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In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï’s formula concerning the error term in the Dirichlet divisor problem.

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

Groups of order 4 are isomorphic to either $${\mathbb {Z}}/4{\mathbb {Z}}$$ or $${\mathbb {Z}}/2{\mathbb {Z}} \times {\mathbb {Z}}/2{\mathbb {Z}}$$ . We give certain sufficient conditions permitting to specify the structure of class groups of order 4 in the family of real quadratic fields $${\mathbb {Q}}{(\sqrt{n^2+1})}$$ as n varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point $$-1$$ . As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of n.

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitiveL-functions, which is supported by some calculations based on Selberg’s conjectures.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet series. In this setting, we obtain new Hal\'asz-type results for the logarithmic mean value of $f$. More precisely, we prove estimates for the sum $\sum_{n=1}^x f(n)/n$ in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums $F_N(s)$. In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field $K$. More precisely, we give some improved results for the number of zeros up to height $T$ as well as new zero density results for the number of zeros up to height $T$, lying to the right of $\Re(s) =\sigma$, where $\sigma > 1/2$., Comment: 27 pages

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions., 50 pages, this is the accepted version for publication in Acta Math., minor changes and corrections based on referees' reports

Given two distinct number fields $K$ and $M$, and finite order Hecke characters $\chi$ of $K$ and $\eta$ of $M$ respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions coincide: $$L(s, \chi, K) = L(s, \eta, M) .$$ When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassman in 1926, who found such fields of degree 180, and by Perlis (1977) and others, who showed that there are no nonisomorphic fields of degree less than $7$. We construct infinitely many such pairs where the fields are quadratic. This gives dihedral automorphic forms induced from characters of different quadratic fields. We also give a classification of such characters of order 2 for the quadratic fields of our examples, all with odd class number., Comment: Added references to work of David Rohrlich. Accepted for publication

This paper is a corrigendum to the article ``On the ideal theorem for number fields''. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

By KMS-classification theorem, the Dedekind zeta function is an invariant of Bost–Connes systems. In this paper, we show that it is in fact an invariant of Bost–Connes C⁎-algebras. We examine second maximal primitive ideals of Bost–Connes C⁎-algebras, and apply K-theory to some quotients.

The Stieltjes constants $\gamma_{n}(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $\zeta_{K}(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $\gamma_{n}(K)$ as Stieltjes obtained in 1885 for $\gamma_{n}(\mathbf{Q})$. We also study the signs of $\gamma_{n}(K)$.

International audience; Using Stechkin's lemma we derive explicit regions of the half complex plane R (s) = 1 - c = logd(K) and vertical bar gamma vertical bar

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 interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV., Comment: This version has substantial improvements in the content and the style. There are more details about the analytic continuation together with new examples of multiple residues. 43 pages

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$.

Let k be a totally real number field. For every odd n≥3, we construct an element in the category MT(k) of mixed Tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group. By a volume calculation, we prove that its period is a rational multiple of $\pi^{n[k:\mathbb{Q}]}\zeta^{*}_{k}(1-n)$ , where $\zeta^{*}_{k}(1-n)$ denotes the special value of the Dedekind zeta function of k. We deduce that the group $\mathrm {Ext}^{1}_{\mathrm {MT}(k)} (\mathbb{Q}(0),\mathbb{Q}(n))$ is generated by the cohomology of a quadric relative to hyperplanes, and that $\zeta^{*}_{k}(1-n)$ is a determinant of volumes of geodesic hyperbolic simplices defined over k.

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over $$\mathbf{Q }$$ with image $$S_5$$ satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to $$(12)(34)$$ . In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and 6, our result follows from known 2-dimensional cases of Artin’s conjecture (proved by Sasaki) as well as the functorial properties of the Asai transfer proved by Ramakrishnan. For the irreducible representations of dimension 5, we encounter the problem of descending an automorphic form from a quadratic extension compatibly with the Galois representation. This problem is partly solved by working instead with a four dimensional representation of some central extension of $$S_5$$ . Our modularity results in this case are contingent on the non-vanishing of a certain Dedekind zeta function on the real line in the critical strip. A result of Booker show that one can (in principle) explicitly verify this non-vanishing, and with Booker’s help we give an example, verifying Artin’s conjecture for representations coming from the (Galois closure) of the quintic field $$K$$ of smallest discriminant (1609).

