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We prove an explicit version of the Chebotarev theorem for the density of prime ideals with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions. In appendix we also give some explicit formulas counting non-trivial zeros of Hecke's L-functions, in that case without assuming the truth of the Riemann hypothesis.

The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly constructed density function, for the case of automorphic L-functions attached to primitive forms with respect to congruence subgroups Gamma_0(N)., Comment: 21pages

In this paper, we give a lower bound for 2k-th moment of the Dedekind zeta-function, where k is any real number greater than 1.

Let gcd ⁡ ( k , j ) be the greatest common divisor of the integers k and j . We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ⁡ ( k , j ) ) with f = id , ϕ , ϕ s , ψ and ψ s for any fixed positive integers r and s , where ϕ , ϕ s , ψ and ψ s are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑ k ≤ x 1 k r + 1 ⋅ ∑ j = 1 k j r ϕ ( gcd ⁡ ( k , j ) ) and ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r ψ ( gcd ⁡ ( k , j ) ) .

B. Sury proved the following Menon-type identity, ∑ a ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) σ r ( n ) , where U ( Z n ) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σ r ( n ) is the sum of r-th powers of positive divisors of n with r being a non-negative integer. Recently, C. Miguel extended this identity from Z to any residually finite Dedekind domain. In this note, we will give a further extension of Miguel's result to the case with many tuples of group of units. For the case of Z , our result reads as follows ∑ a 1 , ⋯ , a s ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a 1 − 1 , ⋯ , a s − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) ∏ i = 1 m ( φ ( p i k i ) s − 1 p i k i r − p i k i ( s + r − 1 ) + σ s + r − 1 ( p i k i ) ) , where n = p 1 k 1 ⋯ p m k m is the prime factorization of n.

In this paper, using Cohen's and Tangedal and Young's theory on the p -adic Hurwitz zeta functions, we construct the analytic Dedekind sums on the p -adic complex plane C p . We show that these Dedekind sums interpolate Carlitz's higher order Dedekind sums p -adically. From Apostol's reciprocity law for the generalized Dedekind sums, we also prove a reciprocity relation for the special values of these p -adic Dedekind sums. Finally, the parallel results for the analytic Dedekind sums on the p -adic complex plane associated with Euler polynomials have also been given.

Let φ be the Euler's totient function and σ k ( n ) = ∑ d | n d k , that is, the sum of the kth powers of the divisors of n. B. Sury showed that ∑ t 1 ∈ U ( Z n ) , t 2 , … , t r ∈ Z n gcd ( t 1 − 1 , t 2 , … , t r , n ) = φ ( n ) σ r − 1 ( n ) , where U ( Z n ) is the group of units in the ring of residual classes modulo n. Here, this identity is extended to residually finite Dedekind domains.

The aim of this paper is to give a simple proof for a reciprocity law of generalized Dedekind sums involving quasi-periodic Euler functions by considering the analytic properties of Euler polynomials which is an analogue of the reciprocity law of the classical Dedekind sums dating back to Dedekind [13] in 1892. We also derive an explicit expression for generalized Dedekind sums involving quasi-periodic Euler functions.

In this paper, we will evaluate the values of the Dedekind zeta functions at s = − 1 of the simplest quartic fields. We first introduce Siegel's formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel's formula to the simplest quartic fields. In the second, we will develop the basic arithmetic properties of the simplest quartic fields which will be necessary in our computation. We will compute the discriminant, ring of integers, and different of the simplest quartic fields. In the third, we will give a full description for a Siegel lattice of the simplest quartic fields, and develop a method of computing sum of divisors function for an ideal. Finally, by combining these results, we compute the values of the Dedekind zeta functions at s = − 1 of the simplest quartic fields.

Assuming the generalized Riemann hypothesis, we prove upper bounds for moments of arbitrary products of automorphic L-functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds to estimate the variance of the coefficients of these zeta- and L-functions in short intervals. We also prove upper bounds for moments of products of central values of automorphic L-functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants.

