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COMPUTING THE RESIDUE OF THE DEDEKIND ZETA FUNCTION
"... Abstract. 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 logΔK/( X logX), where ΔK is the absolute valu ..."
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Abstract. 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 logΔK/( X logX), where ΔK is the absolute
Localization of the first zero of the Dedekind zeta function
 Math. Comp
"... Abstract. Using Weil’s explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known ..."
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Abstract. Using Weil’s explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest
On Equivariant Dedekind ZetaFunctions at s = 1
 DOCUMENTA MATH.
, 2010
"... We study a refinement of an explicit conjecture of Tate concerning the values at s = 1 of Artin Lfunctions. We reinterpret this refinement in terms of Tamagawa number conjectures and then use this connection to obtain some important (and unconditional) evidence for our conjecture. ..."
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We study a refinement of an explicit conjecture of Tate concerning the values at s = 1 of Artin Lfunctions. We reinterpret this refinement in terms of Tamagawa number conjectures and then use this connection to obtain some important (and unconditional) evidence for our conjecture.
Zeros of Dedekind zeta functions in the critical strip
 Math.Comp.66 (1997), 1295–1321. MR 98d:11140 Laboratoire d’Algorithmique Arithmétique, Université BordeauxI,351coursdela Libération, F33405 Talence Cedex France Email address: omar@math.ubordeaux.fr
"... Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff ..."
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Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated
An explicit zerofree regions for the Dedekind zeta functions
, 2011
"... Let K be a number field, nK its degree, and dK the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: Res ≥ 1 − 1logM, Ims  ≥ 1, where logM = 12.55 log dK + 9.69nK log Ims+ 3.03nK + 58.63. Moreover, i ..."
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Let K be a number field, nK its degree, and dK the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: Res ≥ 1 − 1logM, Ims  ≥ 1, where logM = 12.55 log dK + 9.69nK log Ims+ 3.03nK + 58.63. Moreover
Dedekind Zeta Functions and the Complexity of Hilbert’s Nullstellensatz
, 2008
"... Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the ..."
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can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also
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