large and which are separated apart by a given positive number $V$.
n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadrat…
In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for $ζ_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.
The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpre…
It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).
It is proved that Epstein's zeta-function $ζ_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + iγ$ with $ T \leq γ\leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).
We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as defined by Zwegers. Our formula is valid in dimension g=2 at s=1 or s=0. For a choice of parameters obeying a certain symmetry, an indefinite zeta function is a differenced ray class zeta function of a real quadratic field, and its special value at $s=0$ was conjectured by Stark to be a logarithm of an algebraic unit. Our formula also permits practical high…
