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 the Cohn-Elkies sphere packing bound in dimension 1. We compute the bounds through optimization over Schwartz functions using semidefinite programming and also show how semidefinite programming can be used to optimize over functions with bounded support.
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 …
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…
We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.
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.
Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.
This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.
The Northcott property for special values of Dedekind zeta functions and more general motivic $L$-functions was defined by Pazuki and Pengo. We investigate this property for any complex evaluation of Dedekind zeta functions. The results are more delicate and subtle than what was proven for the function field case in previous work of Li and the authors, since they include some surprising behavior in the neighborhood of the trivial zeros. The techniques include a mixture of analytic and computer assisted arguments.
The Stieltjes constants $γ_n(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $ζ_K(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $γ_n(K)$ as Stieltjes obtained in 1885 for $γ_n(\mathbb Q)$. We also study the signs of $γ_n(K)$.
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.
Let F be a local non-archimedean field. We prove a formula relating orbital integrals in GL(n,F) (for the unit Hecke function) and the generating series counting ideals of a certain ring. Using this formula, we give an explicit estimate for such orbital integrals. We also derive an analogous formula for global fields, proving analytic properties of the Dedekind zeta function for orders in global fields.
The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s > 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer--Siegel theorem. As an application we obtain a limit formula for Euler--Kronecker constants in families of number fields.
We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for $x\geq 2$. Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at $s=1$.
We establish new explicit zero-free regions for the Dedekind zeta-function. Two key elements of our proof are a non-negative, even, trigonometric polynomial and explicit upper bounds for the explicit formula of the so-called differenced logarithmic derivative of the Dedekind zeta-function. The improvements we establish over the last result of this kind come from two sources. First, our computations use a polynomial which has been optimised by simulated annealing for a similar problem. Second, we establish sharper upper bounds for the aforementioned explicit formula.
In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bound, resp.
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.
We prove the Kloosterman-Spectral sum formula for PSL(2,Z[i])\PSL(2,C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. This sum formula allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields.
For a number field $K$, let $ζ_{K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $ζ_{K}$ as well as its derivative $ζ_{K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^γ$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].
Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $π^{n[k:\Q]}ζ^*_k(1-n)$, where $ζ^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculat…
Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value over any number field K is a period of a mixed Tate motive. Moreover, if K is a totally real number field, then we can choose a cone C so that every multiple Dedekind zeta associated to the pair (K;C) is unramified over the ring of algebraic integers in K. In [7], the author proves similar statements in the special case of a real quadratic fields for a part…
Let $ψ_\K$ be the Chebyshev function of a number field $\K$. Under GRH we prove an explicit upper bound for $|ψ_\K(x)-x|$ in terms of the degree and the discriminant of $\K$. The new bound improves significantly on previous known results.
In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ ζ_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for $ζ_{K, X} (s)$ and estimate the number of zeros of $ζ_{K, X} (s)$ up to height $T$.
Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log D_K/(\sqrt{X}\log X), where D_K is the absolute value of the discriminant of K. Guided by Weil's explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach's constant to 2.33. This results in substantial speeding of one part of Buchmann's class group algorithm.
In this paper, we compare the two definitions of Bloch group, and survey the elements in Bloch group. We confirm the Lichtenbaum conjecture on the field $Q(ζ_p)$ under the assumption the truth of the base of the Bloch group of $Q(ζ_p)$ and the relations of $K_2$ group. We also study the Lichtenbaum conjecture on non-Galois fields. By PARI, we get some equations of the zeta functions on special values and the structure of tame kernel of these fields.
Let $K$ be an algebraic number field. We construct an additive Markov process $X_t^{K_\mathbb A}$ on the ring of adeles $K_\mathbb A,$ whose coordinates $X_t^{(v)}$ are independent and use this process to give a probabilistic interpretation of the Dedekind zeta function $ζ_K(s),$ for $\re s>1.$ This note extends a recent work of Yasuda [J. Theor. Probab. 23(3):748--769, 2010] where the case of the field $K=\Q$ of rational numbers was considered.
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 …
Given a number field $K$ one associates to it the set $Λ_K$ of Dedekind zeta-functions of finite abelian extensions of $K$. In this short note we present a proof of the following Theorem: for any number field $K$ the set $Λ_K$ determines the isomorphism class of $K$. This means that if for any number field $K'$ the two sets $Λ_K$ and $Λ_{K'}$ coincide, then $K \simeq K'$. As a consequence of this fact we deduce an alternative approach towards the proof of Neukirch-Uchida Theorem for the case of non-normal extensions of number fields.
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…
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…
Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $ζ_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}π \log \Big( d_K \Big( \frac{T}{2πe}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros…
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…
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 …
Distributions of Integral Points and Dedekind Zeta Values
L.C. Academy for Multidisciplinary Studies, Beijing National Center for Applied Mathematics, Capital Normal University（首都师范大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京雁栖湖应用数学研究院）
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $ζ_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely b=9. We also study the rational invariant a/b for single-complex-place cusped manifolds, …
Let K be a number field, n_K its degree, and d_K the absolute value of its discriminant. We prove that, if d_K is sufficiently large, then the Dedekind zeta function associated to K has no zeros in the region: Re(s) > 1 - 1/(12.55 log d_K + 9.69 n_K log|Im s| + 3.03 n_K + 58.63) and |Im s| > 1. Moreover, it has at most one zero in the region: Re (s) > 1- 1/(12.74 log d_K) and |Im s| < 1. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: there is at most one zero in the region Re (s) > 1 - 1/(2 log d_K) and |Im s| < 1/(2 log d_K).
In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper's correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algeb…
Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of GRH thus provides a direct connection between a concrete algebraic geometry problem and the P vs.NP Problem, in a radically different direction from the geometric complexity theory approach to VP vs. VNP. We explore more plausible hypotheses yielding the same speed-up. One minimalist hypothesis we derive involves improving the error term (as a function of the…
We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. 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. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witte…
