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The th harmonic number
is defined as the truncation of the harmonic series
after the
th term
The first ten harmonic numbers are:
.
To compute more harmonic numbers visit .
is integral only for
[1] .
To see the first statement note that the greatest power of 2 not exceeding do not divide any other term if
. Therefore the numerator of
must be odd while the denominator is even. In other words, the denominator of
is a multiple of
where
is determined by relations
.
L.Kürschák [2] proved more: the difference between two distinct harmonic numbers is never an integer. He proved this in the form that with
is never an integer.
T.Nagell [3] proved that even ,
, cannot be an integer. This result was later reproved by P.Erdõs [4] .
The Bertrand postulate says that for every positive integer there is a prime
such that
. Consequently, if
the denominator of
is divisible by at least one odd prime. As we saw above, the denominator of
is divisible by the largest power of 2 less or equal to
. Consequently,
is the only harmonic number with a prime denominator.
An analogous question for which the numerator of
is a prime number is more delicate. The first 10 values of
are (for more details visit
):
L.Theisinger [1] also proved that
![]() | (1) |
A combinatorial interpretation of harmonic numbers says that ( [5] , p.275 )
![]() | (2) |
where denotes the unsigned Stirling number of the first kind, counting the permutation of
elements that are product of
disjoint cycles.
Euler found the following integral representation
![]() | (3) |
It can be shown using the integral comparison criterion applied to the function for
that
![]() | (4) |
This implies that , and that
, that is that
grows about as fast (or better as slowly) as
. Actually more can be shown. Using the fact that
converges in the infinite product
there follows that there exists a non-vanishing finite limit
![]() | (5) |
say . Clearly
. Consequently there exists also the limit
![]() | (6) |
The constant is the Euler-Mascheroni constant
. To see this we can use Euler’s summation formula which gives that
![]() | (7) |
where stands for the
th Bernoulli number. This formula can be used in several ways:
For instance if we get that
without adding up 1000 unit fractions.
Last statement can be extended also in the following form: is monotone increasing and converges to
. Since
we have
Moreover
using the substitution .
Graphical representation of values (red points) of and their asymptotic curve (blue line)
A generating function for harmonic numbers is
![]() | (8) |
If denotes the number of divisors of an integer
then
In the following graph the harmonic numbers are represented by red points and the values of right hand side by blue points
J.C.Lagarias [6] proved an interesting link to the Riemann hypothesis. He proved that the problem of showing that
![]() | (9) |
for all with equality only for
, is equivalent to the Riemann Hypothesis. The proof uses two bounds for the sums-of-divisors function
proved by G. Robin [7] . One of the inequalities says that the inequality
for
is equivalent with Riemann hypotheses (Robin also showed that one can take
).
In the graph the blue points represent the sum-of-divisors functions and the red ones the values of the right hand side of (9)
For the first multiples of 12 we have
It is possible to extend the above definition of a harmonic number also to non-integral values of using the observation that
![]() | (10) |
converges for all complex values of , except when
is a negative number. If
denotes the value of the series, then when
is a positive integer
we get
.
Wolstenholme's Theorem: If is a prime, then [8]
An interesting problem in which solution harmonic numbers unexpectedly occur was discovered by R.T.Sharp [9] [10] : determine how far an overhang we can achieve by stacking dominoes over the table edge, accounting for the force of gravity.
The center of gravity is a geometric property of any object and it is the point where all the weight of the object can be considered to be concentrated. In other words, it is defined as the average location of the weight of an object. Thus the radiusvector of the center of a system of particles is the average of the positions
of center of the
th particle weighted by its mass
![]() | (11) |
Since we shall suppose that edges of the cards are parallel to the edge of the table, we shall only work with the -coordinate. Let
be the shift distance of the extreme edge of the
th domino to the corresponding edge of the
st domino piece counting from the top.
For the sake of simplicity suppose that length of each domino piece is 2 units, the weight is 1 unit with uniformly distributed mass along the length,, and that the pile contains pieces. Then the
-coordinate of the center of gravity of
th piece from the top is
,
. Then the
-coordinate of the center of such complex of
superimposed domino pieces is
![]() | (12) |
while .
