33 results on '"Dedekind zeta function"'
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2. An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups
- Author
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Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, and Melanie Matchett Wood
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,Arbitrarily large ,Number theory ,Discriminant ,Field extension ,0103 physical sciences ,FOS: Mathematics ,symbols ,Torsion (algebra) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
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
- Published
- 2019
3. Zeros of partial sums of L-functions
- Author
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Arindam Roy and Akshaa Vatwani
- Subjects
Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Zero (complex analysis) ,Algebraic number field ,01 natural sciences ,Combinatorics ,symbols.namesake ,Distribution (mathematics) ,Number theory ,Logarithmic mean ,0103 physical sciences ,FOS: Mathematics ,symbols ,11M41 ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Dirichlet series ,Dedekind zeta function ,Mathematics - Abstract
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
- Published
- 2019
4. ON SIMPLE ZEROS OF THE DEDEKIND ZETA‐FUNCTION OF A QUADRATIC NUMBER FIELD
- Author
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Lilu Zhao and Xiaosheng Wu
- Subjects
Pure mathematics ,Mathematics::General Mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Algebraic number field ,01 natural sciences ,Riemann zeta function ,symbols.namesake ,Quadratic equation ,010201 computation theory & mathematics ,Simple (abstract algebra) ,symbols ,Dedekind cut ,Rectangle ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
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].
- Published
- 2019
5. Pair arithmetical equivalence for quadratic fields
- Author
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Zeév Rudnick and Wen Ching Winnie Li
- Subjects
Degree (graph theory) ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Automorphic form ,Order (ring theory) ,Algebraic number field ,16. Peace & justice ,01 natural sciences ,11R42 (Primary) 11F80, 11F11 (Secondary) ,Combinatorics ,Quadratic equation ,Number theory ,0103 physical sciences ,FOS: Mathematics ,Arithmetic function ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Dedekind zeta function ,Mathematics - Representation Theory ,Mathematics - Abstract
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
- Published
- 2020
6. Corrigendum to the paper 'On the ideal theorem for number fields' [Funct. Approximatio, Comment. Math. 53, No. 1, 31--45 (2015)]
- Author
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Olivier Bordellès
- Subjects
11N37, 11R42 ,Pure mathematics ,Ideal (set theory) ,Mathematics - Number Theory ,General Mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic number field ,Dedekind zeta function ,Mathematics ,Term (time) - Abstract
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.
- Published
- 2020
7. Smooth $$L^2$$ L 2 distances and zeros of approximations of Dedekind zeta functions
- Author
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Maria Monica Nastasescu, Arindam Roy, Junxian Li, and Alexandru Zaharescu
- Subjects
Mathematics::General Mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dedekind sum ,010103 numerical & computational mathematics ,Algebraic geometry ,Algebraic number field ,01 natural sciences ,symbols.namesake ,Riemann hypothesis ,Arithmetic zeta function ,Number theory ,symbols ,Dedekind cut ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2017
8. On the Mean Square of the Error Term For Dedekind Zeta Functions
- Author
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O. M. Fomenko
- Subjects
Statistics and Probability ,Mean square ,Pure mathematics ,Residue (complex analysis) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,010305 fluids & plasmas ,Arithmetic zeta function ,Norm (mathematics) ,0103 physical sciences ,Dedekind cut ,0101 mathematics ,Dedekind zeta function ,Meromorphic function ,Mathematics - Abstract
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
- Published
- 2016
9. The signs of the Stieltjes constants associated with the Dedekind zeta function
- Author
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Sumaia Saad Eddin
- Subjects
Dedekind zeta function ,Physics ,Mathematics - Number Theory ,General Mathematics ,Laurent series ,010102 general mathematics ,Stieltjes constants ,Riemann–Stieltjes integral ,010103 numerical & computational mathematics ,Algebraic number field ,01 natural sciences ,Riemann zeta function ,symbols.namesake ,11M06 ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,11R42 ,Mathematical physics - Abstract
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 $\gamma_{n}(K)$.
