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Selberg’s conjectures and Artin Lfunctions
 1–14. zeros of RankinSelberg Lfunctions 31
, 1994
"... In its comprehensive form, an identity between an automorphic Lfunction and a “motivic ” Lfunction is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of ShimuraTaniyama that every elliptic curve over Q is “modular ” is certainly th ..."
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In its comprehensive form, an identity between an automorphic Lfunction and a “motivic ” Lfunction is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of ShimuraTaniyama that every elliptic curve over Q is “modular ” is certainly the most intriguing reciprocity
Group Representations and Harmonic Analysis from Euler to Langlands, Part II
, 1996
"... T he essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. In the seventeenth and eighteenth centuries, the groups that arose in this connection were t ..."
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T he essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. In the seventeenth and eighteenth centuries, the groups that arose in this connection were the circle R/2#Z , the line R , and finite abelian groups. Embedded in applications were decompositions of functions in terms of multiplicative characters, continuous homomorphisms of the group into the nonzero complex numbers. In the case of the circle, the decomposition is just the expansion of a function on (#,#) into its Fourier series Anthony W. Knapp is professor of mathematics at the State University of New York, Stony Brook. His email address is aknapp@ccmail.sunysb.edu. The author expresses his appreciation to Sigurdur Helgason,
Artin Reciprocity And Mersenne Primes
"... On March 3, 1998, the centenary of Emil Artin was celebrated at the Universiteit van Amsterdam. This paper is based on the two morning lectures, entitled `Artin reciprocity and quadratic reciprocity' and `Class field theory in practice', which were delivered by the authors. It provides an element ..."
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On March 3, 1998, the centenary of Emil Artin was celebrated at the Universiteit van Amsterdam. This paper is based on the two morning lectures, entitled `Artin reciprocity and quadratic reciprocity' and `Class field theory in practice', which were delivered by the authors. It provides an elementary introduction to Artin reciprocity and illustrates its practical use by establishing a recently observed property of Mersenne primes.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
On Artin's Conjecture for Primitive Roots
, 1993
"... Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ) valid for the range l < log x is proven. ..."
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Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ) valid for the range l < log x is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are determined. RESUM E Nous considerons di#erentes generalisations de la conjecture d'Artin pour les racines primitives. Nous demontrons que pour au moins la moitie des nombres premiers p, les premiers log p nombres premiers engendrent une racine primitive. Nous demontrons une version uniforme du Theoreme de Densite de Chebotarev pour le corps Q(# l , 2 ) pour l'intervalle l < log x. On etablit une formule asymptotique uniforme pour les nombres de premiers plus petits que x tels qu' il existe une racine primitive plus petite que s. Nous determinons des minorants pour l'exposant du groupe de classe des corps quadratiques imaginaires valides pour ensembles de discriminants de densite 1. Contents
On the Order of Finitely Generated Subgroups of Q*(mod ρ) and Divisors of ρ1
, 1996
"... Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote t ..."
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Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote the subgroup of Q* generated by a 1 , ..., a r and let 1 p  denote the order of such a group 1 (mod p). In the case r=1, 1=(a), let ord p (a) denote the order of a (mod p). The famous Artin Conjecture for primitive roots (see [1]) states that ord p (a)=p&1 for infinitely many primes p. Artin's Conjecture has been proved under the assumption of the Generalized Riemann Hypothesis by C. Hooley (See [13]). In his paper it is implicitly shown (unconditionally) that ord p (a)>p#log p (1.1) for all but O(x#log x) primes p#x. article no. 0044 207 0022314X#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Supported in part by C.I.C.M.A.
unknown title
"... Emil Artin was born on March 3, 1898 in Vienna, as the son of an art dealer and an opera singer, and he died on December 20, 1962 in Hamburg. He was one of the founding fathers of modern algebra. Van der Waerden acknowledged his debt to Artin and to Emmy Noether (1882–1935) on the title page of his ..."
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Emil Artin was born on March 3, 1898 in Vienna, as the son of an art dealer and an opera singer, and he died on December 20, 1962 in Hamburg. He was one of the founding fathers of modern algebra. Van der Waerden acknowledged his debt to Artin and to Emmy Noether (1882–1935) on the title page of his Moderne Algebra (1930–31), which indeed was originally conceived to be jointly written with Artin. The single volume that contains Artin’s collected papers, published in 1965 [1], is one of the other classics of twentieth century mathematics. Artin’s two greatest accomplishments are to be found in algebraic number theory. Here he introduced the Artin Lfunctions (1923) [2], which are still the subject of a major open problem, and he formulated (1923) [2] and proved (1927) [3] Artin’s reciprocity law, to which the present paper is devoted. Artin’s reciprocity law is one of the cornerstones of class field
1.1.2 Some Elementary Results on the Distribution of Primes............... 8
"... April 1, 2002I would like to thank my advisor, Professor Benedict Gross, for all of his advice and suggestions. ..."
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April 1, 2002I would like to thank my advisor, Professor Benedict Gross, for all of his advice and suggestions.
WWW:http://www.cl.cam.ac.uk/~lp15 Lectures on
, 2007
"... We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel ..."
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We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the wellknown correction of Herbrand’s False Lemma by Gödel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand’s Modus Ponens Elimination. Besides Herbrand’s Fundamental Theorem and its relation to the Löwenheim–Skolem Theorem, we carefully investigate Herbrand’s notion of intuitionism in connection with his notion of falsehood in an infinite domain. We sketch Herbrand’s two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the
The Riemann Hypothesis for Elliptic Curves
"... The Riemann zeta function ζ(s) is defined, for Re(s)> 1, by ζ(s) = ..."