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Topic: Prime counting function

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Prime number

Prime number theorem

Number theory

Jiuzhang suanshu

Neil Hamburger

	Prime counting function - ExampleProblems.com
		In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x.
		Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).
		One is Riemann's prime counting function, denoted Π(x) or J(x).
	www.exampleproblems.com /wiki/index.php/Prime_counting_function (683 words)

	Prime number theorem - Wikipedia, the free encyclopedia
		The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1.
		This function is related to the logarithm by the asymptotic expansion
		There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem.
	en.wikipedia.org /wiki/Prime_number_theorem (1493 words)

	Prime number - Wikipedia, the free encyclopedia
		For a long time, prime numbers were thought as having no possible application outside of number theory; this changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem or the Diffie-Hellman key-exchange algorithm.
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.
	en.wikipedia.org /wiki/Prime_number (4575 words)

	Prime counting function - Wikipedia, the free encyclopedia
		In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x.
		Of great interest in number theory is the growth rate of the prime counting function.
		Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).
	en.wikipedia.org /wiki/Prime_counting_function (991 words)

	PlanetMath: Mangoldt summatory function
		A number theoretic function used in the study of prime numbers; specifically it was used in the proof of the prime number theorem.
		is the prime counting function, is equivalent to the statement that:
		This is version 5 of Mangoldt summatory function, born on 2003-02-11, modified 2005-04-14.
	planetmath.org /encyclopedia/MangoldtSummatoryFunction.html (199 words)

	Read This: The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics
		Of course, it is not enough just to state the Hypothesis; the reason for the interest in it is its connection with the distribution of prime numbers.
		So we are left with du Sautoy's assertion that primes are important, and with his assertion that they are somehow connected with the ζ-function.
		John Derbyshire, Prime Obsession: Bernhard Riemann and the greatest unsolved problem in mathematics, Joseph Henry Press, 2003, ISBN: 0-309-08549-7.
	www.maa.org /reviews/musicprimes.html (2464 words)

	PlanetMath: non-multiplicative function
		In number theory, a non-multiplicative function is an arithmetic function which is not multiplicative.
		Cross-references: Mangoldt function, prime counting function, negative, positive, integers, squares, sum, multiplicative, arithmetic function, number theory
		This is version 13 of non-multiplicative function, born on 2002-06-12, modified 2006-09-02.
	planetmath.org /encyclopedia/PartitionFunction.html (114 words)

	Introduction to twin primes and Brun's constant computation
		However among the deeply studied set of primes there is a famous and fascinating subset for which very little is known and has generated some famous conjectures: the twin primes (the term prime pairs was used before [5]).
		Based on heuristic considerations, a law (the twin prime conjecture) was developed, in 1922, by Godfrey Harold Hardy (1877-1947) and John Edensor Littlewood (1885-1977) to estimate the density of twin primes.
		According to this conjecture the density of twin primes is equivalent to the density of cousin primes.
	numbers.computation.free.fr /Constants/Primes/twin.html (1986 words)

	PlanetMath: prime counting function
		The prime counting function is a non-multiplicative function for any positive real number
		This function is closely related with the Chebyshev's functions
		This is version 8 of prime counting function, born on 2002-06-27, modified 2002-11-24.
	www.planetmath.org /encyclopedia/PrimeCountingFunction.html (97 words)

	Functions for Scientific WorkPlace
		Using these functions requires the same knowledge that is necessary for their use in Maple itself, namely, the Maple function name and the exact calling sequence.
		The Prime Number Theorem states that the prime counting function pcf(x) is asymptotic to x/ln(x).
		For example, twinprimes(3..5) returns 2 because 3 and 5 is a pair of twin primes and 5 and 7 is a pair of twin primes, and the first number of each pair is in the interval [3,5].
	www.artsci.wustl.edu /~bblank/swp (2625 words)

	Ivars Peterson's MathTrek - The Mark of Zeta
		There are 4 primes among the first 10 integers; 25 among the first 100; 168 among the first 1,000; 1,229 among the first 10,000; 9,592 among the first 100,000; and 78,498 among the first 1,000,000.
		As you move from one power of 10 to the next, the ratio of the prime count to the stopping-point number is about 1 to 2.3.
		Although the expression N/log N is a good approximation for the prime count, it isn't as accurate as mathematicians would like, and they tried to improve upon it.
	www.maa.org /mathland/mathtrek_6_21_99.html (1349 words)

