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Topic: Mertens function

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Mertens conjecture

Franz Mertens

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39 (number)

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	Mertens function - Wikipedia, the free encyclopedia
		Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no x such that M(x) > x.
		The Mertens conjecture goes even further, stating that there is no x where the absolute value of the Mertens function exceeds the square root of x.
		Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761-830.
	en.wikipedia.org /wiki/Mertens_function (294 words)

	Möbius function
		The classic Möbius function μ(n) is an important multiplicative function considered in number theory and in combinatorics.
		The Möbius function is multiplicative and is of relevance in the theory of multiplicative and arithmetic functions because it appears in the Möbius inversion formula.
		This function is closely linked with the positions of zeroes of the Euler - Riemann ζ- function.
	www.ebroadcast.com.au /lookup/encyclopedia/mo/Moebius_function.html (527 words)

	Mertens biography
		Mertens' family naturally identified with the German aspect of the country while for those of Polish culture the years of the 19th century were a continual battle to struggle for their identity.
		Mertens completed his university studies at the University of Berlin where he attended lectures by Weierstrass, Kronecker and Kummer.
		Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Mertens.html (633 words)

	Möbius function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-01)
		In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by
		This function is closely linked with the positions of zeroes of the Riemann zeta function.
		The classical Möbius function treated in this article is essentially equal to the Möbius function of the set of all positive integers partially ordered by divisibility.
	www.sciencedaily.com /encyclopedia/moebius_function (608 words)

	Franz Mertens - Wikipedia, the free encyclopedia
		Franz Mertens (March 20, 1840 - March 5, 1927) was a German mathematician.
		The Mertens function is the sum function for the Möbius function, in the theory of arithmetic functions.
		The Mertens conjecture concerning its growth, conjecturing it bounded by x
	en.wikipedia.org /wiki/Franz_Mertens (151 words)

	Wim Mertens - The Belgian Pop & Rock Archives
		Mertens aims at a large audience with his music : in an interview with Het Nieuwsblad he said responding to the question "Your debut album was called "For amusement only".
		Mertens' music has its roots in American Minimal Music, a tradition which holds that composition can be achieved through the use of the fewest possible musical devices.
		Often the function of the top-line is to act as a signifier indicative of the meaning of the music.
	houbi.com /belpop/groups/mertens.htm (1175 words)

	Mertens function (Site not responding. Last check: 2007-11-01)
		Because the Möbius function has only the values -1 0 and +1 it's obvious the Mertens function moves slowly and that is no x such that M (x) > x.
		The Mertens conjecture goes even further stating that there no x where the absolute value of the function exceeds the square root of x.
		The Möbius function is built-in to Mathematica the Mertens function is not but can be defined with this command:
	www.freeglossary.com /Mertens_function (309 words)

	Riemann hypothesis - Wikipedia, the free encyclopedia
		for all sufficiently large x, where π(x) is the prime-counting function, and ln(x) is the natural logarithm of x.
		In other words, the importance of 'the Riemann hypothesis' in mathematics today really stems from the importance of the generalized Riemann hypothesis, but it is simpler to refer to the Riemann hypothesis only in its original special case when describing the problem to people outside of mathematics.
		Other functions, such as the Riesz function, have conjectured rates of growth equivalent to the Riemann hypothesis as well.
	www.wikipedia.com /wiki/Riemann_hypothesis (2351 words)

	Open Questions: The Riemann Hypothesis
		Riemann (1826-66) studied the zeta function (including Euler, as we shall see), the notation is Riemann's, and hence the function is commonly known as the zeta function, after the greek letter ζ.
		The numerical values of such functions at special points (and their "residues" at poles) have particular significance in terms of algebraic and arithmetic objects that the functions are associated with.
		It is a consequence of the functional equation of L(s, χ).
	www.openquestions.com /oq-ma014.htm (14106 words)

	800 (number) - Wikipedia, the free encyclopedia
		× 17, Mertens function(850) returns 0, nontotient, the maximum possible Fair Isaac credit score.
		877 prime number, Bell number, Chen prime, Mertens function(877) returns 0, strictly non-palindromic number.
		Also direct dial code for toll-free phone calls in the US.
	en.wikipedia.org /wiki/Number_800 (1286 words)

	Ed Pegg's Math Games - The Möbius Function (and squarefree numbers) (Site not responding. Last check: 2007-11-01)
		The Möbius function, µ(n), is strongly related to the Zeta function ζ(s).
		This is known as Mertens Function, or M(x).
		Weisstein, Eric W. Mertens Function, Möbius function, Primitive Root, Riemann Zeta Function.
	www.maa.org /editorial/mathgames/mathgames_11_03_03.html (1425 words)