We extend the universality theorem for Hecke L-functions attached to ray class characters from the previously known strip \({ \max \{\frac{1}{2}, 1-\frac{1}{d}\} < {\rm Re}\,s < 1}\) for \({d=\left[K:\mathbb{Q}\right]}\) to the maximal strip \({\frac{1}{2} < {\rm Re}\,s < 1}\) under an assumption of a weak version of the density hypothesis. As a corollary, we give a new proof of the universality theorem for the Dedekind zeta function ζK(s) in the case of \({K/\mathbb{Q}}\) finite abelian.

If K is a number field with nk = [k : ℚ], and ξk the symmetrized Dedekind zeta function of the field, the inequalityfor t ≠ 0 is shown to be true for σ ≥ 1 + improving the result of Lagarias where the constant in the inequality was 9. In the case k = ℚ the inequality is extended to σ ≥ 1 for all t sufficiently large or small and to the region σ ≥ 1 + 1/(log t – 5) for all t ≠ 0. This answers positively a question posed by Lagarias.

We prove functional independence and joint functional independence for a set of Hurwitz zeta functions with periodic coefficients and parameters algebraically independent over the field of rational numbers.

Let K be an algebraic number field. We discuss the problem of counting the number of integral ideals below a given norm and obtain effective error estimates. The approach is elementary and follows a classical line of argument of Dedekind and Weber. The novelty here is that explicit error estimates can be obtained by fine tuning this classical argument without too much difficulty. The error estimate is sufficiently strong to give the analytic continuation of the Dedekind zeta function to the left of the line R ( s ) = 1 as well as explicit bounds for the residue of the zeta function at s = 1 .

Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function., 23 pages; v3 corrects typos and improves Theorem 1.3 & Corollary 1.4; accepted at Funct. Approx. Comment. Math. (2017)

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erd\"os., Comment: 13 pages, 4 figures

International audience; We provide, in a wide range of the parameters, an estimate for the mean-value over sifted integers of certain arithmetic functions with Dirichlet series analytically close to $1/\zeta_{\mathbb{K}}(s)$, where $\mathbb{K}$ is a number field and $\zeta_{\mathbb{K}}$ its Dedekind zeta function.

We study oscillations of the remainder term corresponding to the counting functions of the sets of elements with unique factorization length in semigroups of algebraic numbers such as the semigroup of algebraic integers or totally positive algebraic integers in a given normal field K. The results imply existence of oscillations when the exponent of the class group of the semigroup in question is sufficiently large depending on the field's degree. In particular, when K is a quadratic field or a normal cubic field oscillations exist whenever the class group is not isomorphic to C 2 a ⊕ C 3 b ⊕ C 4 c for nonnegative integers a,b,c. The main part of this study is concerned with the problem of multiplicative independence of Hecke zeta functions. We also show that there are infinitely many fields whose Dedekind zeta function has infinitely many nontrivial multiple zeros.

Let S S be a given set of positive rational primes. Assume that the value of the Dedekind zeta function ζ K \zeta _K of a number field K K is less than or equal to zero at some real point β \beta in the range 1 2 > β > 1 {1\over 2} >\beta >1 . We give explicit lower bounds on the residue at s = 1 s=1 of this Dedekind zeta function which depend on β \beta , the absolute value d K d_K of the discriminant of K K and the behavior in K K of the rational primes p ∈ S p\in S . Now, let k k be a real abelian number field and let β \beta be any real zero of the zeta function of k k . We give an upper bound on the residue at s = 1 s=1 of ζ k \zeta _k which depends on β \beta , d k d_k and the behavior in k k of the rational primes p ∈ S p\in S . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields K K which depend on the behavior in K K of the rational primes p ∈ S p\in S . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers., Comment: Corrected equations 4.4 and 4.5, other small changes, added a question about avoiding longer progressions

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

Let K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m2 + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function ζK(s) at s = −1 and get class number 1 criterion for the simplest cubic fields.