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- 2017
10. Simple zeros of Dedekind zeta functions
- Author
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Stéphane R. Louboutin, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)
- Subjects
Dedekind zeta function ,General Mathematics ,Mathematics::Number Theory ,Dedekind sum ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Arithmetic zeta function ,symbols.namesake ,Dedekind cut ,0101 mathematics ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,11R29 ,Algebraic number field ,Siegel zero ,11R42 (11R29) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,010201 computation theory & mathematics ,symbols ,Complex plane ,11R42 - Abstract
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
- Published
- 2017
11. On the Dedekind Zeta Function
- Author
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O. M. Fomenko
- Subjects
Statistics and Probability ,Combinatorics ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Galois group ,Field (mathematics) ,Cubic field ,Algebraic number field ,Cusp form ,Dedekind zeta function ,Prime zeta function ,Mathematics - Abstract
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.
- Published
- 2014
12. Mean values connected with the Dedekind zeta-function of a non-normal cubic field
- Author
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Guangshi Lü
- Subjects
Pure mathematics ,11f30 ,General Mathematics ,11f66 ,number field ,dedekind zeta function ,Algebraic number field ,Cusp form ,11n37 ,Number theory ,Norm (mathematics) ,QA1-939 ,Non normality ,Dedekind cut ,Cubic field ,Dedekind zeta function ,cusp form ,Mathematics ,11r42 - Abstract
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 $$ S_{3,K_3 } (x) $$.
- Published
- 2013
13. Note on the number of integral ideals in Galois extensions
- Author
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YongHui Wang and Guangshi Lü
- Subjects
Normal basis ,Discrete mathematics ,General Mathematics ,Fractional ideal ,Splitting of prime ideals in Galois extensions ,Ideal class group ,Congruence relation ,Algebraic number field ,Dedekind zeta function ,Field norm ,Mathematics - Abstract
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.
- Published
- 2010
14. Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
- Author
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Kevin A. Broughan
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Field (mathematics) ,010103 numerical & computational mathematics ,Extension (predicate logic) ,Algebraic number field ,01 natural sciences ,Combinatorics ,Riemann Xi function ,symbols.namesake ,symbols ,Logarithmic derivative ,0101 mathematics ,Constant (mathematics) ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2009
15. Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers
- Author
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Seung Ju Cheon, Jun Ho Lee, and Hyun Kim
- Subjects
Discrete mathematics ,Mathematics::Number Theory ,General Mathematics ,Algebraic number field ,Riemann zeta function ,Arithmetic zeta function ,symbols.namesake ,Number theory ,Integer ,Quadratic integer ,symbols ,Dedekind cut ,Dedekind zeta function ,Mathematics - Abstract
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 .
- Published
- 2007
16. Some explicit upper bounds for residues of zeta functions of number fields taking into account the behavior of the prime 2
- Author
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Stéphane Louboutin
- Subjects
Combinatorics ,Pure mathematics ,Arithmetic zeta function ,Number theory ,General Mathematics ,Algebraic geometry ,Algebraic number field ,Class number ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2007
17. Bounding the least prime ideal in the Chebotarev Density Theorem
- Author
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Asif Zaman
- Subjects
Dedekind zeta function ,General Mathematics ,Prime ideal ,Mathematics::Number Theory ,010103 numerical & computational mathematics ,least prime ideal ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Conjugacy class ,FOS: Mathematics ,Number Theory (math.NT) ,Galois extension ,Deuring-Heilbronn phenomenon ,0101 mathematics ,Physics ,Mathematics - Number Theory ,010102 general mathematics ,Density theorem ,Algebraic number field ,11R44 ,Norm (mathematics) ,11M41 ,Chebotarev Density Theorem ,11R42 - Abstract
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)
- Published
- 2015
18. Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques
- Author
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Armand Lachand, Gérald Tenenbaum, Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics::General Mathematics ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Mathematical analysis ,Algebraic number field ,16. Peace & justice ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,symbols.namesake ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,0103 physical sciences ,symbols ,Arithmetic function ,010307 mathematical physics ,0101 mathematics ,Dirichlet series ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2015
19. Independence of Hecke zeta functions of finite order over normal fields
- Author
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Maciej Radziejewski
- Subjects
Discrete mathematics ,Pure mathematics ,Arithmetic zeta function ,Quadratic integer ,Applied Mathematics ,General Mathematics ,Ideal class group ,Quadratic field ,Field (mathematics) ,Algebraic number field ,Algebraic number ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2006
20. Explicit Lower bounds for residues at 𝑠=1 of Dedekind zeta functions and relative class numbers of CM-fields
