Computing the residue of the Dedekind zeta function

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log D_K/(\sqrt{X}\log X), where D_K is the absolute value of the discriminant of K. Guided by Weil's explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach's constant to 2.33. This results in substantial speeding of one part of Buchmann's class group algorithm.Comment: 16 page

A Zeta Function for Multicomplex Algebra

In this paper we define and study a Dedekind-like zeta function for the algebra of multicomplex numbers. By using the idempotent representations for such numbers, we are able to identify this zeta function with the one associated to a product of copies of the field of Gaussian rationals. The approach we use is substantially different from the one previously introduced by Rochon (for the bicomplex case) and by Reid and Van Gorder (for the multicomplex case)

The Twisted Second Moment of the Dedekind Zeta Function of a Quadratic Field

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-\epsilon}$

Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

The \emph{Barnes $\zeta$-function} is \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to \C. Specialized at negative integers $-k$, the Barnes $\zeta$-function gives \zeta_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) where B_k(x; \a) is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing B_k(0; \a) gives the \emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes $\zeta$-functions, Bernoulli--Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page

Note on on Dedekind type DC sums

In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.Comment: 13 page

Artin's conjecture, Turing's method and the Riemann hypothesis

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S5 and A5 representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of non-abelian extensions of Q. In addition, we discuss two methods for locating zeros of arbitrary L-functions. The first uses the explicit formula and techniques (developed jointly with Andreas Strombergsson) for computing with trace formulae. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general L-functions on the critical line via the Fast Fourier Transform. Finally, we present some numerical results testing Artin's conjecture for S5 representations, and the Riemann hypothesis for Dedekind zeta functions of S5 and A5 fields.Comment: 37 pages, 5 figure

An analogue of the Rademacher function for generalized Dedekind sums in higher dimension

We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of the lattice cone defining the Dedekind sums. We associate a Laurent polynomial which is added to generalized Dedekind sums of fixed index to make their denominators bounded. We give explicitly the denominator in terms of Bernoulli numbers. This generalizes the role played by the rational function given by the difference of the Rademacher function and the classical Dedekind sums. We associate an exponential sum to the generalized Dedekind sums using the integrality of the generalized Rademacher function. We show that this exponential sum has a nontrivial bound that is sufficient to fulfill Weyl's equidistribution criterion and thus the fractional part of the generalized Dedekind sums are equidistributed. As an example, for a 3 dimensional case and Zagier's higher dimensional generalization of Dedekind sums, we compute the Laurent polynomials associated.Comment: 43 pages, 2 Appendice

Uniform upper bounds of the distribution of relatively r-prime lattice points

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under assuming the Lindel\"of hypothesis. And we also showed unconditional results for abelian extensions with a degree less than or equal to $6$. In this paper we remove all assumption about number fields and improve uniform upper bounds. Throughout this paper we consider estimates for distribution of ideals of the ring of integer $\mathcal{O}_K$ and obtain uniform upper bounds. And when $K$ runs through cubic extension fields we show better uniform upper bounds than that under the Lindel\" of Hypothesis.Comment: 20 page

Hasse-Weil zeta functions of SL2{\rm SL}_2-character varieties of closed orientable hyperbolic 33-manifolds

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.Comment: 25 pages, revision: Content of Section 3.1 (Corollary 3.2, Lemma 3.3) has been revise

Zeta Functions on Arithmetic Surfaces

We use a form of lifted harmonic analysis to develop a two-dimensional adelic integral representation of the zeta functions of simple arithmetic surfaces. Manipulations of this integral then lead to an adelic interpretation of the so-called mean-periodicity correspondence, which is comparable to the better known automorphicity conjectures for the generic fibre.Comment: 28 pages. Numerous corrections from previous versions, and greater emphasis on expositio