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The Wolstenholme numbers are defined as the numerators of the generalized harmonic number appearing in Wolstenholme's theorem. The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (OEIS A007406).By Wolstenholme's theorem, for prime , where is the th Wolstenholme number. In addition, for prime .The first few prime Wolstenholme numbers are 5, 266681, 40799043101, 86364397717734821, ... (OEIS A123751), which occur at indices , 7, 13, 19, 121, 188, 252, 368, 605, 745, ... (OEIS A111354).

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..

A Fermat prime is a Fermat number that is prime. Fermat primes are therefore near-square primes.Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, the only Fermat numbers for for which primality or compositeness has been established are all composite.The only known Fermat primes are(1)(2)(3)(4)(5)(OEIS A019434), and it seems unlikely that any more will be found using current computational methods and hardware. It follows that is prime for the special case together with the Fermat prime indices, giving the sequence 2, 3, 5, 17, 257, and 65537 (OEIS A092506). is a Fermat prime if and only if the period length of is equal to . In other words, Fermat primes are full reptend primes...

Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form(1)where the s are nonnegative and(2)Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.The following table summarizes the coefficients for various special constants.OEISeA0026682, 8, 75, 8949, 119646723, 15849841722437093, ...Euler-Mascheroni constant A0817820, 1, 3, 16, 389, 479403, 590817544217, ...golden ratio A0062671, 4, 76, 439204, 84722519070079276, ...Lehmer's constant A0020650, 1, 3, 13, 183, 33673, ...A0026673, 73, 8599, 400091364,371853741549033970, ...Pythagoras's constant A0026661, 5, 36, 3406, 14694817,727050997716715, ...The expansion for the golden ratio has the beautiful closed form(3)where is a Lucas number.An illustration of a different cotangent expansion for that is not a Lehmer expansion because its coefficients..

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 important binomial theorem states that(1)Consider sums of powers of binomial coefficients(2)(3)where is a generalized hypergeometric function. When they exist, the recurrence equations that give solutions to these equations can be generated quickly using Zeilberger's algorithm.For , the closed-form solution is given by(4)i.e., the powers of two. obeys the recurrence relation(5)For , the closed-form solution is given by(6)i.e., the central binomial coefficients. obeys the recurrence relation(7)Franel (1894, 1895) was the first to obtain recurrences for ,(8)(Riordan 1980, p. 193; Barrucand 1975; Cusick 1989; Jin and Dickinson 2000), so are sometimes called Franel numbers. The sequence for cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160), and therefore has no closed-form hypergeometric expression.Franel (1894, 1895) was also the first to obtain the recurrence..

The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..

There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..

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..

A double sum is a series having terms depending on twoindices,(1)A finite double series can be written as a product of series(2)(3)(4)(5)An infinite double series can be written in terms of a single series(6)by reordering as follows,(7)(8)(9)(10)Many examples exists of simple double series that cannot be computed analytically,such as the Erdős-Borwein constant(11)(12)(13)(OEIS A065442), where is a q-polygamma function.Another series is(14)(15)(OEIS A091349), where is a harmonic number and is a cube root of unity.A double series that can be done analytically is given by(16)where is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).The double series(17)can be evaluated by interchanging and and averaging,(18)(19)(20)(21)(Borwein et al. 2004, p. 54).Identities involving double sums include the following:(22)where(23)is the floor function, and(24)Consider the series(25)over..

The value for(1)can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and Schur 1918, Kortram 1996, Matsuoka 1961, Papadimitriou 1973, Simmons 1992, Stark 1969, 1970, Yaglom and Yaglom 1987). is therefore the definite sum version of the indefinite sum(2)(3)where is a generalized harmonic number (whose numerator is known as a Wolstenholme number) and is a polygamma function.The problem of finding this value analytically is sometimes known as the Basel problem (Derbyshire 2004, pp. 63 and 370) or Basler problem (Castellanos 1988). It was first proposed by Pietro Mengoli in 1644 (Derbyshire 2004, p. 370). The solution(4)was first found by Euler in 1735 (Derbyshire 2004, p. 64) or 1736 (Srivastava 2000).Yaglom and Yaglom (1987), Holme (1970), and Papadimitriou (1973) all derive the result, from de Moivre's identity or related identities. is given by the..

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..

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