Dirichlet eta function. In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s).
The following simple relation holds: Equivalently, we may begin by defining which is also defined in the region of positive real part. This gives the eta function as a Mellin transform. Hardy gave a simple proof of the functional equation for the eta function, which is From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane. Zeros[edit] adds an infinity of complex simple zeros, located at equidistant points on the line , at where n is any nonzero integer. where.
Mathematics | Number Systems. Hilbert's paradox of the Grand Hotel. The paradox[edit] Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.
Finitely many new guests[edit] Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. Infinitely many new guests[edit] It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests. Infinitely many coaches with infinitely many guests each[edit] , and their coach number to be and where.
Hyperreal number. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1] The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity.
The transfer principle states that true first order statements about R are also valid in *R. For all integers n, one also has for all hyperintegers H. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion.[2] In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were.
For an infinitesimal The transfer principle[edit] Square-free integer. In mathematics, a square-free, or quadratfrei integer, is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The smallest positive square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ...
(sequence A005117 in OEIS) Equivalent characterizations[edit] The positive integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. The positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function. The radical of an integer is always square-free: an integer is square-free if it is equal to its radical. Dirichlet generating function[edit] The Dirichlet generating function for the square-free numbers is where ζ(s) is the Riemann zeta function.
This is easily seen from the Euler product Distribution[edit] Let Q(x) denote the number of square-free (quadratfrei) integers between 1 and x. Riemann zeta function. , which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. This function, as a function of a real argument, was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time. Bernhard Riemann in his article "On the Number of Primes Less Than a Given Magnitude" published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.[1] The values of the Riemann zeta function at even positive integers were computed by Euler.
The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ(3). A058303). Leibniz formula for π. In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that Using summation notation: Names[edit] It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and therefore the value β(1) of the Dirichlet beta function. Proof[edit] Considering only the integral in the last line, we have: Therefore, as n → ∞ we are left with the Leibniz series: for a more detailed proof, together with the original geometric proof by Leibniz himself, see.[3] Inefficiency[edit] Leibniz's formula converges slowly.
For However, the Leibniz formula can be used to calculate π to high precision (hundreds of digits or more) using various convergence acceleration techniques. Which can be evaluated to high precision from a small number of terms using Richardson extrapolation or the Euler–Maclaurin formula. Unusual behavior[edit] where the underlined digits are wrong. Where N is an integer divisible by 4. Euler product[edit] See also[edit] Notes[edit] Number theory. Number theory (or arithmetic[note 1]) is a branch of pure mathematics devoted primarily to the study of the integers, sometimes called "The Queen of Mathematics" because of its foundational place in the discipline.[1] Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).
Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., 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, e.g., as approximated by the latter (Diophantine approximation). The older term for number theory is arithmetic. History[edit] Origins[edit] Dawn of arithmetic[edit] such that .
John Cremona's home page. Fundamental theorem of arithmetic. In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1[1] either is prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.[2][3][4] For example, The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.
The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (e.g. 12 = 2 × 6 = 3 × 4). History[edit] Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.
Proof[edit] . Cyclic number. Details[edit] To qualify as a cyclic number, it is required that successive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, even though all cyclic permutations are multiples: The following trivial cases are typically excluded: single digits, e.g.: 5repeated digits, e.g.: 555repeated cyclic numbers, e.g.: 142857142857 If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal, due to the necessary structure given in the next section.
. (106-1) / 7 = 142857 (6 digits) (1016-1) / 17 = 0588235294117647 (16 digits) (1018-1) / 19 = 052631578947368421 (18 digits) (1022-1) / 23 = 0434782608695652173913 (22 digits) (1028-1) / 29 = 0344827586206896551724137931 (28 digits) (1046-1) / 47 = 0212765957446808510638297872340425531914893617 (46 digits) (1058-1) / 59 = 0169491525423728813559322033898305084745762711864406779661 (58 digits) (1060-1) / 61 = 016393442622950819672131147540983606557377049180327868852459 (60 digits) Special values of L-functions. In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet L-function for the Gaussian field. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼.
There are two families of conjectures, formulated for general classes of L-functions (the very general setting being for L-functions L(s) associated to Chow motives over number fields), the division into two reflecting the questions of: (a) how to replace π in the Leibniz formula by some other "transcendental" number (whether or not it is yet possible for transcendental number theory to provide a proof of the transcendence); and All these conjectures are known only in special cases.
Number theory files for David Eppstein.