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- 검색키워드
- 저자 : Louboutin, Sté (검색결과 9 건)
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TWISTED QUADRATIC MOMENTS FOR DIRICHLET L-FUNCTIONS
LOUBOUTIN, STEPHANE R. Korean Mathematical Society 2015 대한수학회보 Vol.52 No.6
Given c, a positive integer, we set. $$M(f,c):=\frac{2}{{\phi}(f)}\sum_{{\chi}{\in}X^-_f}{\chi}(c)|L(1,{\chi})|^2$$, where $X^-_f$ is the set of the $\phi$(f)/2 odd Dirichlet characters mod f > 2, with gcd(f, c) = 1. We point out several mistakes in recently published papers and we give explicit closed formulas for the f's such that their prime divisors are all equal to ${\pm}1$ modulo c. As a Corollary, we obtain closed formulas for M(f, c) for c $\in$ {1, 2, 3, 4, 5, 6, 8, 10}. We also discuss the case of twisted quadratic moments for primitive characters.
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ON THE DENOMINATOR OF DEDEKIND SUMS
Louboutin, Stephane R. Korean Mathematical Society 2019 대한수학회보 Vol.56 No.4
It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).
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ON CHOWLA'S HYPOTHESIS IMPLYING THAT L(s, χ) > 0 FOR s > 0 FOR REAL CHARACTERS χ
Stephane R., Louboutin Korean Mathematical Society 2023 대한수학회보 Vol.60 No.1
Let L(s, χ) be the Dirichlet L-series associated with an f-periodic complex function χ. Let P(X) ∈ ℂ[X]. We give an expression for ∑<sup>f</sup><sub>n=1</sub> χ(n)P(n) as a linear combination of the L(-n, χ)'s for 0 ≤ n < deg P(X). We deduce some consequences pertaining to the Chowla hypothesis implying that L(s, χ) > 0 for s > 0 for real Dirichlet characters χ. To date no extended numerical computation on this hypothesis is available. In fact by a result of R. C. Baker and H. L. Montgomery we know that it does not hold for almost all fundamental discriminants. Our present numerical computation shows that surprisingly it holds true for at least 65% of the real, even and primitive Dirichlet characters of conductors less than 10<sup>6</sup>. We also show that a generalized Chowla hypothesis holds true for at least 72% of the real, even and primitive Dirichlet characters of conductors less than 10<sup>6</sup>. Since checking this generalized Chowla's hypothesis is easy to program and relies only on exact computation with rational integers, we do think that it should be part of any numerical computation verifying that L(s, χ) > 0 for s > 0 for real Dirichlet characters χ. To date, this verification for real, even and primitive Dirichlet characters has been done only for conductors less than 2·10<sup>5</sup>.
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TWISTED QUADRATIC MOMENTS FOR DIRICHLET L-FUNCTIONS
Stéphane R. Louboutin 대한수학회 2015 대한수학회보 Vol.52 No.6
Given c, a positive integer, we set M(f, c) := 2/Φ(f) ∑ χ∈X− f χ(c)|L(1, χ)|2, where X− f is the set of the Φ(f)/2 odd Dirichlet characters mod f > 2, with gcd(f, c) = 1. We point out several mistakes in recently published papers and we give explicit closed formulas for the f’s such that their prime divisors are all equal to ±1 modulo c. As a Corollary, we obtain closed formulas for M(f, c) for c 2 {1, 2, 3, 4, 5, 6, 8, 10}. We also discuss the case of twisted quadratic moments for primitive characters.
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On the denominator of Dedekind sums
St'ephane R. Louboutin 대한수학회 2019 대한수학회보 Vol.56 No.4
It is well known that the denominator of the Dedekind sum $s(c,d)$ divides $2\gcd (d,3)d$ and that no smaller denominator independent of $c$ can be expected. In contrast, here we prove that we usually get a smaller denominator in $S(H,d)$, the sum of the $s(c,d)$'s over all the $c$'s in a subgroup $H$ of order $n>1$ in the multiplicative group $({\mathbb Z}/d{\mathbb Z})^*$. First, we prove that for $p>3$ a prime, the sum $2S(H,p)$ is a rational integer of the same parity as $(p-1)/2$. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of $S(H,d)$ for non necessarily prime $d$'s. We show that its denominator is a divisor of some explicit divisor of $2d\gcd (d,3)$.
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Determination of the orders generated by a cyclic cubic unit that are Galois invariant
Lee, Jun Ho,Louboutin, Sté,phane R. Elsevier 2015 Journal of number theory Vol.148 No.-
<P><B>Abstract</B></P> <P>Let <I>ϵ</I> be a totally real cubic algebraic unit. Assume that the cubic number field Q ( ϵ ) is Galois. In this situation, it is natural to ask when the cubic order Z [ ϵ ] is invariant under the action of the Galois group Gal ( Q ( ϵ ) / Q ) . It seems that this natural problem has never been looked at. We give an answer to this problem (e.g., we show that if <I>ϵ</I> is totally positive, then this happens in only 12 cases).</P>
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Discriminants of cyclic cubic orders
Lee, J.H.,Louboutin, S.R. Academic Press 2016 Journal of number theory Vol.168 No.-
<P>Let alpha be a cubic algebraic integer. Assume that the cubic number field Q(alpha) is Galois. Let alpha(1), alpha(2) and alpha(3) be the real conjugates of a. We give an explicit Z-basis and the discriminant of the Gal(Q(alpha)/Q)-invariant totally real cubic order Z[alpha(1), alpha(2), alpha(3)]. This new result is completely different from the one previously obtained in the case that the cubic field Q(alpha) is not Galois. (C) 2016 Elsevier Inc. All rights reserved.</P>
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The class number one problem for some non-normal CM-fields of degree 2<i>p</i>
Ahn, Jeoung-Hwan,Boutteaux, Gé,rard,Kwon, Soun-Hi,Louboutin, Sté,phane Elsevier 2012 Journal of number theory Vol.132 No.8
<P><B>Abstract</B></P> <P>To date, the class number one problem for non-normal CM-fields is solved only for quartic CM-fields. Here, we solve it for a family of non-normal CM-fields of degree 2<I>p</I>, p ⩾ 3 and odd prime. We determine all the non-isomorphic non-normal CM-fields of degree 2<I>p</I>, containing a real cyclic field of degree <I>p</I>, and of class number one. Here, p ⩾ 3 ranges over the odd primes. There are 24 such non-isomorphic number fields: 19 of them are of degree 6 and 5 of them are of degree 10. We also construct 19 non-isomorphic non-normal CM-fields of degree 12 and of class number one, and 10 non-isomorphic non-normal CM-fields of degree 20 and of class number one.</P>
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