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Wolstenholme number

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).

Trigamma function

Min Max Re Im A special function corresponding to a polygamma function with , given by(1)An alternative function(2)is sometimes called the trigamma function, where(3)Sums and differences of for small integral and can be expressed in terms of Catalan's constant and . For example,(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

Digamma function

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..

Polygamma function

A special function mostly commonly denoted , , or which is given by the st derivative of the logarithm of the gamma function (or, depending on the definition, of the factorial ). This is equivalent to the th normal derivative of the logarithmic derivative of (or ) and, in the former case, to the th normal derivative of the digamma function . Because of this ambiguity in definition, two different notations are sometimes (but not always) used, namely(1)(2)(3)which, for can be written as(4)(5)where is the Hurwitz zeta function.The alternate notation(6)is sometimes used, with the two notations connected by(7)Unfortunately, Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach's "" is equal to in the usual notation. Also note that the function is equivalent to the digamma function and is sometimes known as the trigamma function. is implemented in the Wolfram Language as PolyGamma[n, z] for positive..

Mellin's formula

where is the digamma function and is the gamma function.

Harmonic series

The series(1)is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).Progressions of the form(2)are also sometimes called harmonic series (Beyer 1987).Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,(3)(4)and since an infinite sum of 1/2's diverges, so does the harmonic series.The generalization of the harmonic series(5)is known as the Riemann zeta function.The sum of the first few terms of..

Book stacking problem

How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..

Wolstenholme's theorem

If is a prime , then the numerator of the harmonic number(1)is divisible by and the numerator of the generalized harmonic number(2)is divisible by . The numerators of are sometimes known as Wolstenholme numbers.These imply that if is prime, then(3)

Periodic zeta function

(1)(2)where is the polygamma function.

Gauss's digamma theorem

At rational arguments , the digamma function is given by(1)for (Knuth 1997, p. 94). These give the special values(2)(3)(4)(5)(6)(7)(8)(9)where is the Euler-Mascheroni constant.

Beurling's function

The entire function(1)(2)where is a polygamma function.It satisfies and for all real . Amazingly, it also has the integral(3)Furthermore, among all functions with the first two properties, minimizes the integral (3) (Beurling 1938, Montgomery 2001).

Harmonic number

A harmonic number is a number of the form(1)arising from truncation of the harmonic series.A harmonic number can be expressed analytically as(2)where is the Euler-Mascheroni constant and is the digamma function.The first few harmonic numbers are 1, , , , , ... (OEIS A001008 and A002805). The numbers of digits in the numerator of for , 1, ... are 1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680, ... (OEIS A114467), with the corresponding number of digits in the denominator given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (OEIS A114468). These digits converge to what appears to be the decimal digits of (OEIS A002285).The first few indices such that the numerator of is prime are given by 2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, ... (OEIS A056903). The search for prime numerators has been completed up to by E. W. Weisstein (May 13, 2009), and the following table summarizes the largest known values.decimal digitsdiscoverer6394227795E. W. Weisstein..

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