441 research outputs found
An identity for the Dedekind eta-function involving two independent complex variables
No full textThe authors prove a new identity for the Dedekind eta-function that involves third powers of the eta-function, with each of the two cubes being a function of a different complex variable
Class invariants from a new kind of Weber-like modular equation
No full textA technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely `algebraic' method. They are obtained by means of specialising certain modular equations related to Weber's modular equations of `irrational type'. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant d1 (mod 8)
A new class of theta function identities in two variables
No full textWe describe a new series of identities, which hold for certain general
theta series, in two completely independent variables. We provide explicit
examples of these identities involving the Dedekind eta function, Jacobi theta
functions, and various theta functions of Ramanujan
A realization theorem for almost Dedekind domains
No full textAn integral domain D is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of D provides a natural measure of the deviation of D from being an SP-domain. In the present paper we show that every ordinal number (Formula presented.) can be realized as the SP-rank of an almost Dedekind domain
On general Dedekind sums
No full textsummary:As a far generalization of the Dedekind sum with the product of periodic Bernoulli polynomials, Mikolás introduced the Dedekind type sum with the product of the Hurwitz zeta-functions , . We adopt the motivation suggested by Mikolás that the Dedekind sum is a generalized inner product in the second variable. The Hurwitz zeta-function has a simple pole at and cannot assume the value while its counterpart, the Lerch zeta-function , is more tractable and we study the Dedekind type sum with the product of the Lerch zeta-functions. We establish a striking identity between these Dedekind type sums to the effect that with a correction term is a constant multiple of -- the base change formula. This implies a new expression for the ordinary Dedekind sum in terms of the one with Apostol's generalized Bernoulli polynomial. In another direction, by letting the second variables vary independently with first variables fixed as , we may elucidate the Hecke correspondence in the previous derivations of the general eta transformation formula. We can also establish many interesting properties of which supplement those of . Moreover, we show that also appears in the pseudo-transformation formula for non-modular functions
On generalized Dedekind prime rings
No full textLet R be a maximal order and A, B be R-ideals of R. Clearly (A B)* superset of B*A* is satisfied and if R is a Dedekind prime ring, the equality holds, i.e., (AB)* = B*A*. However, the equality is not true in general. In this paper, we answer the question: If R is a maximal order when is (A B)* = B*A* for all non-zero R-ideals of R? We call prime Noetherian maximal orders satisfying this property, generalized Dedekind prime rings. We give several characterizations of G-Dedekind prime rings and show that being a G-Dedekind prime ring is a Morita invariant. Moreover, we prove that if R is a PI G-Dedekind prime ring then the polynomial ring R[x] and the Rees ring R[Xt] associated to an invertible ideal X are also PI G-Dedekind prime rings. (c) 2008 Elsevier Inc. All rights reserved
[[alternative]]The Dedekind-mertens number and the Polarized Dedekind-mertens number
No full text[[abstract]]We study content ideals of polynomials and their behavior under multiplication. In [HH2] and [CHH], they give a sharpening of the Dedekind-Mertens Lemma relating the contents of two polynomials to the content of their product. Therefore, we study the Dedekind-Mertens number, the Polarized Dedekind-Mertens number and give some examples about them.
On the mean value of Dedekind sum weighted by the quadratic Gauss sum
No full textsummary:Various properties of classical Dedekind sums have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it
Pointwise and correlation bounds on Dedekind sums over small subgroups
No full textWe obtain new bounds, pointwise and on average, for Dedekind sums s(λ,p) modulo a prime p with λ of small multiplicative order d modulo p. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of L(1,χ) over subgroups of characters to some correlations of Dedekind sums, and use recent results of the second and third author to study these correlations
Dedekind σ-complete vector lattice of b-AM-compact operators
No full textWe give several equivalent conditions characterizing the case when Krb-AM(E,F) is Dedekind σ-complete. Moreover, we describe the case when the space of all regular b-AM-compact operators from E to F is complete under the b-AM-norm.Keywords: Banach lattices, b-AM-compact operator, discrete spac
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