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Topic: Euler product

	Euler (Site not responding. Last check: 2007-11-06)
		Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair.
		Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.
		Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Euler.html (4046 words)

	PlanetMath: Euler product
		The product on the right is called the Euler product for the sum on the left.
		This is version 5 of Euler product, born on 2004-02-21, modified 2004-02-22.
		You mention how at $s=2$, the fact that the product of all primes yields an irrational number is a proof of the infinitude of primes.
	planetmath.org /encyclopedia/EulerProduct.html (303 words)

	[No title]
		The theorem states that the product of* * the parametrized Euler characteristic of one fibration with the parametrized* * Reide- meister torsion class of another fibration yields the parametrized Reide* meister torsion class of the product* fibration.
		PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 5 Lemma 2.4.
		PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 21 Lemma 6.5.
	hopf.math.purdue.edu /Dorabiala-Johnson/torsion.txt (8814 words)

	Prime Product Paradox (Site not responding. Last check: 2007-11-06)
		Euler's zeta function, which forms the basis for Riemann's Hypothesis, is the sum of the integers from 1 to infinity raised to a power.
		Euler was also the first to notice that his zeta function can also be expressed in terms of the primes.
		This paradox is concerned with Euler's prime product, and demonstrates a potential loop-hole with the prime product.
	www.users.globalnet.co.uk /~perry/maths/paradox/paradox.htm (464 words)

	Index to On-Line Encyclopedia of Integer Sequences
		Euler transforms: (1) A000070 A000097 A000098 A000237 A000335 A000391 A000417 A000428 A000608 A000710 A000711 A000712
		Euler transforms: (2) A000713 A000714 A000715 A000716 A001372 A001373 A001384 A001385 A001970 A003080 A003094 A004101
		Euler transforms: (3) A004113 A005470 A005750 A007003 A007441 A007562 A007563 A007713 A007714 A007864 A018243 A023871
	www.research.att.com /~njas/sequences/Sindx_Eu.html (367 words)

	Amazon.com: Books: Euler: The Master of Us All (Dolciani Mathematical Expositions, No 22) (Dolciani Mathematical ... (Site not responding. Last check: 2007-11-06)
		For example, Euler invented the field of analytical number theory, and he was the first mathematician to recognize the importance of and to discover the important properties of complex numbers.
		For example, we learn that Euler began to loose the sight in his right eye at the age of 32, and that despite his virtual blindness by the age of 65, he continued his prolific rate of output until his death at age 84.
		Euler's original work continued to be published more than fifty years after his death in 1783 and his Opera Omnia has only begun to be gathered in the 20th century.
	www.amazon.com /exec/obidos/tg/detail/-/0883853280?v=glance (1778 words)

	Homogeneity and Euler's Theorem
		Phillip Wicksteed (1894) stated the "product exhaustion" thesis implied by the marginal productivity theory of distribution - namely, that if all agents were paid their marginal product, then total costs would exhaust the entire product.
		Wicksteed assumed constant returns to scale - and thus employed a linear homogeneous production function, a function which was homogeneous of degree one.
		It was A.W. Flux (1894) who pointed out that Wicksteed's "product exhaustion" thesis was merely a restatement of Euler's Theorem.
	cepa.newschool.edu /het/essays/math/euler.htm (324 words)

	American Mathematical Monthly, The: Quantum Calculus (Site not responding. Last check: 2007-11-06)
		Euler was next motivated to consider the reciprocal of the infinite product for the generating function of p(m).
		Jacobi gave some remarkable applications of the triple product identity to elliptic functions and to number theory-in particular, to the problem of finding the number of representations of a number as a sum of squares.
		We therefore suspect that Euler's formulas must be particular cases of a nonterminating q-binomial theorem, and this is in fact true.
	www.findarticles.com /p/articles/mi_qa3742/is_200308/ai_n9300777 (1383 words)

	Open Questions: The Riemann Hypothesis
		Euler was (apparently) the first to realize that this fact could be expressed as an identity between an infinite sum and an infinite product.
		Given that there are in fact two product formulas for ζ(s), that one involves the prime numbers, and that the other involves the zeros of ζ(s), it's hard to avoid the suspicion that there is some relationship between the distribution of the primes and the distribution of the zeros.
		Using the alternative product formula for ζ(s) that Riemann conjectured and Hadamard proved, it could be shown that the precise distribution of the zeros of ζ(s) affected the goodness of the approximation of π(x) by Li(x).
	www.openquestions.com /oq-ma014.htm (14106 words)

	Euler product -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		The name arose from the case of the (Click link for more info and facts about Riemann zeta function) Riemann zeta function, where such a product representation was proved by (Swiss mathematician (1707-1783)) Euler.
		An important special case is that in which P(p,s) is a (A geometric progression written as a sum) geometric series, because a(n) is (Click link for more info and facts about totally multiplicative) totally multiplicative.
		In the theory of (Click link for more info and facts about modular form) modular forms it is typical to have Euler products with quadratic polynomials in the denominator here.
	www.absoluteastronomy.com /encyclopedia/e/eu/euler_product.htm (342 words)