For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribution criterion for R(i,j)q^{i+j-2} s_{ij}(p,q). As a special case, we recover the equidistribution result of the classical Dedekind sums multiplied by 12 not using the modular weight of the Dedekind's \eta(\tau)., Comment: Some references added and minor modification

In this note we give an extension of the well-known Menonʼs identity to residually finite Dedekind domains.

In this paper, we consider a certain product of double sine functions as an analogue of the Dedekind–Rademacher sum. Its reciprocity formulas are established by decomposition of a certain double zeta function. As their applications, we reconstruct and refine a part of Arakawaʼs work on ray class invariants of real quadratic fields, and prove directly explicit relations between various invariants which are defined in terms of the double sine function and are related to the Stark–Shintani conjecture. Moreover, in some examples, new expressions of the invariants are revealed. As two appendices, we give a new proof of Carlitzʼs three-term relation for the Dedekind–Rademacher sum and a simple proof of Arakawaʼs transformation formula for an analogue of the generalized Eisenstein series originated with Lewittes.

Let R be a Dedekind ring, K its quotient field, L a separable finite extension over K , and O L the integral closure of R in L . In this paper we provide a “practical” criterion that tests when a given α ∈ O L generates a power basis for O L over R (i.e. when O L = R [ α ] ), improving significantly a result in this direction by M. Charkani and O. Lahlou. Applications in the context of cyclotomic, quadratic, biquadratic number fields, and some Dedekind rings are provided.

Text In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression 1 a 0 m 0 + 1 ∑ k = 1 a 0 − 1 ∏ j = 1 d cot ( m j ) ( π a j k a 0 ) , where a 0 is a positive integer, a 1 , … , a d are positive integers prime to a 0 and m 0 , m 1 , … , m d are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking m 0 = ⋯ = m d = 0 , we derive the reciprocity and rationality theorems of Zagier. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg .

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants Ono K are equal to their class numbers h K . Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if Ono K is large enough compared with h K , then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values d K of their discriminants are less than 2.3 ⋅ 10 9 . We determine all these K's with d K ⩽ 10 6 . There are 76 such K's. We prove that there is at most one such K with d K ⩾ 1.8 ⋅ 10 11 .

This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol's formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums.

In this paper, we study on two subjects. We first construct degenerate analogues of Dedekind sums in the sense of Apostol, Carlitz and Takacs, and prove the corresponding reciprocity formulas. Secondly, we define generalized Dedekind character sums, which are explicit extensions of Berndt's definition, and prove the reciprocity laws. From the derived reciprocity laws, we obtain Berndt's reciprocity laws as special cases.

In this paper, by using generalized logarithms of Dedekind eta-functions, generalized logarithms of theta-functions are obtained. Applying these functions, the relations between Hardy sums and Theta-functions are found. The special cases of these relations give Berndt's Theorems 6.1–8.1 (J. Reine Angew. Math. 303/304 (1978) 332) and explicit formulae of Hardy sums. Using derivative of logarithms of the Dedekind eta-function, relations between logarithm of the theta-functions and Eisenstein series are given. Applying connection between Lambert series and generalized Dedekind sums, the relation between theta-functions and Lambert series are obtained.

For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘ -ordering of S is a sequence of elements of S with a certain minimizing property. These ℘ -orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘ -orderings from the viewpoint of the ℘ -adic metric on R. We find that the ℘ -sequences of S depend only on the closure of S in R ℘ . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘ -sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘ -orderings for all prime ideals ℘⊂R , and show that such simultaneous orderings do not exist for imaginary quadratic number rings.

We define a Dedekind symbol associated with a J-form (J-forms generalize the usual Jacobi forms). Then we prove the reciprocity law for Dedekind symbols. As an example, we give an explicit description for the Dedekind symbol associated with Weierstrass ℘ -function. The reciprocity law in this case then yields new trigonometric identities. This in turn gives rise to a “generating function” of Apostol reciprocity law for the generalized Dedekind sums.

