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Topic: Gauss sums

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	Carl Friedrich Gauss - Wikipedia, the free encyclopedia
		Gauss was a child prodigy, of whom there are many anecdotes pertaining to his astounding precocity while a mere toddler, and made his first ground-breaking mathematical discoveries while still a teenager.
		Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone.
		Gauss predicted correctly the position at which it could be found again, and it was rediscovered by Franz Xaver von Zach on December 31, 1801 in Gotha, and one day later by Heinrich Olbers in Bremen.
	en.wikipedia.org /wiki/Carl_Friedrich_Gauss (2363 words)

	Gaussian period - Wikipedia, the free encyclopedia
		Squaring P as a sum leads to a counting problem, about how many quadratic residues are followed by quadratic residues, that can be solved by elementary methods (as we would now say, it computes a local zeta-function, for a curve that is a conic).
		(Gauss sums are in a sense the finite field analogues of the gamma function.)
		Gauss sums can therefore be written as linear combinations of Gaussian periods, with coefficients χ(a); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)
	en.wikipedia.org /wiki/Gaussian_period (733 words)

	PlanetMath: derivation of Gauss sum up to a sign
		The Gauss sum can be easily evaluated up to a sign by squaring the original series
		"derivation of Gauss sum up to a sign" is owned by bbukh.
		This is version 5 of derivation of Gauss sum up to a sign, born on 2003-06-03, modified 2003-09-09.
	planetmath.org /encyclopedia/DerivationOfGaussSumUpToASign.html (85 words)

	Talk Abstract: Gauss sums, Jacobi sums and p-ranks of cyclic difference sets (Site not responding. Last check: 2007-10-20)
		Gauss sums, Jacobi sums and p-ranks of cyclic difference sets
		Stickelberger's theorem for Gauss sums is used to reduce the computation of these 2-ranks to a problem of counting certain cyclic binary strings of length d.
		This counting problem is then solved combinatorially, with the aid of the transfer matrix method.
	www.ima.umn.edu /cc/wkshp_abstracts/xiang1.html (184 words)

	Laurence R. Taylor Preprints (Site not responding. Last check: 2007-10-20)
		We consider Gauss sums associated to functions $T\to Q/Z$ which satisfy some sort of "quadratic'' property and investigate their elementary properties.
		These properties and a Gauss sum formula from the nineteenth century due to Dirichlet enable us to give elementary proofs of many standard results.
		We derive the Milgram Gauss sum formula computing the signature mod 8 of a non{--}singular bilinear form over $Q$ and its generalization to non{--}even lattices.
	www.nd.edu /~taylor/preprints.html (243 words)

	Amazon.ca: Books: Introduction to Analytic Number Theory (Site not responding. Last check: 2007-10-20)
		The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula.This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation.
		Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed.
		The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains.
	www.amazon.ca /exec/obidos/ASIN/0387901639 (1312 words)

	Reciprocity Laws
		Gauss and Jacobi sums are introduced and studied, and we also present a short discussion of Eisenstein sums.
		Apart from giving the proofs of rational quartic reciprocity by comparing the splitting of primes in certain extensions, of the full quartic reciprocity law in Z[i] using Gauss sums, and of the quartic reciprocity law in the field of eighth roots of unity, we discuss several applications.
		The proof uses the octic reciprocity law, whose proof is based on octic elliptic Gauss sums.
	www.rzuser.uni-hd.de /~hb3/cont.html (1095 words)

	Gauss and Jacobi Sums (Canadian Mathematical Society Series of Monographs and Advanced Texts, Vol 21)
		Book Description Devised in the 19th century, Gauss and Jacobi Sums are classical formulas that form the basis for contemporary research in many of today's sciences.
		Synopsis Gauss and Jacobi sums are the result of work by 19th century number theorists.
		These classical formulae form the basis for contemporary research in the solution of large-scale computational problems.This text provides an introduction to topics covered with exercises which are a combination of a computational nature and drawn from research papers.
	www.uni-protokolle.de /buecher/isbn/0471128074 (203 words)

	The Mathematical Institute Eprints Archive - Kummer's conjecture for cubic Gauss sums (Site not responding. Last check: 2007-10-20)
		This improves on the estimate established by Heath-Brown and Patterson in demonstrating the uniform distribution of the cubic Gauss sums around the unit circle.
		Heath-Brown, D.R. Kummer's conjecture for cubic Gauss sums.
		The proof uses a cubic analogue of the author's mean value estimate for quadratic character sums.
	eprints.maths.ox.ac.uk /archive/00000158 (165 words)

