Sort by:

There exist a variety of formulas for either producing the th prime as a function of or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers ), there exists a constant (OEIS A051021) known as Mills' constant such that(1)where is the floor function, is prime for all (Ribenboim 1996, p. 186). The first few values of are 2, 11, 1361, 2521008887, ... (OEIS A051254). It..

The distinct prime factors of a positive integer are defined as the numbers , ..., in the prime factorization(1)(Hardy and Wright 1979, p. 354).A list of distinct prime factors of a number can be computed in the Wolfram Language using FactorInteger[n][[All, 1]], and the number of distinct prime factors is implemented as PrimeNu[n].The first few values of for , 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (OEIS A001221; Abramowitz and Stegun 1972, Kac 1959). This sequence is given by the inverse Möbius transform of , where is the characteristic function of the prime numbers (Sloane and Plouffe 1995, p. 22). The prime factorizations and distinct prime factors of the first few positive integers are listed in the table below.prime factorizationdistinct prime factors (A027748)1--0--221233134125515622, 377178129131022, 511111111222, 313131131422, 71523, 51612The numbers consisting only of distinct..

The least common multiple of two numbers and , variously denoted (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), (Gellert et al. 1989, p. 25; Graham et al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West 2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325; Wolfram Language), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that(1)The least common multiple of more than two numbers is similarly defined.The least common multiple of , , ... is implemented in the Wolfram Language as LCM[a, b, ...].The least common multiple of two numbers and can be obtained by finding the prime factorization of each(2)(3)where the s are all prime factors of and , and if does not occur in one factorization, then the corresponding exponent is taken as 0. The least..

The Mangoldt function is the function defined by(1)sometimes also called the lambda function. has the explicit representation(2)where denotes the least common multiple. The first few values of for , 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).The Mangoldt function is implemented in the WolframLanguage as MangoldtLambda[n].It satisfies the divisor sums(3)(4)(5)(6)where is the Möbius function (Hardy and Wright 1979, p. 254).The Mangoldt function is related to the Riemann zeta function by(7)where (Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).The summatory Mangoldt function, illustratedabove, is defined by(8)where is the Mangoldt function, and is also known as the second Chebyshev function (Edwards 2001, p. 51). is given by the so-called explicit formula(9)for and not a prime or prime power (Edwards 2001, pp. 49, 51, and 53), and the sum is over all nontrivial..

The so-called explicit formulagives an explicit relation between prime numbers and Riemann zeta function zeros for and not a prime or prime power. Here, is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros of the Riemann zeta function , i.e., those in the critical strip so (Montgomery 2001).

A number theoretic character, also called a Dirichlet character (because Dirichlet first introduced them in his famous proof that every arithmetic progression with relatively prime initial term and common difference contains infinitely many primes), modulo is a complex function for positive integer such that(1)(2)(3)for all , and(4)if . can only assume values which are roots of unity, where is the totient function.Number theoretic characters are implemented in the Wolfram Language as DirichletCharacter[k, j, n], where is the modulus and is the index.

A linear transformation is hyperbolic if none of its eigenvalues has modulus 1. This means that can be written as a direct sum of two -invariant subspaces and (where stands for stable and for unstable) such that there exist constants , , and with(1)(2)for , 1, ....

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.For example, the integral(1)is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values(2)and then taking the limit as ,(3)(4)(5)Improper integrals of the form(6)with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If decreases at least as fast as , then let(7)(8)(9)(10)and(11)(12)If diverges as for , let(13)(14)(15)(16)(17)and(18)If diverges as for , let(19)(20)(21)and(22)(23)If the integral diverges exponentially, then let(24)(25)(26)and(27)..

The Pippenger product is an unexpected Wallis-like formula for given by(1)(OEIS A084148 and A084149; Pippenger 1980). Here, the th term for is given by(2)(3)where is a double factorial and is the gamma function.

Consider the Euler product(1)where is the Riemann zeta function and is the th prime. , but taking the finite product up to , premultiplying by a factor , and letting gives(2)(3)where is the Euler-Mascheroni constant (Havil 2003, p. 173). This amazing result is known as the Mertens theorem.At least for , the sequence of finite products approaches strictly from above (Rosser and Schoenfeld 1962). However, it is highly likely that the finite product is less than its limiting value for infinitely many values of , which is usually the case for any such inequality due to the presence of zeros of on the critical line . An example is Littlewood's famous proof that the sense of the inequality , where is the prime counting function and is the logarithmic integral, reverses infinitely often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend [this] result to show that [the Mertens inequality] fails for large ; we have not investigated..