	The Zeta Function (Site not responding. Last check: 2007-11-06)
		Euler expressed the zeta function as a remarkable infinite product, 1/ζ(s) = Π(1 - p
		It is typical of Euler's brilliance to have seen this.
		The variable s can be complex, and ζ(s) is an analytic function of s (it is a uniformly convergent series of analytic functions), but we shall confine ourselves here only to real values of s.
	www.du.edu /~jcalvert/math/zeta.htm (266 words)

	Read This: Euler: The Master Of Us All
		Euler had already solved the "Basel problem," summing the reciprocals of the square integers.
		The factors in the product representation have all the prime numbers in their numerators, and, for a product to diverge, it must be an infinite product.
		The Number Theory chapter tells us about Euler's sigma function, the sum of all the divisors of n, about his work with perfect and amicable numbers and his discovery that the fifth Fermat number is composite.
	www.maa.org /reviews/master.html (722 words)

	Rotations3D (Site not responding. Last check: 2007-11-06)
		A 3D rotation R moves a 3D point P1 in a rigid manner to a 3D point P2, around a 3D axis Δ and with an amplitude equal to an angle θ, such that P2 = R[Δ,θ](P1).
		Euler angles representation suffers from the so-called "gimbal lock", that you can experience in PERM, that consists, for the rotated object, in not rotating around initial Y or X axes when theta or psi are modified.
		Quaternions were imagined by Hamilton in 1866, on a sunny morning in the deep mountains.
	www-timc.imag.fr /Antoine.Leroy/tutoriaux/Math/Rotations3D.html (718 words)

	Review of "Prime Obsession"
		In 1737 the Swiss mathematician Leonhard Euler announced that he had found that the sum of the reciprocals of the squares of positive integers is pi-squared / 6.
		Euler went on, finding similar expressions for the sum of the reciprocals of even powers of the positive integers and studying the general properties of the sum of the powers of the positive integers, for an arbitrary exponent.
		Primes and the zeta function remained separate for a while: The former were classified as belonging to arithmetic, and the latter was considered an object in analysis to be studied by calculus.
	www.olimu.com /Riemann/Reviews/AmericanScientist.htm (2391 words)

	Amazon.com: Books: Gamma : Exploring Euler's Constant (Site not responding. Last check: 2007-11-06)
		In an age when a 'computer' is taken to mean a machine rather than a person and calculations of fantastic complexity are routine and executed at lightning speed, constricting difficulties with ordinary arithmetic seem (and are) extremely remote.
		The most frustrating example is the "proof" of Euler's zeta function formula, one of the prettiest pieces of mathematics.
		Product offered violates Amazon.com's policy on items that can be listed for sale.
	www.amazon.com /exec/obidos/tg/detail/-/0691099839?v=glance (1837 words)

	Download Euler Script - Linotype.com (Site not responding. Last check: 2007-11-06)
		The Euler family, which currently consists of nine separate fonts, was designed by Hermann Zapf during the early 1980s.
		AMS Euler Text was a special alphabet family developed for the composition of mathematics, which was commissioned by the American Mathematical Society (AMS) in Providence, Rhode Island, and was created in collaboration with Donald E. Knuth (Department of Computer Sciences, Stanford University).
		The fonts in the Euler family are named in honor of the Swiss mathematician Leonard Euler (1707-1783).
	www.linotype.com /59110/eulerscript-font.html (683 words)

	Read This: Prime Obsession
		A finite product is everywhere convergent, so we have another proof that there are infinitely many primes.
		After describing the Euler product and its implications, the author discusses Riemann's explicit formula for π(x) in terms of the complex zeros ρ = β + iγ of ζ(s).
		He was able to read the whole book, and, although he is not ready to start research in analytic number theory, he did gain some appreciation of the Riemann Hypothesis and of the Euler product formula.
	www.maa.org /reviews/primeobsession.html (1911 words)

	[No title]
		Generalized orbifold Euler characteristics A generalization of physicists' orbifold Euler characteristic (1-2) was give* *n in the introduction in (1-3).
		We recall that for a finite dimensional manifold admi* tting a torus action, it is well known that Euler* characteristic of the fixed point s* ub- manifold is the same as the Euler* characteristic of the original manifold.
		The Euler characteristic of 12 HIROTAKA TAMANOI the fixed point subset under this torus action is precisely given by O(d)p(M; G* *), as in (2-20).
	hopf.math.purdue.edu /Tamanoi/orbifold.txt (7112 words)

	Mathutils (Site not responding. Last check: 2007-11-06)
		Roatate a euler by an amount in degrees around an axis.
		A new quaternion representing the cross product of the two quaternions.
		A new vector representing the cross product of the two vectors.
	www.blender.org /modules/documentation/236PythonDoc/Mathutils-module.html (401 words)

