Concept
Prime number theorem
Related concepts (31)
Tweedie distribution
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.
Arithmetic progression
An arithmetic progression or arithmetic sequence ( ()) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.
Natural density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, ] as n grows large. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides.
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long.
Jacques Hadamard
Jacques Salomon Hadamard (adamaʁ; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique.
Mertens function
In number theory, the Mertens function is defined for all positive integers n as where is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows: Less formally, is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number.
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression and Dirichlet's theorem states that this sequence contains infinitely many prime numbers.
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number with A less restrictive formulation is: for every , there is always at least one prime such that Another formulation, where is the -th prime, is: for This statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all integers . His conjecture was completely proved by Chebyshev (1821–1894) in 1852 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem.
Chebyshev's bias
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. Let π(x; n, m) denote the number of primes of the form nk + m up to x. By the prime number theorem (extended to arithmetic progression), That is, half of the primes are of the form 4k + 1, and half of the form 4k + 3.
Perron's formula
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. Let be an arithmetic function, and let be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for . Then Perron's formula is Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an integer.