The Smarandache function is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that gives the smallest value for a given at which (i.e., divides factorial). For example, the number 8 does not divide , , , but does divide , so .For , 2, ..., is given by 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (OEIS A002034), where it should be noted that Sloane defines , while Ashbacher (1995) and Russo (2000, p. 4) take . The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A046022), which occur at the values where . The incrementally smallest values of relative to are = 1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 3/40, 1/15, 1/16, 1/24, 1/30, ... (OEIS A094404 and A094372), which occur at , 6, 12, 20, 24, 40, 60, 80, 90, 112, 120, 180, ... (OEIS A094371).Formulas exist for immediately computing for special forms of . The simplest cases are(1)(2)(3)(4)(5)where is a prime,..

The binomial coefficient is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number. The symbols and are used to denote a binomial coefficient, and are sometimes read as " choose ." therefore gives the number of k-subsets possible out of a set of distinct items. For example, The 2-subsets of are the six pairs , , , , , and , so . The number of lattice paths from the origin to a point ) is the binomial coefficient (Hilton and Pedersen 1991).The value of the binomial coefficient for nonnegative and is given explicitly by(1)where denotes a factorial. Writing the factorial as a gamma function allows the binomial coefficient to be generalized to noninteger arguments (including complex and ) as(2)For nonnegative integer arguments, the gamma function reduces to factorials, leading to(3)which is Pascal's triangle. Using the symmetryformula(4)for integer , and complex , this..

A subspace of is called a retract of if there is a continuous map (called a retraction) such that for all and all , 1. , and 2. . Equivalently, a subspace of is called a retract of if there is a continuous map (called a retraction) such that for all ,

The th Ramanujan prime is the smallest number such that for all , where is the prime counting function. In other words, there are at least primes between and whenever . The smallest such number must be prime, since the function can increase only at a prime.Equivalently,Using simple properties of the gamma function, Ramanujan (1919) gave a new proof of Bertrand's postulate. Then he proved the generalization that , 2, 3, 4, 5, ... if , 11, 17, 29, 41, ... (OEIS A104272), respectively. These are the first few Ramanujan primes.The case for all is Bertrand's postulate.

The exponential factorial is defined by the recurrencerelation(1)where . The first few terms are therefore(2)(3)(4)(5)... (OEIS A049384). The term has digits.The exponential factorial is therefore a kind of "factorial powertower."The sum of the reciprocals of the exponential factorials is given by(6)(7)(OEIS A080219). This sum is a Liouvillenumber and is therefore transcendental.

The Wallis formula follows from the infinite productrepresentation of the sine(1)Taking gives(2)so(3)(4)(OEIS A052928 and A063196).An accelerated product is given by(5)(6)where(7)(Guillera and Sondow 2005, Sondow 2005). This is analogous to the products(8)and(9)(Sondow 2005).A derivation of equation (◇) due to Y. L. Yung (pers. comm., 1996; modified by J. Sondow, pers. comm., 2002) defines(10)(11)(12)where is a polylogarithm and is the Riemann zeta function, which converges for . Taking the derivative of (11) gives(13)which also converges for , and plugging in then gives(14)(15)(16)Now, taking the derivative of the zeta function expression (◇) gives(17)and again setting yields(18)(19)(20)(21)where(22)(OEIS A075700) follows from the Hadamard product for the Riemann zeta function. Equating and squaring (◇) and (◇) then gives the Wallis formula.This derivation of the..

Integrals over the unit square arising in geometricprobability are(1)which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.Unit square integrals involving the absolute valueare given by(2)(3)for and , respectively.Another simple integral is given by(4)(Bailey et al. 2007, p. 67). Squaring the denominator gives(5)(6)(7)(8)(9)(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. A related integral is given by(10)which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives(11)(12)(13)(14)(OEIS A093753), where is Catalan's constant.A collection of beautiful integrals over the unit squareare given by Guillera and Sondow..