Let E be an elliptic curve defined over Q and without complex multiplication. For a prime p of good reduction, let Ē be the reduction of E modulo p . Assuming that certain Dedekind zeta functions have no zeros in Re( s )>3/4, we determine how often Ē ( F p ) is a cyclic group. This result was previously obtained by J.-P. Serre using the full Generalized Riemann Hypothesis for the same Dedekind zeta functions considered by us.

Resume The article uses a generalization of the structure of sequences appearing on the continued fraction development of algebraic numbers given by the markoff theory. Thanks to Dedekind sums and their reciprocity law, these sequences give birth to new diophantine equations and a solution for them. Their sets of solutions can then be described with the same methods as used in the classical theory. In the general case, the transposition of the conjecture of Cassels and Zagier is not valid. Some complement are given for the modular invariant of the group SL (2, Z ). It is shown how to extend to GL (2, Z ).

Given a subsetXof a Dedekind domainD, and a polynomialF∈D[x], thefixed divisor d(X, F) ofFoverXis defined to be the ideal inDgenerated by the elementsF(a),a∈X. In this paper we derive a simple expression ford(X, F) explicitly in terms of the coefficients ofF, using a generalized notion of “factorial” introduced by the author in a previous paper. WhenX=D= Z , this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Polya, Gunji, and McQuillan relatingd( Z , F) and the usual factorial function.

LetRbe a domain andKits quotient-field. For a subsetSofK, let F R(S) be the set of polynomialsf∈K[x] withf(S)⊆Rand define theR-closure ofSas the set of thoset∈Kfor whichf(t)∈Rfor allf∈ F R(S). The concept ofR-closure was introduced by McQuillan (J. Number Theory39(1991), 245–250), who gave a description in terms of closure inP-adic topology, whenRis a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker thanP-adic topology, which allows us to treat ideals of infinite index, and derive a characterization ofR-closure whenRis a Krull ring. This gives us a criterion for F R(S)= F R(T), whereSandTare subsets ofK. As a corollary we get a generalization to Krull rings ofR. Gilmer's result (J. Number Theory33(1989), 95–100) characterizing those subsetsSof a Dedekind ring with finite residue fields for which F R(S)= F R(R).

A necessary and sufficient condition is given for a positive integer to appear as the denominator of some reduced Dedekind sum.

If h , k ∈ Z , k > 0, the Dedekind sum is given by s(h,k) = ∑ μ=1 k μ k hμ k , with ((x)) = x − [x] − 1 2 , x∉Z , =0 , x∈Z . The Hecke operators T n for the full modular group SL (2, Z ) are applied to log η(τ) to derive the identities ( n ∈ Z + ) ∑ ∑ s(ah+bk,dk) = σ(n)s(h,k) , ad=n b( mod d) d>0 where ( h , k ) = 1, k > 0 and σ ( n ) is the sum of the positive divisors of n . Petersson had earlier proved (∗) under the additional assumption k ≡ 0, h ≡ 1 (mod n ). Dedekind himself proved (∗) when n is prime.

