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Explicit upper bounds for the average order of dn (m) and application to class number
 J. Inequal. Pure and Appl. Math
"... ABSTRACT. In this paper, we prove some explicit upper bounds for the average order of the generalized divisor function, and, according to an idea of Lenstra, we use them to obtain bounds for the class number of number fields. ..."
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ABSTRACT. In this paper, we prove some explicit upper bounds for the average order of the generalized divisor function, and, according to an idea of Lenstra, we use them to obtain bounds for the class number of number fields.
Effective results for hyper and superelliptic equations over number
"... Abstract. Let f be a polynomial with coefficients in the ring OS of Sintegers of a given number field K, b a nonzero Sinteger, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) bym = f(x) in x, y ∈ OS. In the present paper we give explicit upper bounds in te ..."
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Abstract. Let f be a polynomial with coefficients in the ring OS of Sintegers of a given number field K, b a nonzero Sinteger, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) bym = f(x) in x, y ∈ OS. In the present paper we give explicit upper bounds in terms of K,S, b, f,m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K,S, b, f such that if m> C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over Z. 1.
BrauerSiegel like results for relative class numbers of CMfields
, 2001
"... We prove BrauerSiegel like results about the asymptotic behavior of relative class numbers of CMfields. ..."
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We prove BrauerSiegel like results about the asymptotic behavior of relative class numbers of CMfields.
CMFIELDS WITH RELATIVE CLASS NUMBER ONE
, 2005
"... We will show that the normal CMfields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CMfields with relative class number one are of degrees ≤ 96, and the CMfields with class number one are of degrees ≤ 104. By many ..."
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We will show that the normal CMfields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CMfields with relative class number one are of degrees ≤ 96, and the CMfields with class number one are of degrees ≤ 104. By many authors all normal CMfields of degrees ≤ 96 with class number one are known except for the possible fields of degree 64 or 96. Consequently the class number one problem for normal CMfields is solved under the Generalized Riemann Hypothesis except for these two cases.
EFFECTIVE RESULTS FOR DIOPHANTINE EQUATIONS OVER FINITELY GENERATED DOMAINS
"... Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over Z. We consider Thue equations F (x, y) = δ in x, y ∈ A, where F is a binary form with coefficients from A and δ is a nonzero element from A, and hyper and superelliptic equations f(x) = δym in ..."
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Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over Z. We consider Thue equations F (x, y) = δ in x, y ∈ A, where F is a binary form with coefficients from A and δ is a nonzero element from A, and hyper and superelliptic equations f(x) = δym in
An inequality for the class number
, 2006
"... We prove in an elementary way a new inequality for the average order of the Piltz divisor function with application to class number of number fields. ..."
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We prove in an elementary way a new inequality for the average order of the Piltz divisor function with application to class number of number fields.
www.elsevier.de/exmath Counting integral ideals in a number field
, 2005
"... Let K be an algebraic number field. We discuss the problem of counting the number of integral ideals below a given norm and obtain effective error estimates. The approach is elementary and follows a classical line of argument of Dedekind and Weber. The novelty here is that explicit error estimates c ..."
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Let K be an algebraic number field. We discuss the problem of counting the number of integral ideals below a given norm and obtain effective error estimates. The approach is elementary and follows a classical line of argument of Dedekind and Weber. The novelty here is that explicit error estimates can be obtained by fine tuning this classical argument without too much difficulty. The error estimate is sufficiently strong to give the analytic continuation of the Dedekind zeta function to the left of the line R(s) = 1 as well as explicit bounds for the residue of the zeta function at s = 1.