	PlanetMath: Gauss sum
		derivation of Gauss sum up to a sign
		This is version 4 of Gauss sum, born on 2002-06-22, modified 2003-10-18.
		(Number theory :: Exponential sums and character sums :: Gauss and Kloosterman sums; generalizations)
	planetmath.org /encyclopedia/GaussSum.html (103 words)

	Katz, N.M.: Exponential Sums and Differential Equations. (AM-124).
		This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them.
		These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields.
		The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.
	www.pupress.princeton.edu /titles/4732.html (151 words)

	Exponential Sums (Site not responding. Last check: 2007-10-20)
		We will be focusing on Exponential sums in Number Theory and, in particular, on rational exponential sums which are sums of roots of unity.
		We will discuss how they are used and the basic questions about exponential sums, for instance, estimates for their absolute values (both complex and p-adic).
		Exponential sums, L-series and the Riemann hypothesis for curves.
	www.ma.utexas.edu /users/voloch/expsums.html (127 words)

	Cyclotomic units, Gauss sums, and Iwasawa invariants of certain real abelian fields (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Using cyclotomic units and Gauss sums, we compute Iwasawa invariants, especially # p (k) of certain real abelian fields.
		Mathematics Subject Classification (2000): 11R23 1 Introduction Let p be a prime number and k a finite extension of the rational number field Q.
		On the Gauss equation in the exterior algebra - Agaoka (1984)
	citeseer.ist.psu.edu /406180.html (460 words)

	Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets (ResearchIndex)
		5 Gauss and Jacobi sums (context) - Berndt, Evans et al.
		1 On congruences arising from relative Gauss sums (context) - Yamamoto
	citeseer.ist.psu.edu /484377.html (515 words)

	The Asymptotic Distribution of Exponential Sums, I, S. J. Patterson
		It has been possible, for a long time, to estimate these sums efficiently.
		On the other hand, when the degree of {$f(x)$} is greater than 2 very little is known about their asymptotic distribution, even though their history goes back to C. Gauss and E.
		We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1067634728 (184 words)

	Errata and remarks for Gauss and Jacobi sums, by Berndt, Evans, Williams
		Errata and remarks for Gauss and Jacobi sums, by Berndt, Evans, Williams
		41, line 13, replace "The inner sum on y vanishes" by "The inner sum on x vanishes".
		Throughout the text, the possessive form Gauss' should be replaced by Gauss's.
	math.ucsd.edu /~revans/errata/errata.html (452 words)

	Gauss Sums over Quasi-Frobenius Rings
		These sums have been extensively studied in the context of the finite fields.
		is a Galois ring of characteristic four, we get a non-archimedian estimate of these sums that generalizes the congruences of Stickelberger.
		is larger than the sum of the binary digits of
	www.univ-tln.fr /~langevin/EXPOSES/GSQF/abstract.html (153 words)

	Alibris: Nicholas M Katz (Site not responding. Last check: 2007-10-20)
		His idea was to study "nth order linear differential equations by studying the rank "n local systems (of local holomorphic solutions) to which they gave rise.
		His first application was to study the classical Gauss hypergeometric function, which...
		It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions).
	www.alibris.com /search/books/author/Nicholas_M_Katz (392 words)

	Carl Friedrich Gauss - Computing Reference - eLook.org
		Gauss did it in seconds, having noticed that 1+...+100 = 100+...+1 = (101+...+101)/2.
		He did important work in almost every area of mathematics.
		He nearly went into architecture rather than mathematics; what decided him on mathematics was his proof, at age 18, of the startling theorem that a regular N-sided polygon can be constructed with ruler and compasses if and only if N is a power of 2 times a product of distinct Fermat primes.
	www.elook.org /computing/carl-friedrich-gauss.htm (265 words)

	Gauss and Jacobi Sums (Site not responding. Last check: 2007-10-20)
		Devised in the 19th century, Gauss and Jacobi Sums are classical formulas that form the basis for contemporary research in many of today's sciences.
		This book offers readers a solid grounding on the origin of these abstract, general theories.
		Though the main focus is on Gauss and Jacobi, the book does explore other relevant formulas, including Cauchy.
	store.totalbookstore.com /0471128074.html (57 words)

