- Harmonic number
:"The term "harmonic number" has multiple meanings. For other meanings, see
harmonic number (disambiguation) ".In
mathematics , the "n"-th harmonic number is the sum of the reciprocals of the first "n" natural numbers:::This also equals "n" times the inverse of the
harmonic mean of these natural numbers.Harmonic numbers were studied in antiquity and are important in various branches of
number theory . They are sometimes loosely termed harmonic series, are closely related to theRiemann zeta function , and appear in various expressions for variousspecial function s.Calculation
An integral representation is given by
Euler ::
This representation can be easily shown to satisfy the recurrence relation by the formula
:
and then
:
inside the integral.
"H""n" grows about as fast as the
natural logarithm of "n". The reason is that the sum is approximated by theintegral :
whose value is ln("n"). More precisely, we have the limit:
:
(where γ is the
Euler-Mascheroni constant ), and the corresponding asymptotic expansion::
pecial values for fractional arguments
There are the following special analytic values for fractional arguments between 0 and 1, given by the integral:More may be generated from the recurrence relation .
:
:
:
:
:
Generating functions
A
generating function for the harmonic numbers is:
where is the
natural logarithm . An exponential generating function is:
where is the entire
exponential integral . Note that:
where is the
incomplete gamma function .Applications
The harmonic numbers appear in several calculation formulas, such as the
digamma function ::
This relation is also frequently used to define the extension of the harmonic numbers to non-integer "n". The harmonic numbers are also frequently used to define γ, using the limit introduced in the previous section, although
:
converges more quickly.
In 2002
Jeffrey Lagarias proved that theRiemann hypothesis is equivalent to the statement that :is true for everyinteger "n" ≥ 1 with strict inequality if "n" > 1; here σ("n") denotes the sum of the divisors of "n".See also
Watterson estimator ,Tajima's D ,coupon collector's problem .Generalization
Generalized harmonic numbers
The generalized harmonic number of order of "m" is given by
:
Note that the limit as "n" tends to infinity exists if .
Other notations occasionally used include
:
The special case of is simply called a harmonic number and is frequently written without the superscript, as
:
In the limit of , the generalized harmonic number converges to the
Riemann zeta function :
The related sum occurs in the study of
Bernoulli number s; the harmonic numbers also appear in the study ofStirling number s.A
generating function for the generalized harmonic numbers is:
where is the
polylogarithm , and . The generating function given above for is a special case of this formula.Generalization to the complex plane
Euler's integral formula for the harmonic numbers follows from the integral identity
:
which holds for general complex-valued "s", for the suitably extended
binomial coefficient s. By choosing "a"=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating theNewton series :
which is just the Newton's generalized
binomial theorem . The interpolating function is in fact just thedigamma function ::
where is the digamma, and is the Euler-Mascheroni constant. The integration process may be repeated to obtain
:
References
* Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn, " [http://www.math.hmc.edu/~benjamin/papers/harmonic.pdf A Stirling Encounter with Harmonic Numbers] ", (2002) Mathematics Magazine, 75 (2) pp 95-103.
*Donald Knuth . "The Art of Computer Programming", Volume 1: "Fundamental Algorithms", Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.7: Harmonic Numbers, pp.75–79.
* Ed Sandifer, " [http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf How Euler Did It -- Estimating the Basel problem] " (2003)
*
* Peter Paule and Carsten Schneider, " [http://www.risc.uni-linz.ac.at/publications/download/risc_200/HarmonicNumberIds.pdf Computer Proofs of a New Family of Harmonic Number Identities] ", (2003) Adv. in Appl. Math. 31(2), pp. 359-378.
* Wenchang CHU, " [http://www.combinatorics.org/Volume_11/PDF/v11i1n15.pdf A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers] ", (2004) "The Electronic Journal of Combinatorics", 11, #N15.
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