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Maier's Theorem
In number theory, Maier's theorem is a theorem due to Helmut Maier about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if ''π'' is the prime-counting function and ''λ'' > 1, then :\frac does not have a limit as ''x'' tends to infinity; more precisely the limit superior is greater than 1, and the limit inferior is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when ''λ'' ≥ 2 (using the Borel–Cantelli lemma). Proofs Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound z = x^ , u fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher. gave another proof, and also showed that most probabilistic models of primes incorrectly predict the mean square error :\int_2^Y\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Maier
Helmut Maier (born 17 October 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matrix method as well as Maier's theorem for primes in short intervals. He has also done important work in exponential sums and trigonometric sums over special sets of integers and the Riemann zeta function. Education Helmut Maier graduated with a Diploma in Mathematics from the University of Ulm in 1976, under the supervision of Hans-Egon Richert. He received his PhD from the University of Minnesota in 1981, under the supervision of J. Ian Richards. Research and academic positions Maier's PhD thesis was an extension of his paper ''Chains of large gaps between consecutive primes''. In this paper Maier applied for the first time what is now known as Maier's matrix method. This method later on led him and other mathematicians to the discovery ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal to if is exactly a prime number, and equal to otherwise. That is, the number of prime numbers less than , plus half if equals a prime. Growth rate Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately \frac where is the natural logarithm, in the sense that \lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is \lim_\frac=1 where is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence (x_n) is denoted by \liminf_x_n\quad\text\quad \varliminf_x_n, and the limit superior of a sequence (x_n) is deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel–Cantelli Lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century. A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law. Statement of lemma for probability spaces Let ''E''1, ''E''2, ... be a sequence of events in some probability space. The Borel–Cantelli lemma states: Here, "lim sup" denotes limit supremum of the sequence of events. That is, lim sup ''E''''n'' is the outcome that infinitely many of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Buchstab
Aleksandr Adol'fovich Buchstab (October 4, 1905 – February 27, 1990;. , variously transliterated as Bukhstab, Buhštab, or Bukhshtab) was a Soviet mathematician who worked in number theory and was "known for his work in sieve methods". He is the namesake of the Buchstab function, which he wrote about in 1937. Life and career Buchstab was born in Stavropol; his father was a physician. He studied at the Rostov Polytechnic Institute and Rostov University before moving to the faculty of mechanics and mathematics at Moscow State University, where he earned a degree in 1928. He worked at the Moscow Higher Technical College from 1928 until 1930, and then from 1930 to 1939 at Azerbaijan University, where was the chair of algebra and function theory and then dean of physics and mathematics. During this period, Buchstab also did graduate studies at Moscow State under the supervision of Aleksandr Khinchin. He defended his candidacy in 1939, and at that time was appointed to a profe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patrick X
Patrick may refer to: *Patrick (given name), list of people and fictional characters with this name * Patrick (surname), list of people with this name People *Saint Patrick (c. 385–c. 461), Christian saint * Gilla Pátraic (died 1084), Patrick or Patricius, Bishop of Dublin * Patrick, 1st Earl of Salisbury (c. 1122–1168), Anglo-Norman nobleman * Patrick (footballer, born 1983), Brazilian right-back * Patrick (footballer, born 1985), Brazilian striker *Patrick (footballer, born 1992), Brazilian midfielder * Patrick (footballer, born 1994), Brazilian right-back *Patrick (footballer, born May 1998), Brazilian forward *Patrick (footballer, born November 1998), Brazilian attacking midfielder *Patrick (footballer, born 1999), Brazilian defender * Patrick (footballer, born 2000), Brazilian defender * John Byrne (Scottish playwright) (born 1940), also a painter under the pseudonym Patrick * Don Harris (wrestler) (born 1960), American professional wrestler who uses the ring name Patrick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Square Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maier's Matrix Method
Maier's matrix method is a technique in analytic number theory by Helmut Maier Helmut Maier (born 17 October 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matri ... that is used to demonstrate the existence of intervals of natural numbers within which the prime numbers are distributed with a certain property. In particular, it has been used to prove Maier's theorem and also the existence of chains of large gaps between consecutive primes . The method uses estimates for the distribution of prime numbers in arithmetic progressions to prove the existence of a large set of intervals where the number of primes in the set is well understood and hence that at least one of the intervals contains primes in the required distribution. The method The method first selects a primorial and then constructs an interval in which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