	The Hawk Eye Newspaper (Site not responding. Last check: 2007-11-01)
		Job recruitment right now is a function of BADCO, the economic development division of the Chamber of Commerce -- an otherwise special interest organization established to promote business issues and to benefit its dues-paying members.
		The city's tab would be considerable, however, it if took on the job recruitment function itself, which would be wrongheaded.
		Their chief function is to see that the community infrastructure -- streets, sewers and sidewalks, for example -- is in quality shape.
	www.thehawkeye.com /COLUMNS/mertens/2001/bme62401.html (596 words)

	Funzione di Mertens (Site not responding. Last check: 2007-11-01)
		La funzione di Mertens M(n) è calcolata per un dato intero n come la somma dei risultati funzione di Möbius per ogni numero intero da 1 n o per dirla algebricamente
		Dove ci sono lotti di numeri primi e dei numeri sfenici la funzione Mertens potrebbe tuffarsi nel territorio negativo rimbalzante territorio positivo dopo una serie di numeri 2- o 4-fattori e rimanente allo stesso in cui là a lotti dei numeri pieno di quadrati successivi.
		Mertens egli stesso è andato per quanto che ci è no
	it.freeglossary.com /Funzione_di_Mertens (247 words)

	Prosogram
		This tonal perception model was validated in listening experiments using stimuli resynthesized using the stylized contour (Mertens et al, 1997).
		Changes in the spectral properties of the signal tend to function as boundaries (House, 1990), breaking up a voiced continuum into a sequence of syllabic nuclei.
		Mertens, Piet (2004) Un outil pour la transcription de la prosodie dans les corpus oraux.
	bach.arts.kuleuven.be /pmertens/prosogram (1485 words)

	Riemann_hypothesis (Site not responding. Last check: 2007-11-01)
		The Riemann zeta function along the critical line is sometimes studied in terms of the Z function, whose real zeros correspond to the zeros of the zeta function on the critical line.
		Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper ''On the Number of Primes Less Than a Given Magnitude'', but as it was not essential to his central purpose in that paper, he did not attempt a proof.
		The zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the ''explicit formulae'' which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
	goc.subdomain.de /Riemann_hypothesis (1752 words)

	Dr Shaun Stevens
		The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function.
		Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.
		The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.
	www.mth.uea.ac.uk /~h008/research/abstracts/orbits.html (128 words)

	Mertens (print-only) (Site not responding. Last check: 2007-11-01)
		Mertens worked on a number of different topics including potential theory, geometrical applications to determinants, algebra and analytic number theory, publishing 126 papers.
		The result is important since a proof of Mertens' conjecture would imply the truth of the Riemann hypothesis.
		Among Mertens other papers we mention: Invariante Gebilde ternärer Formen (1887); Invariante Gebilde quaternärer Formen (1889); Dirichletscher Reihen (1895); Zur linearen Transformation der q-Reihen (1901); and Beweis der Galois'schen Fundamentalsatzes (1902).
	www-groups.dcs.st-and.ac.uk /history/Printonly/Mertens.html (639 words)

	Problem G - Riemann vs Mertens
		We'll leave Riemann and the zeta function and concern ourselves with the closely related, but much easier to calcultate Mertens's function.
		The zeta function has many zeros (values of s for which zeta(s)=0).
		One way to prove the hypothesis, would be to prove that the Mertens's function is bounded by the square root of it's argument: M(N) = O(sqrt(N)), where O stands for the big-oh notation, which means that for big enough N, the absolute value of M(N) will never exceed sqrt(N).
	acm.uva.es /p/v107/10738.html (587 words)

	600 (number) - Wikipedia, the free encyclopedia
		607 prime number, sum of three consecutive primes (197 + 199 + 211), Mertens function(607) = 0, balanced prime, strictly non-palindromic number, also area code of Ithaca, New York
		635 = 5 × 127, sum of nine consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89), Mertens function(635) = 0
		661 prime number, sum of three consecutive primes (211 + 223 + 227), Mertens function sets new low of -11 which stands until 665, star number
	en.wikipedia.org /wiki/Number_600 (1334 words)

	Definition of 800 (number)
		811 prime number, sum of five consecutive primes (151 + 157 + 163 + 167 + 173), Mertens function(811) returns 0
		877 prime number, Bell number, Mertens function(877) returns 0
		This is the number of nxn magic squares for n = 4.
	www.wordiq.com /definition/800_%28number%29 (1046 words)

	PlanetMath:
		marginal density function (=marginal distribution) owned by mathcam
		marginal probability function (=marginal distribution) owned by mathcam
		mass function (=continuous density function) owned by mathcam
	planetmath.org /encyclopedia/M (1445 words)