The stability criterion is that the -coordinate of the center of gravity of pieces
must lie exactly above the right edge of piece
. Since the right edge of piece
has
-coordinate 2, we get from (12)
![]() | (13) |
This implies that for
. This implies that the total extension for block of
dominoes is
. In the following picture the
th triangle shaped point from top always shows the position of the center of gravity of top
dominoes.
A bit surprising consequence of the solution is that given sufficiently large number of dominoes, arbitrary large overhangs can be achieved.
In the above way of stacking of dominoes we achieved an overhang of , We stacked here all the dominoes one on top of another with the
th domino from the top displaced by
beyond the block below. This solution is however not optimal. M.Paterson and U.Zwick [11] showed that it is exponentially far from optimal and gave explicit constructions with an overhang of
.
Generalized harmonic number of order
is defined as truncation of the series of reciprocals of the
th powers of positive integers
Clearly, .
The first ten generalized harmonic numbers are:
To compute more generalized harmonic numbers visit .
Comparison graph of values of for
.
The following identity goes back to Euler ( [5] , p.278 )
![]() | (14) |
Similarly to (10) we can take the series
to get a generalization to non-integral values of indices.
The generalized harmonic number converges to the Riemann zeta function in the following sense
J.H.Conway and R.K.Guy [12] defined the hyperharmonic numbers, the th harmonic number of order
:
, and
for
.
The first ten harmonic numbers of order 2 are:
The first ten harmonic numbers of order 3 are:
We have [13]
![]() | (15) |
where is the number of permutations of the set
having
disjoint non-empty cycles in which the elements
are restricted to appear in different cycles (the so-called
-Stirling numbers [14] , where
), or [12]
![]() | (16) |
To compute more hyperharmonic numbers visit .
[1] | Theisinger, L. (1915). Bemerkung über die harmonische Reihe. (German). Monatsh. f. math., 26, 132-134. |
[2] | Kürschák, J. (1918). Über die harmonische Reihe. (Hungarian). Math. és phys. lapok, 27, 299-300. |
[3] | Nagell, T. (1923 (1924)). Eine Eigenschaft gewisser Summen. Skr. Norske Vid. Akad. Kristiania, I(13), 10-15. |
[4] | Erdõs, P. (1932). Generalization of an elementary number-theoretic theorem of Kürschák (Hungarian). Mat. Fiz. Lapok, 39, 17-24. |
[5] | Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete mathematics: a foundation for computer science (2nd ed.). Amsterdam: Addison-Wesley Publishing Group. |
[6] | Lagarias, J. C. (2002). An elementary problem equivalent to the Riemann hypothesis. Amer. Math. Montly, 109(6), 534-543. |
[7] | Robin, G. (1984). Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl., IX. Sér. 63, 187-213. |
[8] | Wolstenholme, J. (1862). On certain properties of prime numbers. Quarterly Journal of Mathematics, 5, 35-39. |
[9] | Sharp, R. T. (1954). Problem 52: Overhanging dominoes. Pi Mu Epsilon Journal, 1(10), 411-412. |
[10] | Trigg, C. W. (1954). Problem 52: Overhanging dominoes, (proposer: R. T. Sharp). Pi Mu Epsilon Journal, 1(10), 411-412. |
[11] | Paterson, M., & Zwick, U. (2006). Overhang. In: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm (pp. 231-240). New York: ACM. |
[12] | Conway, J. H., & Guy, R. K. (1996). The Book of Numbers. Berlin: Springer Verlag. |
[13] | Benjamin, A. T., Gaebler, D., & Gaebler, R. (2003). A combinatorial approach to hyperharmonic numbers. Integers, 3, Paper A15. |
[14] | Broder, A. Z. (1984). The r-Stirling numbers. Discrete Math., 49, 241-259. |
Cite this web-page as:
Štefan Porubský: Harmonic Number.