“Apéry's constant”的意思、由来-开放百科全书

In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number

where {{math|ζ}} is the Riemann zeta function. It has an approximate value of^[1]

This constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees^[2] and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.

Irrational number

Beuker's simplified irrationality proof involves approximating the integrand of the known triple integral for

In particular, van der Poorten's article chronicles this approach by noting that

where

are the Legendre polynomials, and the subsequences

are integers or almost integers.

Series representations

Classical

Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of {{math|ζ(3)}}. Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "Known digits").

The following series representation was found by Hjortnaes in 1953,^[9] then rediscovered and widely advertised by Apéry in 1979:^[3]

The following series representation, found by Amdeberhan in 1996,^[10] gives (asymptotically) 1.43 new correct decimal places per term:

The following series representation, found by Amdeberhan and Zeilberger in 1997,^[11] gives (asymptotically) 3.01 new correct decimal places per term:

The following series representation, found by Sebastian Wedeniwski in 1998,^[12] gives (asymptotically) 5.04 new correct decimal places per term:

It was used by Wedeniwski to calculate Apéry's constant with several million correct decimal places.^[13]

The following series representation, found by Mohamud Mohammed in 2005,^[14] gives (asymptotically) 3.92 new correct decimal places per term:

Digit by digit

In 1998, Broadhurst^[15] gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.

Others

Integral representations

There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

Simple formulas

For example, this one follows from the summation representation for Apéry's constant:

The next two follow directly from the well-known integral formulas for the Riemann zeta function:

This one follows from a Taylor expansion of {{math|χ₃(e^ix)}} about {{math|x {{=}} ±{{sfrac|π|2}}}}, where {{math|χ_ν(z)}} is the Legendre chi function:

More complicated formulas

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma-functions.^[21]

Known digits

The number of known digits of Apéry's constant {{math|ζ(3)}} has increased dramatically during the last decades. This is due both to the increasing performance of computers and to algorithmic improvements.

Reciprocal

The reciprocal of {{math|ζ(3)}} is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as {{math|N}} goes to infinity, the probability that three positive integers less than {{math|N}} chosen uniformly at random will be relatively prime approaches this value).{{sfnp|Mollin|2009}}

Extension to {{math|ζ(2n + 1)}}

Many people have tried to extend Apéry's proof that {{math|ζ(3)}} is irrational to other odd zeta values. In 2000, Tanguy Rivoal showed that infinitely many of the numbers {{math|ζ(2n + 1)}} must be irrational.^[26] In 2001, Wadim Zudilin proved that at least one of the numbers {{math|ζ(5)}}, {{math|ζ(7)}}, {{math|ζ(9)}}, and {{math|ζ(11)}} must be irrational.^[27]

See also

Notes

1. ^¹See {{harvnb|Wedeniwski|2001}}.
2. ^See {{harvnb|Frieze|1985}}.
3. ^¹See {{harvnb|Apéry|1979}}.
4. ^See {{harvnb|van der Poorten|1979}}.
5. ^¹See {{harvnb|Beukers|1979}}.
6. ^See {{harvnb|Zudilin|2002}}.
7. ^See {{harvnb|Euler|1773}}.
8. ^See {{harvnb|Srivastava|2000|loc=p. 571 (1.11)}}.
9. ^See {{harvnb|Hjortnaes|1953}}.
10. ^See {{harvnb|Amdeberhan|1996}}.
11. ^See {{harvnb|Amdeberhan|Zeilberger|1997}}.
12. ^See {{harvnb|Wedeniwski|1998}} and {{harvnb|Wedeniwski|2001}}. In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from {{harvnb|Amdeberhan|Zeilberger|1997}}. The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).
13. ^See {{harvnb|Wedeniwski|1998}} and {{harvnb|Wedeniwski|2001}}.
14. ^See {{harvnb|Mohammed|2005}}.
15. ^See {{harvnb|Broadhurst|1998}}.
16. ^See {{harvnb|Berndt|1989|loc=chapter 14, formulas 25.1 and 25.3}}.
17. ^See {{harvnb|Plouffe|1998}}.
18. ^See {{harvnb|Srivastava|2000}}.
19. ^See {{harvnb|Jensen|1895}}.
20. ^See {{harvnb|Blagouchine|2014}}.
21. ^See {{harvnb|Evgrafov|Bezhanov|Sidorov|Fedoriuk|1969|loc=exercise 30.10.1}}.
22. ^See {{harvnb|Gourdon|Sebah|2003}}.
23. ^¹See {{harvnb|Yee|2009}}.
24. ^¹²See {{harvnb|Yee|2015}}.
25. ^See {{harvnb|Nag|2015}}.
26. ^See {{harvnb|Rivoal|2000}}.
27. ^See {{harvnb|Zudilin|2001}}.

References

}} (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).