A plane partition is a two-dimensional array of integers that are nonincreasing both from left to right and top to bottom and that add up to a given number . In other words,(1)(2)and(3)Implicit in this definition is the requirement that the array be flush on top and to the left and contain no holes.(4)For example, one plane partition of 22 is illustrated above.The generating function for the number of planar partitions of is(5)(OEIS A000219, MacMahon 1912b, Speciner 1972,Bender and Knuth 1972, Bressoud and Propp 1999).Writing , a recurrence equation for is given by(6)where is a divisor function. It also has generating function(7)MacMahon (1960) also showed that the number of plane partitions whose Young tableaux fit inside an rectangle and whose integers do not exceed (in other words, with all ) is given by(8)(Bressoud and Propp 1999, Fulmek and Krattenthaler 2000). Expanding out the products gives(9)(10)where is the Barnes G-function...

The constant in the Laurent series(1)of about a point is called the residue of . If is analytic at , its residue is zero, but the converse is not always true (for example, has residue of 0 at but is not analytic at ). The residue of a function at a point may be denoted . The residue is implemented in the Wolfram Language as Residue[f, z, z0].Two basic examples of residues are given by and for .The residue of a function around a point is also defined by(2)where is counterclockwise simple closed contour, small enough to avoid any other poles of . In fact, any counterclockwise path with contour winding number 1 which does not contain any other poles gives the same result by the Cauchy integral formula. The above diagram shows a suitable contour for which to define the residue of function, where the poles are indicated as black dots.It is more natural to consider the residue of a meromorphic one-form because it is independent of the choice of coordinate. On a Riemann..

The prime number theorem shows that the th prime number has the asymptotic value(1)as (Havil 2003, p. 182). Rosser's theorem makes this a rigorous lower bound by stating that(2)for (Rosser 1938). This result was subsequently improved to(3)where (Rosser and Schoenfeld 1975). The constant was subsequently reduced to (Robin 1983). Massias and Robin (1996) then showed that was admissible for and . Finally, Dusart (1999) showed that holds for all (Havil 2003, p. 183). The plots above show (black), (blue), and (red).The difference between and is plotted above. The slope of the difference taken out to is approximately .

Mertens' second theorem states that the asymptotic form of the harmonic series for the sum of reciprocal primes is given bywhere is a prime, is a constant known as the Mertens constant, and is a Landau symbol.

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

Just as the ratio of the arc length of a semicircle to its radius is always , the ratio of the arc length of the parabolic segment formed by the latus rectum of any parabola to its semilatus rectum (and focal parameter) is a universal constant(1)(2)(3)(4)(OEIS A103710). This can be seen from the equation of the arc length of a parabolic segment(5)by taking and plugging in and .The other conic sections, namely the ellipse and hyperbola, do not have such universal constants because the analogous ratios for them depend on their eccentricities. In other words, all circles are similar and all parabolas are similar, but the same is not true for ellipses or hyperbolas (Ogilvy 1990, p. 84).The area of the surface generated by revolving for about the -axis is given by(6)(7)(Love 1950, p. 288; OEIS A103713) and the area of the surface generated by revolving for about the -axis is(8)(9)(Love 1950, p. 288; OEIS A103714).The expected distance..

The prime number theorem gives an asymptotic form for the prime counting function , which counts the number of primes less than some integer . Legendre (1808) suggested that for large ,(1)with (where is sometimes called Legendre's constant), a formula which is correct in the leading term only,(2)(Nagell 1951, p. 54; Wagon 1991, pp. 28-29; Havil 2003, p. 177).In 1792, when only 15 years old, Gauss proposed that(3)Gauss later refined his estimate to(4)where(5)is the logarithmic integral. Gauss did not publish this result, which he first mentioned in an 1849 letter to Encke. It was subsequently posthumously published in 1863 (Gauss 1863; Havil 2003, pp. 174-176).Note that has the asymptotic series about of(6)(7)and taking the first three terms has been shown to be a better estimate than alone (Derbyshire 2004, pp. 116-117).The statement (4) is often known as "the" prime number theorem and was proved..