	Riemann's Zeta Function - Harold M. Edwards (Site not responding. Last check: 2007-10-19)
		Riemann does this by allowing the variable s of the zeta function to be complex, which enables him to prove the functional equation of the zeta function and the product representation of the xi function defined through it.
		Riemann feels that all nontrivial zeros have real part 1/2, but this doesn't really matter right now since the term in the prime density expression depending on the zeros is "periodic" in any case and Riemann thus discards it without much harm when he derives his expression for the number of primes less than x.
		Chapter 12 "Miscellany" includes a proof of the prime number theorem that "is 'elementary' in the technical sense", but, as Edwards admits, it is neither straightforward, nor natural, nor insightful.
	www.bookswap.ws /Content/findonamazonus-Asin-0486417409.html (675 words)

	prime numbers
		If the prime numbers are truly uniform random then, for an infinite amount of them, their Benford counterparts should be evenly distributed among the digits 1 to 9.
		And the prime numbers are the key to quantum chaos because of the connection between the Riemann Zeta function and random matrix theory.
		Thus no one can call the prime numbers random anymore, according to the definition of randomness that there is no deterministic process (Note: but for the prime numbers there are the n-formula's) which can predict better than change what the next element of a given "random" sequence will be.
	www.home.zonnet.nl /galien8/prime/prime.html (1619 words)

	Animation - the prime counting function pi(x) (Site not responding. Last check: 2007-10-19)
		At each step, the current function (shown in yellow) is modified by adding a waveform whose frequency and amplitude are related to the next pair of nontrivial (complex) zeros in a simple and direct way.
		The lower animated graph is the derivative of the function above, and we see the positions of the primes emerging as Dirac delta-type spikes.
		, a logarithmically-weighted prime counting function of great importance (for example in the proof of the prime number theorem.
	www.secamlocal.ex.ac.uk /~mwatkins/zeta/pianim.htm (155 words)

	G-Systems
		The number of all instances that are primes relative to r -1 is φ(r), where φ is the Euler function.
		If g(i) is prime relative to r -1, then its class must be the self class (not nested) and such class has k instances.
		His theorem of enumeration [Preparata,1974] was designed for counting of classes with regard to arbitrary operations of symmetry.
	www.sweb.cz /vladimir_ladma/english/music/articles/dide99.htm (1781 words)

	Catalogue of GP/PARI Functions: Arithmetic functions
		This function also allows vector and matrix arguments, in which case the operation is recursively applied to each component of the vector or matrix.
		The content of a rational function is the ratio of the contents of the numerator and the denominator.
		If x is of type integer, rational, polynomial or rational function, the result is a two-column matrix, the first column being the irreducibles dividing x (prime numbers or polynomials), and the second the exponents.
	pari.math.u-bordeaux.fr /dochtml/html/Arithmetic_functions.html (5134 words)

	Review of "Prime Obsession"
		The Riemann Hypothesis states that "all nontrivial zeros of the Zeta function have real part one-half." Understanding the statement of the hypothesis is Derbyshire's first mission for the reader.
		In short, most functions with a dependent variable, say f(x)=x^2-2x+1, have a value for which if you replace x with this value, the function returns zero.
		The Zeta function has an infinite number of these zeroes and an infinite number of these is "non-trivial." The non-trivial zeroes come from complex number values.
	www.olimu.com /Riemann/Reviews/Slashdot.htm (639 words)

	Improved prime counting function
		A new and improved implementation of the Extended Meissel-Lehmer Algorithm for the calculation of pi (N), the number of primes <= N written in C++ can be found at my webpage at www.cbau.freeserve.co.uk This implementation comes with a test program that calculates various values of pi (N) for N from 10^3 to 10^18.
		All the values from the original paper by Lagarias, Miller and Odlyzko are tested and agree with the published results.
		A very short overview of the mathematics involved: Let phi (N, k) be the number of integers <= N which are not divisible by any of the first primes.
	www.usenet.com /newsgroups/sci.math/msg05666.html (422 words)

	Formulas for and the th Prime
		Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers.
		Ruiz, A functional recurrence to obtain the prime numbers using the Smarandache Prime Function,
		Ruiz, The general term of the prime number sequence and the Smarandache Prime Function.
	www.mathematicsmagazine.com /corresp/Formulaspipn2.htm (293 words)

	[No title]
		Sieves can be used to study statistical data on prime numbers, like counting the number of primes, looking for small and large gaps between consecutive primes, counting the number of twin primes (primes p for which p+2 is also a prime), counting primes with a given form, etc.
		At the end of the 1930's, Cramer proposed a model [4] to construct conjectures on distribution of prime numbers.
		The Cramer model suggest that the prime numbers behave like a random sequence with the same growth constraint (in other words, each number n have a probability 1/log(n) to be prime in this model).
	numbers.computation.free.fr /Constants/Primes/countingPrimes.html (1433 words)

	math lessons - Prime number theorem
		In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers.
		This notation means only that the limit of the quotient of the two functions π(x) and x/ln(x) as x approaches infinity is 1; it does not mean that the limit of the difference of the two functions as x approaches infinity is zero.
		Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.
	www.mathdaily.com /lessons/Prime_number_theorem (481 words)