The properties of discriminants and differents were studied first by Dedekind and Hilbert in finite algebraic extensions of fields of algebraic numbers. From a local point of view, that is equivalent to a study of the p-adic case, where the results of Dedekind and Hilbert can be formulated as follows. Dedekind's theorem: The g.c.d. Δ(Kk) of differents of integral bases of a finite algebraic extension Kk (which I call an algebraic different if Kk and the g.c.d δ(Kk) of differents of integral elements of Kk (which I call an arithmetic different of Kk) coincide; Hilbert's theorem (which is the basis of Herbrand's ramification theory of intermediate extensions): If K ⊃ L ⊃ k, δ(Kk) = δ(KL) δ(Lk). These results are easily generalizable to the “classical case” of henselian valued basic fields, i.e., the case when the valuation is discrete and the residual extension Kk of Kk is separable. But, in the general case of extensions Kk of valued fields (where k may be assumed to be henselian), Dedekind's and Hilbert's theorems are not always true: the algebraic different Δ(Kk) divides the arithmetic different δ(Kk), but generally δ(Kk) ≠ Δ(Kk), and Hilbert's theorem holds only for the algebraic different. When the valuation is discrete, I call an extension Kk dedekindian when δ(Kk) = Δ(Kk) and hilbertian if, for every intermediate field L of Kk (i.e., K ⊇ L ⊇ k), Hilbert's theorem δ(Kk) = δ(KL) δ(Lk) for arithmetic differents holds. When the valuation is dense, the situation is more complicated, because of the existence of two kinds of ideals (principal and other), and it is convenient to define dedekindian and hilbertian extensions in a slightly different manner and to introduce somewhat wider classes of extensions called quasi-dedekindian and quasi-hilbertian. I study the relations between Δ(Kk) and δ(Kk), and, in particular, I give a complete characterization of dedekindian extensions for both discrete and dense valuations; I also give examples of non-dedekindian and non-hilbertian extensions. In Section 4, some connections with the ramification theory (both for normal and non-normal extensions) are studied and a weak analog of Hilbert's theorem [δ(Kk) δ(Lk) divides δ(Kk)] is proved.

Let R be a Dedekind domain, G a finite group of automorphisms of R, and A an ambiguous ideal of R i.e., σA = A for all σ ∈ G. The Tate groups Hn(G, A) are considered as RG-modules. A localization theorem is proved and the precise RG-module structure determined in a particular case. In addition some remarks are made concerning cohomological triviality.

Roiter proved that for a given Z -order Λ in a semisimple Q -algebra, there is a positive integer n = n(Λ) such that there are at most n nonisomorphic Λ-lattices which are all locally isomorphic. The main result of this article is to extend this result to finitely generated Λ-modules. One can work over other Dedekind domains D. If it is only assumed that the residue fields of D are finite, then it is shown that M ⊕ X ≅ N ⊕ X implies e copies of M are isomorphic to e copies of N for some e = e(Λ) for M, N, and X finitely generated Λ-Modules. Jacobinski proved there is an extension field k (depending only on Λ) such that two Λ-lattices become isomorphic under an extension of scalars to the integers of k if and only if they are locally isomorphic. This result is extended to modules, and also an analog of this result is proved for arbitrary Dedekind domains.

An elementary proof is given of the author's transformation formula for the Lambert series G p (x) = Σ n−1 ∞ n −p x n (1−x n ) relating Gp(e2πiτ) to Gp(e2πiAτ), where p > 1 is an odd integer and Aτ = (aτ + b) (cτ + d) is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function log η(τ) = πiτ 12 − G 1 (e 2πiτ ) , and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.

Sums of Dedekind type are defined by the formula f(h, k) = Σ μ( mod k) A( μ k ) B( hμ k ) , where each of A(x) and B(x) satisfies a multiplication formula of the form Σ b( mod q) F(x + b q ) = q ν(F) F(qx) . Properties of these sums are obtained, including an extension of M. I. Knopp's identity for the classical Dedekind sums s(h, k).

Generalized reciprocity formulas and Dedekind-Petersson-Knopp-type formulas are given to generalized Dedekind sums attached to Dirichlet characters, defined on a certain congruence subgroup of SL2(Z). In addition, these formulas are respectively construed as transformational and eigen properties of those sums redefined on a certain set of cusps.

A description of the quasigenus, which was introduced into the theory of integral quadratic forms by B. W. Jones, is given in terms of the general theory of spinor general of lattices over Dedekind domains. As a consequence several questions raised by Jones regarding the relationship between the partitionings of a genus induced by the splittings into spinor genera and quasigenera are answered.

Let K/ Q be an algebraic number field and ζK(s) be the associated Dedekind ζ function. A quantitative estimate is proved which shows that the average order of the coefficients of ζkm(s) (for m ∈ Z +) arises from infrequent occurrences of very large values of these coefficients. This leads to new Ω-estimates for the associated error terms, improving results of Szego and Walfisz.