Suppose that in some neighborhood of ,(1)for some function (say analytic or integrable) . Then(2)These functions form a forward/inverse transform pair. For example, taking for all gives(3)and(4)which is simply the usual integral formula for the gammafunction.Ramanujan used this theorem to generate amazing identities by substituting particular values for .

Let be any function, say analytic or integrable. Then(1)and(2)where is the Dirichlet lambda function and is the gamma function. Equation (◇) is obtained from (◇) by defining(3)These formulas give valid results only for certain classes of functions, and are connected with Mellin transforms (Hardy 1999, p. 15).

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted in modern texts, although the notations and (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where and and are related by(1)Let the multivalued function be interpreted to stand..

Twin primes are pairs of primes of the form (, ). The term "twin prime" was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 and A006512).All twin primes except (3, 5) are of the form .It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes,(1)converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201)...

The number obtained by adding the reciprocals of the odd twinprimes,(1)By Brun's theorem, the series converges to a definite number, which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1989, p. 201). By contrast, the series of all prime reciprocals diverges to infinity, as follows from the Mertens second theorem by letting (which provides a stronger characterization of the divergence than Euler's proof that , obtained more than a century before Mertens' proof).Shanks and Wrench (1974) used all the twin primes among the first 2 million numbers. Brent (1976) calculated all twin primes up to 100 billion and obtained (Ribenboim 1989, p. 146)(2)assuming the truth of the first Hardy-Littlewood conjecture. Using twin primes up to , Nicely (1996) obtained(3)(Cipra 1995, 1996), in the process discovering a bug in Intel's® PentiumTM microprocessor. Using twin primes up to , Nicely..

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function(1)is always even (except for ), all primes gaps are also even. The notation is commonly used to denote the smallest prime corresponding to the start of a prime gap of length , i.e., such that is prime, , , ..., are all composite, and is prime (with the additional constraint that no smaller number satisfying these properties exists).The maximal prime gap is the length of the largest prime gap that begins with a prime less than some maximum value . For , 2, ..., is given by 4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ... (OEIS A053303).Arbitrarily large prime gaps exist. For example, for any , the numbers , , ..., are all composite (Havil 2003, p. 170). However, no general method..

The prime counting function is the function giving the number of primes less than or equal to a given number (Shanks 1993, p. 15). For example, there are no primes , so . There is a single prime (2) , so . There are two primes (2 and 3) , so . And so on.The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it is not my fault. You'll just have to put up with it."Letting denote the th prime, is a right inverse of since(1)for all positive integers. Also,(2)iff is a prime number.The first few values of for , 2, ... are 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ... (OEIS A000720). The Wolfram Language command giving the prime counting function for a number is PrimePi[x], which works up to a maximum value of .The notation..

The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function(1)These numbers arise in the series expansions of trigonometric functions, and areextremely important in number theory and analysis.There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written , while the Bernoulli numbers encountered in older literature are written (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials or with and .The Bernoulli number and polynomial should not be confused with the Bell numbers and Bell polynomial, which are also commonly denoted and , respectively.Bernoulli numbers defined by the modern definition are denoted and sometimes called "even-index" Bernoulli numbers...

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

Bertrand's postulate, also called the Bertrand-Chebyshev theorem or Chebyshev's theorem, states that if , there is always at least one prime between and . Equivalently, if , then there is always at least one prime such that . The conjecture was first made by Bertrand in 1845 (Bertrand 1845; Nagell 1951, p. 67; Havil 2003, p. 25). It was proved in 1850 by Chebyshev (Chebyshev 1854; Havil 2003, p. 25; Derbyshire 2004, p. 124) using non-elementary methods, and is therefore sometimes known as Chebyshev's theorem. The first elementary proof was by Ramanujan, and later improved by a 19-year-old Erdős in 1932.A short verse about Bertrand's postulate states, "Chebyshev said it, but I'll say it again; There's always a prime between and ." While commonly attributed to Erdős or to some other Hungarian mathematician upon Erdős's youthful re-proof the theorem (Hoffman 1998), the quote is actually..