	400 (number) (Site not responding. Last check: 2007-11-01)
		401 prime number tetranacci number sum of seven consecutive primes (43 47 + 53 + 59 + 61 67 + 71) sum of nine consecutive (29 + 31 + 37 + 41 43 + 47 + 53 + 59 61) Mertens function returns 0.
		Also area code for Rhode Island also HTTP status code for an request also in the name of a plan 401(k)
		× 101 Mertens function returns 0 Also HTTP status code for file not perhaps the most famous HTTP status code all time
	www.freeglossary.com /Number_400 (1130 words)

	39 (number) (Site not responding. Last check: 2007-11-01)
		It also finds the pi function (given a number, this function returns the number of primes below that number).
		Jewish Encyclopedia: Forty, the Number Reviews the occurrences of the number forty in the Bible, where it is second to the number seven in its frequency of occurrence.
		Forbes Celebrity 100 Find the top celebrities based on such measures as the number of visits to their Web sites, the number of press clips, and the number of times they've appeared on magazine covers, as well as their earnings.
	www.serebella.com /encyclopedia/article-39_(number).html (1603 words)

	The Effect of Reducing Alfalfa Haylage Particle Size on Cows in Early Lactation -- Kononoff and Heinrichs 86 (4): 1445 ...
		Mertens, D. Creating a system for meeting the fiber requirements of dairy cattle.
		Mertens, D. Physically effective NDF and its use in dairy rations explored.
		Effect of physical form of forage on chewing activity, dry matter intake, and rumen function of dairy cows in early lactation.
	jds.fass.org /cgi/content/full/86/4/1445 (5942 words)

	The Life Cycle of Coagulation Factor VIII in View of Its Structure and Function -- Lenting et al. 92 (11): 3983 -- Blood
		One functional aspect of factor VIII-vWF complex formation may be to prevent premature binding of factor VIII to components
		Sudkahar K, Fay PJ: Effects of copper on the structure and function of factor VIII subunits: Evidence for an auxiliary role for copper ions in cofactor activity.
		Mertens K, Bertina RM: Activation of human blood coagulation factor VIII by activated factor X, the common product of the intrinsic and the extrinsic pathway of blood coagulation.
	www.bloodjournal.org /cgi/content/full/92/11/3983 (8449 words)

	400 (number) (Site not responding. Last check: 2007-11-01)
		403 = 13 × 31, Mertens function returns 0.
		Also, HTTP status code for "file not found", perhaps the most famous HTTP status code of all time.
		432 sum of four consecutive primes (103 + 107 + 109 + 113), a highly totient number, sum of totient function for first 37 integers and a Harshad number, Zuckerman number.
	nba.servegame.org /en/451_(number).htm (1828 words)

	zeta.html
		The 10^22-nd zero of the Riemann zeta function, A.
		A nonlinear equation and its application to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices, P. Forrester and A. Odlyzko, in Organic Mathematics, J.
		On the distribution of spacings between zeros of the zeta function, A.
	www.dtc.umn.edu /~odlyzko/doc/zeta.html (409 words)

	id:A100765 - OEIS Search Results
		Numbers n for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both -1.
		This sequence is a subset of A100306, Numbers n for which the values of the Moebius function and the Mertens function agree.
		a(5) = 102 because it is a sphenic (exactly 3 distinct prime factors, A007304) number, so the Mobius function yields -1, and the sum of that value and the previous Mobius values (the Mertens function) is also -1.
	www.research.att.com /~njas/sequences/A100765 (197 words)

	PrimeFan's Listing of Esoteric Integer Sequences
		Numbers n for which the value of Möbius function and Mertens function are the same.
		On another page I list the values of Möbius function and Mertens function for the first 2500 integers.
		Numbers n for which the value of Möbius function and Mertens function are both 0.
	www.geocities.com /primefan/EsotericIntegerSequences.html (4668 words)

	600 (number)
		× 19, Mertens function(608) = 0, nontotient ---- 609 = 3 × 7 × 29, sphenic number ---- 610 = 2 × 5 × 61, sphenic number, nontotient.
		× 13, Mertens function(637) = 0, decagonal number ---- 638 = 2 × 11 × 29, sphenic number, sum of four consecutive primes (151 + 157 + 163 + 167), nontotient, centered heptagonal number ---- 639 = 3
		× 83 ---- 665 = 5 × 7 × 19, sphenic number, Mertens function sets new low of -12 which stands until 1105 ---- 666 has its own article.
	en.mcfly.org /600+%28number%29 (856 words)

	Number Theory - Physics Forums Library
		Dirichlet series inversion and prime number coutnign function...
		Prove that an integer with digits '1' is not a perfect square.
		Integral equation of second kind and prime number counting function
	www.physicsforums.com /archive/index.php/f-80.html (554 words)

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