Five new exceptional values of the Dedekind symbol are presented, and a conjecture is proposed on the necessary and sufficient conditions for integers to be exceptional values.

The truth of the Riemann hypothesis for Dedekind zeta functions over certain Kummer fields is shown to imply that the density of prime divisors of an infinite set of second-order linear recurrences is greater than 0. A rough estimate, suggested by R. R. Laxton, for calculating the density is also shown to be a consequence of the Riemann hypothesis holding over certain Kummer fields.

Let V be a nondefective quadratic space over a function field F of characteristic 2. Let S be a Dedekind set of spots consisting of almost all spots on F and let o be the corresponding ring of integers. If L is an o lattice on V we let g denote the number of classes in the genus of L . We prove that g is finite if dim V ≥ 10 or if dim V = 8 and o is a principal ideal domain. The finiteness of g is also established in all cases where V does not contain a quaternary anisotropic subspace. The number of spinor genera in a genus is also examined and shown to be finite in all cases.

Abelian differentials of four elliptic function fields may be constructed from Dedekind's function η. Each of the fields mentioned belongs to a congruence subgroup of index 12 of the modular group. A method involving Weierstrass' ℘-functions then yields the respective absolute invariants. Two of the fields are thus seen not to admit of complex multiplication.

Let F be a field with a Dedekind set of spots S, let D F be the integers of F determined by S. Let V be a regular quadratic space over F with associated bilinear form B(x, y). We find necessary and sufficient conditions on the volume of a lattice L so that (1) if x, y ∈ L, B(x, y) ∈ D F and (2) L is maximal with respect to (1). This result is applied to the case where V is an algebraic number field over the rationals and the bilinear form is given by Tr(xy).

Erich Hecke first showed that the Dedekind zeta-function for an ideal class in an imaginary quadratic field has an infinite number of zeros on the critical line. Recently, K. Chandrasekharan and Raghavan Narasimhan proved the result for both real and imaginary quadratic fields. In this paper we give a quantitative result. Namely, let N0(T) denote the number of zeros of the Dedekind zeta-function ζK(12 + it T) for 0 < t < T. Then, for every ε > 0, there exists a positive constant A such that N0(T) > AT12−ϵ.

In this paper we give a simple proof for the reciprocity formula for the generalized Dedekind sums and derive an explicit expression for these sums.

Let o be a Dedekind domain with quotient field F and let M be an o -lattice on an n-dimensional vector space V over F. Let G be any one of the linear congruence groups TL n (M;a), Sl n (M;a),GL n (M;a) determined by an integral ideal a. If n ≥ 3, then every automorphism Λ of G can be uniquely expressed in exactly one of the forms ∧=P x °Φ g or ∧=P x °Psi; h where Pχ is a radial automorphism, Φg is associated with certain semilinear isomorphisms of V onto V and Ψh is associated with certain semilinear isomorphisms of V onto the dual space of V. For background and motivation see O. T. O'Meare, The automorphisms of the linear groups over any integral domain, J. reine angew. Math. 223 (1966), pp. 56–100. The proof uses results on the characterization of transvections by O. T. O'Meara (Group-theoretic characterization of transvections using CDC subgroups) and by H. Zassenhaus (Characterization of unipotent matrices).

Let p ⩾3 be a prime number, b ⩾2 a primitive root mod p and z an integer, 1⩽ z ⩽ p −1. The digit expansion of z / p with respect to the basis b has a period consisting of the first p −1 digits c 1 , …, c p −1 . We express the variance σ 2 of c 1 , …, c p −1 in terms of the Dedekind sum s ( p , b ) and investigate the behaviour of σ 2 for b fixed and p →∞. The reciprocity law for Dedekind sums is the most important tool in this investigation.

A simple proof of the identity D(a, c) = − D( a , c ) for the Dedekind sums D(a, c) introduced by Sczech will be given.