For any algebraic number of degree , a rational approximation to must satisfyfor sufficiently large . Writing leads to the definition of the irrationality measure of a given number. Apostol (1997) states the theorem in the slightly modified but equivalent form that there exists a positive constant depending only on such that for all integers and with ,

The Mellin transform is the integral transformdefined by(1)(2)It is implemented in the Wolfram Language as MellinTransform[expr, x, s]. The transform exists if the integral(3)is bounded for some , in which case the inverse exists with . The functions and are called a Mellin transform pair, and either can be computed if the other is known.The following table gives Mellin transforms of common functions (Bracewell 1999, p. 255). Here, is the delta function, is the Heaviside step function, is the gamma function, is the incomplete beta function, is the complementary error function erfc, and is the sine integral.convergenceAnother example of a Mellin transform is the relationship between the Riemann function and the Riemann zeta function ,(4)(5)A related pair is used in one proof of the primenumber theorem (Titchmarsh 1987, pp. 51-54 and equation 3.7.2)...

Thomae's theorem, also called Thomae's transformation, is the generalizedhypergeometric function identity(1)where is the gamma function, is a generalized hypergeometric function,(2)and (Bailey 1935, p. 14). It is a generalization of Dixon's theorem (Slater 1966, p. 52).An equivalent formulation is given by(3)(Hardy 1999, p. 104). The symmetry of this form was used by Ramanujan in his proof of the identity, which is essentially the same as Thomae's. Interestingly, this is one of the few cases in which Ramanujan gives an explicit proof of one of his propositions (Hardy 1999, p. 104).A special case of the theorem is given by(4)(J. Sondow, pers. comm., May 25, 2003).

Min Max Re Im A special function which is given by the logarithmic derivative of the gamma function (or, depending on the definition, the logarithmic derivative of the factorial).Because of this ambiguity, two different notations are sometimes (but not always) used, with(1)defined as the logarithmic derivative of the gamma function , and(2)defined as the logarithmic derivative of the factorial function. The two are connected by the relationship(3)The th derivative of is called the polygamma function, denoted . The notation(4)is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation for . The digamma function is returned by the function PolyGamma[z] or PolyGamma[0, z] in the Wolfram Language, and typeset using the notation .The digamma function arises in simple sums such as(5)(6)where is a Lerch transcendent.Special cases are given by(7)(8)(9)(10)Gauss's digamma theorem states..

The natural logarithm is the logarithm having base e, where(1)This function can be defined(2)for .This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(3)The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .Common and natural logarithms can be expressed in terms of each other as(4)(5)The natural logarithm is especially useful in calculusbecause its derivative is given by the simple equation(6)whereas logarithms in other bases have the more complicated..

A subspace of is called a deformation retract of if there is a homotopy (called a retract) such that for all and , 1. , 2. , and 3. . A tightening of the last condition gives a so-called strongdeformation retract (Bredon 1993, pp. 45-46).Note that a deformation retract is also a retract, because the homotopy defines a continuous map

Bowling, known as "ten pins" throughout most of the world, is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in the form of a triangle with its vertex oriented towards the bowler. The arrangement of the 10 bowling pins is that of a tetractys and is also triangular number .Up to two balls (or "bowls") are allowed per "frame," and a game consists of ten frames (with a special rule being used for the number of balls awarded in the last frame). If all pins are knocked down on the first ball, the result is called a "strike," no second ball is awarded for that frame (except in the case of a strike being obtained in the tenth and last frame, in which case two extra balls are awarded), and the number of points tallied is 10 plus the number of pins knocked down on the next two balls. If some or none of the pins are knocked down on the first bowl, then a second ball is awarded...

A sequence of real numbers is equidistributed on an interval if the probability of finding in any subinterval is proportional to the subinterval length. The points of an equidistributed sequence form a dense set on the interval .However, dense sets need not necessarily be equidistributed. For example, , where is the fractional part, is dense in but not equidistributed, as illustrated above for to 5000 (left) and to (right)Hardy and Littlewood (1914) proved that the sequence , of power fractional parts is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio (Finch 2003), and the golden ratio .The top set of above plots show the values of for equal to e, the Euler-Mascheroni constant , the golden ratio , and pi. Similarly, the bottom set of above plots show a histogram of the distribution of for these constants. Note that while most..

Let be a real number, and let be the set of positive real numbers for which(1)has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,(2)where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :(3)where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality(4)holds for any and all integers and with sufficiently large.The..

