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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.

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

We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

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.

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

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.

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.

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.

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 .

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 .

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.

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.

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 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.

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

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.

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

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually proveThis may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

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).