- Author
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Stéphane Louboutin
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,MathematicsofComputing_GENERAL ,Algebraic number field ,Upper and lower bounds ,Riemann zeta function ,Combinatorics ,symbols.namesake ,Arithmetic zeta function ,Number theory ,symbols ,Dedekind cut ,CM-field ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 2003
21. Geometric-progression-free sets over quadratic number fields
- Author
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Kimsy Tor, Jasmine Powell, Nathan McNew, Madeleine Weinstein, Karen Huan, Steven J. Miller, and Andrew Best
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,General Mathematics ,Unique factorization domain ,Natural number ,010103 numerical & computational mathematics ,0102 computer and information sciences ,Algebraic number field ,01 natural sciences ,Infimum and supremum ,Ring of integers ,Geometric progression ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Combinatorics (math.CO) ,0101 mathematics ,Algebraic number ,Dedekind zeta function ,Mathematics - Abstract
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
- Published
- 2014
- Full Text
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22. On the geometric side of the Arthur trace formula for the symplectic group of rank 2
- Author
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Werner Hoffmann and Satoshi Wakatsuki
- Subjects
Pure mathematics ,Symplectic group ,Mathematics - Number Theory ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Algebraic number field ,Unipotent ,01 natural sciences ,Shintani zeta function ,11F72, 11S90 (Primary) 11R42, 11E45, 22E30, 22E35 (Secondary) ,0103 physical sciences ,FOS: Mathematics ,Binary quadratic form ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Dedekind zeta function ,Mathematics - Representation Theory ,Mathematics ,Symplectic geometry - Abstract
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.
- Published
- 2013
23. Asymptotic properties of the Dedekind zeta-function in families of number fields
- Author
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Aleksei I Zykin
- Subjects
Pure mathematics ,General Mathematics ,Algebraic number field ,Dedekind zeta function ,Mathematics - Published
- 2009
24. Determination of all Quaternion Octic CM-Fields with Class Number 2
- Author
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Stéphane Louboutin
- Subjects
Combinatorics ,General Mathematics ,Quartic function ,Galois group ,Quaternion group ,Algebraic number field ,Abelian group ,Hilbert class field ,Quaternion ,Dedekind zeta function ,Mathematics - Abstract
It is known that there are only finitely many normal CM-fields with class number one or with given class number (see [9, Theorem 2; 11, Theorem 2]) and J. Hoffstein showed that the degree of any normal CMfield with class number one is less than 436 (see [2, Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM-fields with class number one: there are 172 non-isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non-abelian but normal octic CM-fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM-fields, then they focussed on dihedral octic CM-fields and proved that there are 17 dihedral octic CM-fields with class number one. The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM-field with class number 2, namely: Q(V
- Published
- 1996
25. Lower bounds for relative class numbers of CM-fields
- Author
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Stéphane Louboutin
- Subjects
Combinatorics ,Discriminant ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Zero (complex analysis) ,Ideal (ring theory) ,Absolute value (algebra) ,Totally real number field ,Algebraic number field ,Abelian group ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 1994
26. Sum formula for Kloosterman sums and the fourth moment of the Dedekind zeta-function over the Gaussian number field
- Author
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Bruggeman, R.W., Motohashi, Y., Analyse, Universiteit Utrecht, and Dep Wiskunde
- Subjects
Pure mathematics ,Spectral theory ,sum formula ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::Number Theory ,Gaussian number field ,Automorphic form ,Wiskunde en computerwetenschappen ,automorphic forms ,Kloosterman sums ,Mathematics::Group Theory ,FOS: Mathematics ,Dedekind cut ,Number Theory (math.NT) ,fourth moment ,Representation Theory (math.RT) ,11M41 ,11F72 ,11L05 ,22E30 ,Mathematics ,Mathematics - Number Theory ,Fourth power ,spectral theory ,Algebraic number field ,Mathematics::Geometric Topology ,Moment (mathematics) ,Landbouwwetenschappen ,Wiskunde: algemeen ,Dedekind zeta-function ,Kloosterman sum ,Mathematics - Representation Theory ,Dedekind zeta function ,11R42 - Abstract
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
- Published
- 2003
27. A note on the mean value of the zeta and $L$-functions. X
- Author
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Bruggeman, R.W., Motohashi, Y., Analyse, Universiteit Utrecht, and Dep Wiskunde
- Subjects
sum formula ,General Mathematics ,Wiskunde en computerwetenschappen ,Picard group ,automorphic representation ,Type (model theory) ,Combinatorics ,imaginary quadratic number field ,symbols.namesake ,Arithmetic zeta function ,Zeta-function ,Kloosterman sum ,Mathematics ,Hyperbolic space ,Mathematical analysis ,Algebraic number field ,11F72 ,Riemann zeta function ,11M06 ,Landbouwwetenschappen ,Wiskunde: algemeen ,spectral decomposition ,symbols ,Wiskunde en Informatica (WIIN) ,Dedekind zeta function - Abstract
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.