Jacobi's imaginary transformations relate elliptic functions to other elliptic functions of the same type but having different arguments. In the case of the Jacobi elliptic functions , , and , the transformations are(1)(2)(3)where is the elliptic modulus, and is the complementary modulus (Abramowitz and Stegun 1972; Whittaker and Watson 1990, p. 505).In the case of the Jacobi theta functions,Jacobi's imaginary transformation gives(4)(5)(6)(7)where(8)and is interpreted as satisfying (Whittaker and Watson 1990, p. 475).Equation (6) can be written as the functional equation(9)where and is the half-period ratio (Davenport 1980, p. 62). This form is useful for computing for small , since then the series for converges much faster than that for . In his paper of 1859, Riemann used this functional equation for the theta function in one of his proofs of the functional equation for the Riemann zeta function (Davenport..

The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of , . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.The set of complex numbers is implemented in the Wolfram Language as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative..

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies(1)where is the complex argument. Written explicitly in terms of real and imaginary parts,(2)An explicit example of complex exponentiation is given by(3)A complex number taken to a complex number can be real. In fact, the famous example(4)shows that the power of the purely imaginary to itself is real. Min Max Re Im In fact, there is a family of values such that is real, as can be seen by writing(5)This will be real when , i.e., for(6)for an integer. For positive , this gives roots or(7)where is the Lambert W-function. For , this simplifies to(8)For , 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ......

An Argand diagram is a plot of complex numbersas pointsin the complex plane using the x-axis as the real axis and y-axis as the imaginary axis. In the plot above, the dashed circle represents the complex modulus of and the angle represents its complex argument.While Argand (1806) is generally credited with the discovery, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line...

A generalization of the product rule for expressingarbitrary-order derivatives of products of functions,where is a binomial coefficient. This can also be written explicitly as(Roman 1980), where is the differential operator.

There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series(1)into a series with more rapid convergence to the same value to(2)where the forward difference is defined by(3)(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when is taken in the second of Euler's hypergeometric transformations(4)where is a hypergeometric function, it gives Euler's convergence improvement transformation of the series (Abramowitz and Stegun 1972, p. 555).The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21)...

The constant is base of the natural logarithm. is sometimes known as Napier's constant, although its symbol () honors Euler. is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(1)With the possible exception of , is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of is(2)(OEIS A001113). can be defined by the limit(3)(illustrated above), or by the infinite series(4)as first published by Newton (1669; reprinted in Whiteside 1968, p. 225). is given by the unusual limit(5)(Brothers and Knox 1998).Euler (1737; Sandifer 2006) proved that is irrational by proving that has an infinite simple continued fraction (; Nagell 1951), and Liouville proved in 1844 that does not satisfy any quadratic equation with integral coefficients (i.e., if it is..

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..

A digit sum is a sum of the base- digits of , which can be implemented in the Wolfram Language as DigitSum[n_, b_:10] := Total[IntegerDigits[n, b]]The following table gives for , 2, ... and small .OEIS for , 2, ...2A0001201, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, ...3A0537351, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, ...4A0537371, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, ...5A0538241, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, ...6A0538271, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, ...7A0538281, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, ...8A0538291, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, ...9A0538301, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, ...10A0079531, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, ...Plotting versus and gives the plot shown above.The digits sum satisfies the congruence(1)In base 10, this congruence is the basis of casting out nines and of fast divisibility tests such as those for 3 and 9. satisfies the following unexpected identity(2)the case of which was given in the 1981 Putnam competition..

Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..

The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a map f of a manifold to itself such that the tangent bundle of is hyperbolic with respect to .A trivial example is to map all of to a single point of . Here, the eigenvalues are all zero. A less trivial example is an expanding map on the circle , e.g., , where is identified with the real numbers (mod 1). Here, all the eigenvalues equal 2 (i.e., the eigenvalue at each point of ). Note that this map is not a diffeomorphism because , so it has no inverse.A nontrivial example is formed by taking Arnold's cat map on the 2-torus , and crossing it with an expanding map on to form an Anosov map on the 3-torus , where denotes the Cartesian product. In other words,..

