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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 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 family of approximations of the Dedekind zeta function ζK(s) of a number field K/Q. Weighted L^2-norms of the difference of two such approximations of ζK(s) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of ζK(s) can be obtained from such weighted L^2-norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.

Let Kn be a number field of degree n over ℚ. By D(x,Kn) denote the number of all nonzero integral ideals in Kn with norm ≤ x. The Dedekind zeta function ζKn(s) is a meromorphic function with a simple pole at s = 1 and with residue, say, Λn. Define

Let Kn be a number field of degree n over Q. By $$ {A}_{K_n}(x) $$ denote the number of integral ideals with norm ≤ x. Landau’s classical estimate is $$ {A}_{K_n}(x)={\varLambda}_n x+ O\left({x}^{\left( n-1\right)/\left( n+1\right)}\right). $$ In this paper, the error term is improved for the non-normal field $$ {K}_4=\mathrm{Q}\left(\sqrt[4]{m}\right) $$ and for K6, the normal closure of a cubic field K3 with the Galois group S3. Bibliography: 25 titles.

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.

Let K be an algebraic number field of finite degree over the rational filed ℚ. Let ak be the number of integral ideals in K with norm k. In this paper we study the l-th integral power sum of ak, i.e., Σk⩽αakl(l = 2, 3, ...). We are able to improve the classical result of Chandrasekharan and Good. As an application we consider the number of solutions of polynomial congruences.

The author uses analytic methods to study the distribution of integral ideals and Hecke Grossencharacters in algebraic number fields. Nowak’s results on the distribution of integral ideals, and Chandrasekharan and Good’s results on the distribution of Hecke Grossencharacters are improved.

For the mean square value of the error term associated with the Dedekind zeta function of a cubic field K 3, an asymptotic formula (instead of the upper bounds of Chandrasekharan and Narasimhan (1964) and Lau (1999) ) is obtained. Modular analogs of the classical divisor problems are also studied. Bibliography: 21 titles.

For a cubic extension K3/ℚ, which is not normal, new results on the behavior of mean values of the Dedekind zeta function of the field K3 in the critical strip are obtained.

The Demǰanenko matrices are generalized with the averaged Bernoulli polynomials, and their determinants are computed by the Dedekind-Frobenius formula. 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. We utilize the structural properties of the group of reduced residue classes modm and those of (averaged) Bernoulli polynomials, thus appealing to the predominance of Galois theory over the fields themselves, which makes it possible to present very lucid reasoning of the whole picture.

Let {K m } m ≥ 4 be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial f m (x) = x 3 − mx 2 − (m + 1)x − 1, where m is an integer with m ≥ 4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to K m , and compute the values of the Dedekind zeta function of K m .

We recall the known explicit upper bounds for the residue at s = 1 of the Dedekind zeta function of a number field K. Then, we improve upon these previously known upper bounds by taking into account the behavior of the prime 2 in K. We finally give several examples showing how such improvements yield better bounds on the absolute values of the discriminants of CM-fields of a given relative class number. In particular, we will obtain a 4,000-fold improvement on our previous bound for the absolute values of the discriminants of the non-normal sextic CM-fields with relative class number one.

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 .

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

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 .

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).

Let\(K = \mathbb{Q}(\sqrt d )\) be any quadratic number field with discriminantd. ζK(s) denotes the Dedekind zeta-function. The purpose of this note is to prove the following asymptotic formula: $$\int\limits_0^T {|\zeta _K ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + it)|^2 dt = ({6 \mathord{\left/ {\vphantom {6 {\pi ^2 }}} \right. \kern-\nulldelimiterspace} {\pi ^2 }})} \prod\limits_{p/d} {(1 + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})^{ - 1} \cdot R_K^2 \cdot T \cdot \log ^2 T + O_\varepsilon \left\{ {\left| d \right|1 + \varepsilon \cdot T \cdot \log T} \right\},} $$ where the implied constant depends only on e. HereRK, denotes the residue of ζK(s) ats=1.