We give upper bounds on the modulus of the values at s = 1 of Artin L-functions of abelian extensions unramified at all the infinite places.We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for CM-fields. For example, we will reduce the determination of all the non-abelian normal CM-fields of degree 24 with Galois group SL2(F3) (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such CM-fields.

We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s=2-p. We use this generalization to give a statement on the non-vanishing modulo p of this relative zeta function at the point s=1 under a mild condition., Comment: 6 pages

We obtain a determinant expression of the zeta function of a generalized permutation over a finite set. As a corollary we prove the functional equation for the zeta function. In view of absolute mathematics, this is an extension from $GL(n,\mathbf{F}_{1})$ to $GL(n,\mathbf{F}_{1^{m}})$, where $\mathbf{F}_{1}$ and $\mathbf{F}_{1^{m}}$ denote the imaginary objects “the field of one element” and “its extension of degree $m$”, respectively. As application we obtain a certain product formula for the zeta function, which is analogous to the factorization of the Dedekind zeta function into a product of Dirichlet $L$-functions for an abelian extention.

Let K {\mathbf {K}} be a CM-field that is a quadratic extension of a totally real number field k {\mathbf {k}} . Under a technical assumption, we show that the relative class number of K {\mathbf {K}} is large compared with the absolute value of the discriminant of K {\mathbf {K}} , provided that the Dedekind zeta function of k {\mathbf {k}} has a real zero s s such that 0 > s > 1 0 > s > 1 . This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

International audience; Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée

We construct a quantum statistical mechanical system which generalizes the Bost-Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explict class field theory for such fields. This system admits the Dedekind zeta function as partition function and the Idele class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. We also give an interpretation of the original BC system and of the GL(2) system in terms of Shimura varieties, which motivates the construction for imaginary quadratic fields. The geometric notion underlying the construction is that of commensurability of K-lattices., Comment: 29 pages LaTeX

The purpose of this paper is to study an analogue of Euler's constant for the Selberg zeta functions of a compact Riemann surface and the Dedekind zeta function of an algebraic number field. Especially, we establish similar expressions of such Euler's constants as de la Vall\'ee-Poussin obtained in 1896 for the Riemann zeta function. We also discuss, so to speak, higher Euler's constants and establish certain formulas concerning the power sums of essential zeroes of these zeta functions similar to Riemann's explicit formula.

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., Comment: 64 pages; Plain TeX

The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$ of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$, and on a new version of the sum formula of Kuznetsov type for $\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$. Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$ gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting on the three dimensional hyperbolic space (the $K$-trivial situation). To solve this problem, it was necessary to include the $K$-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.

The zeta function ζ ( A , s ) \zeta (A,s) associated with a narrow ideal class A A for a real quadratic field can be decomposed into ∑ Q Z Q ( s ) \sum \nolimits _Q {{Z_Q}(s)} , where Z Q ( s ) {Z_Q}(s) is a Dirichlet series associated with a quadratic form Q ( x , y ) = a x 2 + b x y + c y 2 Q(x,y) = a{x^2} + bxy + c{y^2} , and the summation is over finite reduced quadratic forms associated to the narrow ideal class A A . The values of Z Q ( s ) {Z_Q}(s) at nonpositive integers were obtained by Zagier [16] and Shintani [12] via different methods. In this paper, we shall obtain the values of Z Q ( s ) {Z_Q}(s) at negative half-integers s = − 1 / 2 , − 3 / 2 , … , − m + 1 / 2 , … s = - 1/2, - 3/2, \ldots , - m + 1/2, \ldots . The values of Z Q ( s ) {Z_Q}(s) at nonpositive integers were also obtained by our method, and our results are consistent with those given in [16].