An Anosov diffeomorphism is a diffeomorphism of a manifold to itself such that the tangent bundle of is hyperbolic with respect to . Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's cat map.A hyperbolic linear map with integer entries in the transformation matrix and determinant is an Anosov diffeomorphism of the -torus. Not every manifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are expansive, and there are no Anosov diffeomorphisms on the circle.It is conjectured that if is an Anosov diffeomorphism on a compact Riemannian manifold and the nonwandering set of is , then is topologically conjugate to a finite-to-one factor of an Anosov automorphism of a nilmanifold. It has been proved that any Anosov diffeomorphism on the -torus is topologically conjugate to an Anosov automorphism, and also that Anosov diffeomorphisms are structurally stable...

The Lerch transcendent is generalization of the Hurwitz zeta function and polylogarithm function. Many sums of reciprocal powers can be expressed in terms of it. It is classically defined by(1)for and , , .... It is implemented in this form as HurwitzLerchPhi[z, s, a] in the Wolfram Language.The slightly different form(2)sometimes also denoted , for (or and ) and , , , ..., is implemented in the Wolfram Language as LerchPhi[z, s, a]. Note that the two are identical only for .A series formula for valid on a larger domain in the complex -plane is given by(3)which holds for all complex and complex with (Guillera and Sondow 2005).The Lerch transcendent can be used to express the Dirichletbeta function(4)(5)A special case is given by(6)(Guillera and Sondow 2005), where is the polylogarithm.Special cases giving simple constants include(7)(8)(9)(10)where is Catalan's constant, is Apéry's constant, and is the Glaisher-Kinkelin constant..

For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..

The line in the complex plane on which the Riemann hypothesis asserts that all nontrivial (complex) Riemann zeta function zeros lie. The plot above shows the first few zeros of the Riemann zeta function, with the critical line shown in red. The zeros with and that do not line on the critical line are the trivial zeros of at , , .... Although it is known that an infinite number of zeros lie on the critical line and that these comprise at least 40% of all zeros, the Riemann hypothesis is still unproven.An attractive poster plotting the Riemann zeta function zeros on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes discusses the critical line after realizing that character Ethan's daughter has been kidnapped because he is close to solving..

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..

Landau (1911) proved that for any fixed ,as , where the sum runs over the nontrivial Riemann zeta function zeros and is the Mangoldt function. Here, "fixed " means that the constant implicit in depends on and, in particular, as approaches a prime or a prime power, the constant becomes large.Landau's formula is roughly the derivative of the explicitformula.Landau's formula is quite extraordinary. If is not a prime or a prime power, then and the sum grows as a constant times . But if is a prime or a prime power, then and the sum grows much faster, like a constant times . This exhibits an amazing connection between the primes and the s; somehow the zeros "recognize" when is a prime and cause large contributions to the sum.

The Hurwitz zeta function is a generalization of the Riemann zeta function that is also known as the generalized zeta function. It is classically defined by the formula(1)for and by analytic continuation to other , where any term with is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].The slightly different form(2)is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for .The plot above shows for real and , with the zero contour indicated in black.For , a globally convergent series for (which, for fixed , gives an analytic continuation of to the entire complex -plane except the point ) is given by(3)(Hasse 1930).The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].For , reduces to the Riemann zeta function ,(4)If the singular term is excluded from the sum definition of , then as well.The Hurwitz zeta function is given by the integral(5)for and .The..

Euler's series transformation is a transformation that sometimes accelerates the rate of convergence for an alternating series. Given a convergent alternating series with sum(1)Abramowitz and Stegun (1972, p. 16) define Euler's transformation as(2)where is the forward difference operator(3)and is a binomial coefficient.An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as(4)where is the backward difference operator(5)Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:(6)gives faster convergence,(7)gives same rate of convergence, and(8)gives slower convergence.To see why the Euler transformation works, consider Knopp's convention for difference operator and write(9)(10)Now repeat the process on the series in brackets to obtain(11)and continue to infinity. This proves each finite step in..

The derivative identity (1)(2)(3)(4)where denotes the derivative of . The Leibniz identity extends the product rule to higher-order derivatives.

A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one.A complex number can be tested to see if it is transcendental using the Wolfram Language command Not[Element[x, Algebraics]].Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a geometric construction using the ancient Greek rules, it must be either rational or a very special kind of algebraic number known as a Euclidean number. Because the number is transcendental, the construction..