	S.O.S. Mathematics CyberBoard :: View topic - Problem of the Week - "Basel Problem" - 19/11/2004
		The article also states that Euler later devised another proof using elementary calculus, Taylor series and integration by parts, but even that proof does not meet modern standards of rigor.
		On the other hand, jazzmaster's link states that Euler used a numerical method to estimate the value of zeta(2); he recognized the value as being approximately (pi^2)/6, so he subsequently restricted his search to trigonometric functions.
		Euler was able to solve this problem in 1735, when he caused a major sensation by showing that the sum had the unexpected value
	www.sosmath.com /CBB/viewtopic.php?t=11387&start=30 (1956 words)

	chap2
		This chapter opens with the ``Euler product", a remarkable identity that unlocks a whole vista of further ideas.
		This identity encapsulates in analytic form the fact that integers are uniquely expressible as products of primes.
		The definitions of these functions seem a little strange if presented without motivation, but the Euler product leads us to them in a natural way.
	www.maths.lancs.ac.uk /~jameson/chap2 (232 words)

	Some number-theoretical constants: the transformation
		The basic idea, which seems to have been rediscovered independently many times, is to rewrite the original factors as products of factors of the simpler shape
		If the resulting product of powers of zeta values converges, it does so a lot faster than the original Euler product, since the values
		still has a finite product whose rational value can be written down explicitly, but we are outside the domain of convergence of the decomposition, and the individual factors do not approach 1), and while the original formula converges poorly,
	www.gn-50uma.de /alula/essays/Moree/Moree-txfm.en.shtml (513 words)

	Multiplicative relations in powers of Euler's product (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		In a recent paper, Cooper and Hirschhorn conjecture relations among the coe#- cients of certain products of powers of Euler's product.
		Introduction and statement of results Define Euler's product by (q) # := # Y n=1 (1 - q n).
		In a recent paper [C-H], Cooper and Hirschhorn study identities between the coe#cients of products of powers of this product.
	citeseer.ist.psu.edu /331722.html (312 words)

	PlanetMath: Riemann zeta function
		We list here some of the key properties [1] of the zeta function.
		where the product is taken over all positive integer primes
		The Euler product formula (2) given above expresses the zeta function as a product over the primes
	planetmath.org /encyclopedia/RiemannZetaFunction.html (785 words)

	Welcome to Euler
		This is the GTK+ based version of EULER for Unix / Linux systems.
		EULER is a program for quickly and interactively computing with real and complex numbers and matrices, or with intervals, in the style of MatLab, Octave,...
		EULER can read these data form the file and produce plots of these data, fit polynomials to it, do further computations etc.
	euler.sourceforge.net (302 words)

	REU 2002 Robinson
		Following du Sautoy, the group studied the well-known Igusa local zeta function for $f(x,y)=y^2-x^3$ for all $p$ and formed its Euler product $\prod_p Z_p(t)$.
		However, they were unable to show that Re$(s)=-4/5$ was the natural boundary for the Euler product and can only conjecture that it is. These students are all beginning their junior year in college.
		Next we determine the natural boundary for the Euler product of this zeta function.
	www.mtholyoke.edu /~robinson/reu/reu02/reu02.htm (736 words)

	nrich.maths.org::Mathematics Enrichment::NRICH
		According to the message, this was guessed by Euler because sin has roots at n*pi.
		Now continuing on in the crazy spirit Euler started out in why not differentiate both sides with respect to x using (and you're going to love this) the product rule extended to infinite products.
		Now using Euler's infinite product and applying the difference of two squares rule, sin x =...f(-2)f(-1)f(0)f(1)f(2)...
	www.nrich.maths.org.uk /askedNRICH/edited/963wp_l2h.html (390 words)

	[No title]
		In retrospect, given our knowledge that $pi$ is transcendental (this was only proved about a century ago), Euler's evaluation (1.4) of $zt (2)$ explains why Mengoli failed with his attempts; since Mengoli's techniques were based on telescoping series and related elementary methods, he could not obtain an answer such as (1.4).
		In 1737 Euler showed that $zeta (s)$, which is defined by (1.1) for $rRe (s) > 1$, can also be written as.DS 2.EQ (1.5) zt (s) ~=~ PI from p ~ (1-p sup -s) sup -1 ~~~~~rRe (s) > 1 ^,.EN.DE where $p$ runs over the primes.
		This expansion, now called the Euler product of $zeta (s)$, follows from the fact that every positive integer has a unique expansion as a product of prime powers, and so when one expands the product in (1.5) and rearranges the terms, the sum (1.1) results.
	www.dtc.umn.edu /~odlyzko/doc/arch/primes.quantum.troff (4279 words)

	L-functions and elliptic curves
		This is well-defined precisely because n has a unique representation as a product of primes.
		The next simplest kind of Euler product ("quadratic") has factors which are the reciprocals of polynomials that are quadratic in
		A Dirichlet series need not have an Euler product form at all, of course.
	www.mbay.net /~cgd/flt/flt06.htm (2077 words)

	Some number-theoretical constants
		In its original form, the Euler product converges abysmally slow.
		It has been folklore knowledge for some time that it can be transformed into a product of powers of values of the Riemann zeta function
		The trick which makes these computations feasible is to compute the contributions from the small and larger primes separately.
	www.gn-50uma.de /alula/essays/Moree/Moree.en.shtml (1108 words)

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