	Supplement to the paper "Gauss Sums, Jacobi Sums, and $p$-ranks of Cyclic Difference Sets" (Site not responding. Last check: 2007-10-20)
		Supplement to the paper "Gauss Sums, Jacobi Sums, and p-ranks of Cyclic Difference Sets" by Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang
		Below we provide Maple and Mathematica inputs for generating the adjacency matrices which are used in the proofs of Theorems 4.6 and 4.8, and we provide the proof of the fact (mentioned in the Remark on page 30) that the number A_{\sigma+\gamma}(3u) is never a power of 3.
		Here is the proof of the above mentioned fact in the Remark on page 33, together with the computer data on which it is based.
	www.mat.univie.ac.at /~kratt/artikel/glynn.html (199 words)

	Gauss and Jacobi Sums - Reviewscout.co.uk (Site not responding. Last check: 2007-10-20)
		Devised in the 19th century, Gauss and Jacobi Sums are classical formulas
		Gauss and Jacobi sums are the result of work by 19th century number
		The main focus is on Gauss and Jacobi, but the text includes others,
	www.reviewscout.co.uk /0471128074 (186 words)

	2.1 Jacobi and Gauss' sums (Site not responding. Last check: 2007-10-20)
		A convolution of multiplicative characters evaluated at a field element is called a Jacobi sum.
		We will need two more facts on the Jacobi sums: the representation of the Jacobi sum as a product of Gauss sums, and the variance with the tower of field extensions above
		A Gauss sum is by definition a Fourier transform of a multiplicative character:
	bes.homepage.t-online.de /v4/v4node10.html (131 words)

	VARIOUS NUMBER THEORY ANNOUNCEMENTS (Site not responding. Last check: 2007-10-20)
		His proof of the irrationality of the sum of the inverse of the cubes of integers by an exceptionally clever method worthier of his Greek ancestors than of Bourbaki, made him a legend.
		S.J. Patterson: Gauss sums Proofs of the sign of quadratic Gauss sums from Gauss to Hecke; connections with modern results.
		Schwermer: Raeumliche Anschauung in the work of Minkowski Gauss suggested in 1840 to study quadratic forms using lattices; this was taken up by Dirichlet and later by Minkowski.
	www.numbertheory.org /ntw/announcements.html (8562 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Gauss Sums, Kloosterman Sums and Monodromy Groups (Annals of Mathematics Studies) Nicholas M. Katz ISBN: 0691084335
		Gauss Sums, Kloosterman Sums and Monodromy Groups (Annals of Mathematics Studies)
		Please wait while we find you the best price for Gauss Sums, Kloosterman Sums and Monodromy Groups (Annals of Mathematics Studies), this should take no more than 30 seconds.
	www.bookhead.co.uk /0691084335.aspx (82 words)

	CiteULike: Quantum phase uncertainty in mutually unbiased measurements and Gauss sums (Site not responding. Last check: 2007-10-20)
		CiteULike: Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
		Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
		We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters.
	www.citeulike.org /article/116367 (254 words)

	A simple improvement of a theorem of Pólya (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: > are known, due to Burgess [2]; however, the proof replies on Weil's proof, in an essential way, of the Riemann hypothesis for finite fields.
		Our proof is more elementary; it uses a change of order in summation involving Gauss sums, and a simple estimation of sums by integrals.
		Then the Gauss sum G is defined as P p\Gamma1 s=0 i s p j ¸ s.
	citeseer.lcs.mit.edu /cai95simple.html (278 words)

	Atlas: Semi-local Units Modulo Gauss Sums by Tatiana Beliaeva (Site not responding. Last check: 2007-10-20)
		In this talk we give a "character by character" relation between the Iwasawa characteristic series of the "+"-part of the class group (on the infinite level of the cyclotomic Z
		This generalize a result obtained by H.Ichimura in the semi-simple case under a certain pseudo-monogenity assumption on the class group ("Local Units Modulo Gauss Sums", J.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakl-64.
	atlas-conferences.com /cgi-bin/abstract/cakl-64 (158 words)

	Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets (Site not responding. Last check: 2007-10-20)
		A 87 (1999), 74-119, the only definitive repository of the content that has been certified and accepted after peer review.
		We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2
		We give further applications of the 2-rank formulas, including the determination of the nonzeros of certain binary cyclic codes, and a criterion in terms of the trace function to decide for which \beta in F
	www.mat.univie.ac.at /~kratt/artikel/binary.html (243 words)

	List of C. Krattenthaler's papers
		Xiang), Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J.
		(with Heng Huat Chan) Recent progress in the study of representations of integers as sums of squares, Bull.
		HYP and HYPQ - Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, J. Symbol.
	www.mat.univie.ac.at /~kratt/papers.html (1640 words)

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