Min Max Min Max Re Im The Dirichlet eta function is the function defined by(1)(2)where is the Riemann zeta function. Note that Borwein and Borwein (1987, p. 289) use the notation instead of . The function is also known as the alternating zeta function and denoted (Sondow 2003, 2005). is defined by setting in the right-hand side of (2), while (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of except (Sondow 2003).The eta function is related to the Riemann zeta function and Dirichlet lambda function by(3)and(4)(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithmfunction,(5)The value may be computed by noting that the Maclaurin series for for is(6)Therefore, the natural logarithm of 2 is(7)(8)(9)(10)The derivative of the eta function is given by(11)or in the special case , by(12)(13)(14)(15)This latter fact provides a remarkable..

Min Max Min Max Re Im The Dirichlet beta function is defined by the sum(1)(2)where is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function by(3)The beta function can be defined over the whole complexplane using analytic continuation,(4)where is the gamma function.The Dirichlet beta function is implemented in the WolframLanguage as DirichletBeta[x].The beta function can be evaluated directly special forms of arguments as(5)(6)(7)where is an Euler number.Particular values for are(8)(9)(10)(11)where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).It is involved in the integral(12)(Guillera and Sondow 2005).Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.The derivative can also be computed analytically at a number of integer values of including(13)(14)(15)(16)(17)(18)(19)(OEIS..

The number of digits in the base- representation of a number is called the -ary digit count for . The digit count is implemented in the Wolfram Language as DigitCount[n, b, d].The number of 1s in the binary representation of a number , illustrated above, is given by(1)(2)where is the greatest dividing exponent of 2 with respect to . This is a special application of the general result that the power of a prime dividing a factorial (Vardi 1991, Graham et al. 1994). Writing for , the number of 1s is also given by the recurrence relation(3)(4)with , and by(5)where is the denominator of(6)For , 2, ..., the first few values are 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120; Smith 1966, Graham 1970, McIlroy 1974).For a binary number, the count of 1s is equal to the digit sum . The quantity is called the parity of a nonnegative integer . and satisfy the beautiful identities(7)(8)where is the Euler-Mascheroni constant and (OEIS A094640) is its "alternating analog"..

The product of primes(1)with the th prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .The zeta-regularized product over allprimes is given by(2)(3)(Muñoz Garcia and Pérez-Marco 2003, 2008), answering the question posed by Soulé et al. (1992, p. 101). A derivation proceeds by algebraic manipulation of the prime zeta function and gives the more general results(4)and(5)(Muñoz Garcia and Pérez-Marco 2003).Mertens theorem states that(6)where is the Euler-Mascheroni constant, and a closely related result is given by(7)There are amazing infinite product formulas forprimes given by(8)(Ramanujan 1913-1914; Le Lionnais 1983, p. 46) and(9)(OEIS A082020; Ramanujan 1913-1914).More general formulas are given by(10)where is the Riemann zeta function and by the Euler product(11)Named prime..

The Hadamard product is a representation for the Riemann zeta function as a product over its nontrivial zeros ,(1)where is the Euler-Mascheroni constant and is the Gamma function (Titchmarsh 1987, Voros 1987). The constant in the exponent is given by(2)(3)(OEIS A077142). Hadamard used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.The product can also be stated in the alternate form(4)where is the xi-function and(5)(Havil 2003, p. 204).

The first few values of (known as a superfactorial) for , 2, ... are given by 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).The first few positive integers that can be written as a product of factorials are1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, ... (OEIS A001013).The number of ways that is a product of smaller factorials, each greater than 1, for , 2, ... is given by 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, ... (OEIS A034876), and the numbers of products of factorials not exceeding are 1, 2, 4, 8, 15, 28, 49, 83, ... (OEIS A101976).The only known factorials which are products of factorials in an arithmeticprogression of three or more terms are(1)(2)(3)(Madachy 1979).The only solutions to(4)are(5)(6)(7)(Cucurezeanu and Enkers 1987).There are no nontrivial identities of the form(8)for with for for except(9)(10)(11)(12)(Madachy 1979; Guy 1994, p. 80). Here, "nontrivial" means that identities with , or equivalently are excluded, since..

Check the price

for your project

for your project

we accept

Money back

guarantee

guarantee

Price calculator

We've got the best prices, check out yourself!