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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,
Functional independence of the singularities of a class of Dirichlet series
 Amer. J. Math
, 1998
"... Functional independence of the singularities of a class of Dirichlet series ..."
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Functional independence of the singularities of a class of Dirichlet series
Artin Reciprocity and Mersenne primes
, 2000
"... 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
Artin’s conjecture; Unconditional approach and elliptic analogue
 Master’s thesis
, 2008
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, I have explored the different approach ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, I have explored the different approaches towards proving Artin’s ‘primitive root ’ conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific set of 13 distinct numbers such that for at least one of them, the conjecture is true. Along the same line, using sieve theory, D. R. HeathBrown reduced this set down to 3 distinct primes in the year 1986. This is
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 ..."
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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
The Riemann Hypothesis for Elliptic Curves
"... The Riemann zeta function ζ(s) is defined, for Re(s)> 1, by ζ(s) = ..."
SOME DENSITY QUESTIONS AND AN APPLICATION
"... Abstract. Let S = {a1, a2, · · · , a`} be a finite set of nonzero integers. Recently, R. Balasubramanian et al., ([2], 2010) computed the density of those primes p such that ai is a quadratic residue (respectively, nonresidue) modulo p for every i. As an application of this result, the proved ..."
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Abstract. Let S = {a1, a2, · · · , a`} be a finite set of nonzero integers. Recently, R. Balasubramanian et al., ([2], 2010) computed the density of those primes p such that ai is a quadratic residue (respectively, nonresidue) modulo p for every i. As an application of this result, the proved an exact formula for the degree of the multiquadratic field Q(√a1,√a2,...,√a`) over Q. In this lecture notes, we give an expository of the above result together with all the preliminaries that needed. 1.