	Amazon.com: Riemann's Zeta Function: Books: Harold M. Edwards (Site not responding. Last check: 2007-10-19)
		It includes a translation of Riemann's original paper (On the Number of Primes...) which is very nice and most authors now seem to forget to mention (mainly because of the obscure way in which it was written).
		The fifth one includes an error estimation due to Poussin for the prime number theorem, and the equivalent of the Riemann Hypothesis in terms of prime distributions.
		Finally we have my favourite chapter, counting zeros: Hardy's theorem, which says that there are infinitely many zeros in the critical line, which was improved by Littlewood, then later by Selberg, and then by Levinson.
	www.amazon.com /Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409 (1967 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		This function is well defined for those integral vectors b that lie in the nonnegative span of the columns of A (of which we assume that it does not contain a line).
		The asymptotic methods and the shape of 38 of the 40 identities suggest the influence of the 5-dissection of the generating function for the crank of partitions.
		The diagrammatic procedure of calculation of the generating function $\Phi({\bf d}^3;z)$ for the set $\Delta({\bf d}^3)$ is developed.
	www.theoryofnumbers.com /CANT/2005/cant2005_abstract.doc (2488 words)

	F. Conjectures (Math 413, Number Theory)
		Def: Riemann's Zeta function, Z(s), is defined as the analytic extension of
		Tables of the earliest occurrence of each prime gap up to at least 804 have also been tabulated (and can be added to).
		There is also a distributed search for larger twin primes which has a page at http://www.serve.com/cnash/twinsearch.html.
	www.math.umbc.edu /~campbell/Math413Fall98/Conjectures.html (485 words)

	Math 571 Analytic Number Theory I (Site not responding. Last check: 2007-10-19)
		Our objective, starting only from the most elementary considerations, is to study the behaviour of the prime numbers and, in particular, their distribution.
		Here the the sum on the left is over the prime powers p^k not exceeding x and the sum on the right is over the zeros r of the Riemann z-function.
		Chebychev's inequalities for the prime counting function, and Merten's theorem.
	www.math.psu.edu /rvaughan/Math571.htm (222 words)

	Riemann Prime Counting Function -- from Wolfram MathWorld
		jumps by 1; when it is the square of a prime, it jumps by 1/2; when it is a cube of a prime, it jumps by 1/3; and so on (Derbyshire 2004, pp.
		in the Riemann function with the logarithmic integral
		Riemann's function is related to the prime counting function by
	mathworld.wolfram.com /RiemannPrimeCountingFunction.html (497 words)

	Lecture 21 (Site not responding. Last check: 2007-10-19)
		Primes may * pop up in clumps, such as the numbers 101, 103, 107, 109, and 113, or * at huge intervals.
		If n is a prime number, then * the gap to the next prime will, on average, be the natural logarithm of * n, or log n.
		What's more, * there is no upper limit to the number of primes that can squeeze into * the space "allotted" for one.
	www.math.tamu.edu /~mpilant/math646/lecture21.html (430 words)

	Animation - the prime counting function pi(x) (Site not responding. Last check: 2007-10-19)
		At each step, the current function (shown in yellow) is modified by adding a waveform whose frequency and amplitude are related to the next pair of nontrivial (complex) zeros in a simple and direct way.
		The lower animated graph is the derivative of the function above, and we see the positions of the primes emerging as Dirac delta-type spikes.
		, a logarithmically-weighted prime counting function of great importance (for example in the proof of the prime number theorem.
	secamlocal.ex.ac.uk /~mwatkins/zeta/pianim.htm (155 words)

	Prime counting function-- no error. Text - Physics Forums Library
		10-03-2005, 08:36 AM I have developed a prime counting function with no error; it returns the exact number of primes equal to or less than any number one chooses.
		There are unwieldy asymptotics for the nth prime, but the use of p_n in the algorithms you saw there is a notational one.
		If you look at Meissel's basic formula it involves a sum over the primes less than sqrt(x), the time it will take to find these will be inconsequential to the overall time of the algorithm.
	www.physicsforums.com /archive/index.php/t-92067.html (748 words)

	How many primes are there?
		Even though the distribution of primes seems random (there are (probably) infinitely many twin primes and there are (definitely) arbitrarily large gaps between primes), the function pi(x) is surprisingly well behaved: In fact, it has been proved (see the next section) that:
		The Prime Number Theorem: The number of primes not exceeding x is asymptotic to x/log x.
		Gauss was also studying prime tables and came up with a different estimate (perhaps first considered in 1791), communicated in a letter to Encke in 1849 and first published in 1863.
	primes.utm.edu /howmany.shtml (1731 words)

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