- Published
- 2001
28. The mean square of the Dedekind zeta function in quadratic number fields
- Author
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Wolfgang H. Müller
- Subjects
Discrete mathematics ,Fourth power ,General Mathematics ,Algebraic number field ,Dirichlet character ,Riemann zeta function ,Combinatorics ,symbols.namesake ,Arithmetic zeta function ,Mean value theorem (divided differences) ,symbols ,Dedekind zeta function ,Prime zeta function ,Mathematics - Abstract
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.
- Published
- 1989
29. Analytic formulas for the regulator of a number field
- Author
-
Eduardo Friedman
- Subjects
Maxima and minima ,Discrete mathematics ,Root of unity ,General Mathematics ,Analytic element method ,Regulator ,Algebraic number field ,Dedekind zeta function ,Mathematics - Abstract
LetR=Rk andw=wk be the regulator and the number of roots of unity in the number fieldk. We determine allk for whichR/w
- Published
- 1989
30. A remark on zeta functions
- Author
-
Jun-ichi Igusa
- Subjects
Pure mathematics ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Computer Science::Computational Geometry ,Algebraic number field ,Riemann zeta function ,Zeta distribution ,Algebra ,Arithmetic zeta function ,symbols.namesake ,symbols ,Locally compact space ,Computer Science::Data Structures and Algorithms ,Dedekind zeta function ,Prime zeta function ,Haar measure ,Mathematics - Abstract
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
- Published
- 1978
31. On the distribution of ideals in cubic number fields
- Author
-
Wolfgang H. Müller
- Subjects
Discrete mathematics ,Riemann hypothesis ,symbols.namesake ,Distribution (number theory) ,General Mathematics ,symbols ,Ideal norm ,Square-free integer ,Algebraic number field ,Dedekind zeta function ,Mathematics - Abstract
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).
- Published
- 1988
32. A mean value theorem for the Dedekind zeta-function of a quadratic number field
- Author
-
Jürgen G. Hinz
- Subjects
Discrete mathematics ,Quadratic equation ,General Mathematics ,Asymptotic formula ,Quadratic field ,Algebraic number field ,Dedekind zeta function ,Mathematics - Abstract
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.
- Published
- 1979
33. Cropping Euler factors of modular L-functions
- Author
-
Joan-Carles Lario, Jorge Jiménez-Urroz, Josep González, Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya. TN - Grup de Recerca en Teoria de Nombres, and Universitat Politècnica de Catalunya. MAK - Matemàtica Aplicada a la Criptografia
- Subjects
Weil restriction ,Abelian variety ,Pure mathematics ,Endomorphism ,11G18 ,11M41 ,14G35 ,General Mathematics ,Mathematics::Number Theory ,Frobenius algebras ,Matemàtica aplicada ,L-functions ,Abelian varieties ,FOS: Mathematics ,Order (group theory) ,Number Theory (math.NT) ,Abelian group ,Arithmetic of abelian varieties ,Mathematics ,Mathematics - Number Theory ,Applied Mathematics ,Algebraic number field ,Matemàtiques i estadística::Àlgebra [Àrees temàtiques de la UPC] ,Varietats abelianes ,Distribution of Frobenius elements ,Dedekind zeta function ,Frobenius, Àlgebra de - Abstract
According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety then its L-function must capture substantial part of the arithmetic properties of A. The smallest number field L where A has all its endomorphisms defined must also have a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties A_f/Q associated to weight 2 newforms for the modular group Gamma_1(N). Specifically, our goal is to relate the order of L(A_f/Q,s) at s = 1 with Euler products cropped by the set of primes that split completely in L. The results we obtain for the case when f has complex multiplication are complete, while in the absence of CM, our results depend on the rate of convergence in Sato-Tate distributions., 24 pages
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