as x --, oc. In [7] two versions of the Chebotarev density theorem were proved, one unconditional and the other on the assumption of the Generalized Riemann Hypothesis (GRH), each of which expressed ~Zc(X ) as the sum of the main term ICI IC]_lGt Li(x) and an error term which is an effectively computable function of x, i-~-' and the associated field constants n K = [ K : Q ], h i = [L :Q] and dK,dz, (the absolute values of the discriminants of the two fields). Assuming the truth of the GRH for ~L(s), that paper also proved the existence of an effectively computable constant b (independent of K and L) such that for any conjugacy class C, there exists a prime

Denote byV(V(Δ K ,X) the number of sign-changes in [0,X] (X>0) of the remainder-termΔ K (X):=ψ K (x)−x of the prime-ideal theorem $$\mathop {\lim }\limits_{x \to \infty } \psi _K (x)/x = 1$$ , where $$\psi _K (x): = \sum\nolimits_{N\mathfrak{p}^r \leqslant x} {log N} \mathfrak{p}$$ stands for the generalized Chebyshev function of an algebraic number fieldK. Under certain conditions some effective estimates of the kind $$V(\Delta _K ,X) \geqslant c_K \log X (X \geqslant X_K )$$ are obtained, wherec K >0 andX K ≥2 depend in an explicit way on the parameters of the fieldK and the zeros of the Dedekind zeta function ζ K .

The hyperbolic three-manifold which results from (5,1) Dehn surgery on the complement of a figure-eight knot in S3 is arithmetic. I. Introduction. Let M be the complete orientable hyperbolic three-manifoldwhich results from (5,1) Dehn surgery on the complement of the figure-eight knotK in S3. In this note we will prove THEOREM 1. M is arithmetic. A precise description of M as an arithmetic manifold is given in the summaryat the end of this paper. One consequence of Theorem 1 and the results of Borel in [1] is thatVolume(M) = 12 • 2833/2cfc(2)(27r)-6,where Cfc(s) denotes the Dedekind zeta function of the unique quartic field fc of discriminant —283. Another consequence of Theorem 1 and Borel's work is thatthere exist infinitely many minimal elements in the set of manifolds commensurable to M. By work of J0rgenson and Thurston (see [7, §6.6]), the set of volumes of completeorientable hyperbolic three-manifolds is a well-ordered subset of R of order typeww. In particular, there is a minimal element v\ in this set. In [4] Meyerhoffconjectured that M has volume v%, but this is shown to be not true by Weeks [9].Weeks proved that the manifold M' obtained by (5,1), (5,2) Dehn surgery on the

LetR=Rk andw=wk be the regulator and the number of roots of unity in the number fieldk. We determine allk for whichR/w

In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as "test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin's conjecture is true. Introduction. We shall first review the definition of the zeta distribution associated with a number field K: let AK resp. A' denote the adele resp. idele groups of K, d Xx a Haar measure on A', and S (AK) the Schwartz-Bruhat space of the locally compact additive group AK; the topological dual S (AK)' of S (AK) is then the space of tempered distributions on AK. Let {xl denote the modulus of an idele x; then

LetK be a cubic number field. Denote byA K (x) the number of ideals with ideal norm ≤x, and byQ K (x) the corresponding number of squarefree ideals. The following asymptotics are proved. For every e>0 e>0 $$\begin{gathered} {\text{ }}A_K (x) = c_1 x + O(x^{43/96 + \in } ), \hfill \\ Q_K (x) = c_2 x + O(x^{1/2} \exp {\text{ }}\{ - c(\log {\text{ }}x)^{3/5} (\log \log {\text{ }}x)^{ - 1/5} \} ). \hfill \\ \end{gathered}$$ Herec 1,c 2 andc are positive constants. Assuming the Riemann hypotheses for the Dedekind zeta function ζ K , the error term in the second result can be improved toO(x 53/116+e).