# Harmonic number

For other uses, see Harmonic number (disambiguation). Harmonic number_sentence_0

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Harmonic number_sentence_1

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. Harmonic number_sentence_2

Harmonic numbers have been studied since antiquity and are important in various branches of number theory. Harmonic number_sentence_3

They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. Harmonic number_sentence_4

The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. Harmonic number_sentence_5

In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. Harmonic number_sentence_6

His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers. Harmonic number_sentence_7

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. Harmonic number_sentence_8

This leads to a variety of surprising conclusions regarding the long tail and the theory of network value. Harmonic number_sentence_9

Bertrand's postulate implies that, except for the case n = 1, the harmonic numbers are never integers. Harmonic number_sentence_10

Harmonic number_table_infobox_0

The first 40 harmonic numbersHarmonic number_table_caption_0
1Harmonic number_cell_0_2_0 1Harmonic number_cell_0_2_1 1Harmonic number_cell_0_2_3 1Harmonic number_cell_0_2_4
2Harmonic number_cell_0_3_0 3Harmonic number_cell_0_3_1 /2Harmonic number_cell_0_3_2 1.5Harmonic number_cell_0_3_3 1.5Harmonic number_cell_0_3_4
3Harmonic number_cell_0_4_0 11Harmonic number_cell_0_4_1 /6Harmonic number_cell_0_4_2 ~1.83333Harmonic number_cell_0_4_3 1.83333Harmonic number_cell_0_4_4
4Harmonic number_cell_0_5_0 25Harmonic number_cell_0_5_1 /12Harmonic number_cell_0_5_2 ~2.08333Harmonic number_cell_0_5_3 2.08333Harmonic number_cell_0_5_4
5Harmonic number_cell_0_6_0 137Harmonic number_cell_0_6_1 /60Harmonic number_cell_0_6_2 ~2.28333Harmonic number_cell_0_6_3 2.28333Harmonic number_cell_0_6_4
6Harmonic number_cell_0_7_0 49Harmonic number_cell_0_7_1 /20Harmonic number_cell_0_7_2 2.45Harmonic number_cell_0_7_3 2.45Harmonic number_cell_0_7_4
7Harmonic number_cell_0_8_0 363Harmonic number_cell_0_8_1 /140Harmonic number_cell_0_8_2 ~2.59286Harmonic number_cell_0_8_3 2.59286Harmonic number_cell_0_8_4
8Harmonic number_cell_0_9_0 761Harmonic number_cell_0_9_1 /280Harmonic number_cell_0_9_2 ~2.71786Harmonic number_cell_0_9_3 2.71786Harmonic number_cell_0_9_4
9Harmonic number_cell_0_10_0 7 129Harmonic number_cell_0_10_1 /2 520Harmonic number_cell_0_10_2 ~2.82897Harmonic number_cell_0_10_3 2.82897Harmonic number_cell_0_10_4
10Harmonic number_cell_0_11_0 7 381Harmonic number_cell_0_11_1 /2 520Harmonic number_cell_0_11_2 ~2.92897Harmonic number_cell_0_11_3 2.92897Harmonic number_cell_0_11_4
11Harmonic number_cell_0_12_0 83 711Harmonic number_cell_0_12_1 /27 720Harmonic number_cell_0_12_2 ~3.01988Harmonic number_cell_0_12_3 3.01988Harmonic number_cell_0_12_4
12Harmonic number_cell_0_13_0 86 021Harmonic number_cell_0_13_1 /27 720Harmonic number_cell_0_13_2 ~3.10321Harmonic number_cell_0_13_3 3.10321Harmonic number_cell_0_13_4
13Harmonic number_cell_0_14_0 1 145 993Harmonic number_cell_0_14_1 /360 360Harmonic number_cell_0_14_2 ~3.18013Harmonic number_cell_0_14_3 3.18013Harmonic number_cell_0_14_4
14Harmonic number_cell_0_15_0 1 171 733Harmonic number_cell_0_15_1 /360 360Harmonic number_cell_0_15_2 ~3.25156Harmonic number_cell_0_15_3 3.25156Harmonic number_cell_0_15_4
15Harmonic number_cell_0_16_0 1 195 757Harmonic number_cell_0_16_1 /360 360Harmonic number_cell_0_16_2 ~3.31823Harmonic number_cell_0_16_3 3.31823Harmonic number_cell_0_16_4
16Harmonic number_cell_0_17_0 2 436 559Harmonic number_cell_0_17_1 /720 720Harmonic number_cell_0_17_2 ~3.38073Harmonic number_cell_0_17_3 3.38073Harmonic number_cell_0_17_4
17Harmonic number_cell_0_18_0 42 142 223Harmonic number_cell_0_18_1 /12 252 240Harmonic number_cell_0_18_2 ~3.43955Harmonic number_cell_0_18_3 3.43955Harmonic number_cell_0_18_4
18Harmonic number_cell_0_19_0 14 274 301Harmonic number_cell_0_19_1 /4 084 080Harmonic number_cell_0_19_2 ~3.49511Harmonic number_cell_0_19_3 3.49511Harmonic number_cell_0_19_4
19Harmonic number_cell_0_20_0 275 295 799Harmonic number_cell_0_20_1 /77 597 520Harmonic number_cell_0_20_2 ~3.54774Harmonic number_cell_0_20_3 3.54774Harmonic number_cell_0_20_4
20Harmonic number_cell_0_21_0 55 835 135Harmonic number_cell_0_21_1 /15 519 504Harmonic number_cell_0_21_2 ~3.59774Harmonic number_cell_0_21_3 3.59774Harmonic number_cell_0_21_4
21Harmonic number_cell_0_22_0 18 858 053Harmonic number_cell_0_22_1 /5 173 168Harmonic number_cell_0_22_2 ~3.64536Harmonic number_cell_0_22_3 3.64536Harmonic number_cell_0_22_4
22Harmonic number_cell_0_23_0 19 093 197Harmonic number_cell_0_23_1 /5 173 168Harmonic number_cell_0_23_2 ~3.69081Harmonic number_cell_0_23_3 3.69081Harmonic number_cell_0_23_4
23Harmonic number_cell_0_24_0 444 316 699Harmonic number_cell_0_24_1 /118 982 864Harmonic number_cell_0_24_2 ~3.73429Harmonic number_cell_0_24_3 3.73429Harmonic number_cell_0_24_4
24Harmonic number_cell_0_25_0 1 347 822 955Harmonic number_cell_0_25_1 /356 948 592Harmonic number_cell_0_25_2 ~3.77596Harmonic number_cell_0_25_3 3.77596Harmonic number_cell_0_25_4
25Harmonic number_cell_0_26_0 34 052 522 467Harmonic number_cell_0_26_1 /8 923 714 800Harmonic number_cell_0_26_2 ~3.81596Harmonic number_cell_0_26_3 3.81596Harmonic number_cell_0_26_4
26Harmonic number_cell_0_27_0 34 395 742 267Harmonic number_cell_0_27_1 /8 923 714 800Harmonic number_cell_0_27_2 ~3.85442Harmonic number_cell_0_27_3 3.85442Harmonic number_cell_0_27_4
27Harmonic number_cell_0_28_0 312 536 252 003Harmonic number_cell_0_28_1 /80 313 433 200Harmonic number_cell_0_28_2 ~3.89146Harmonic number_cell_0_28_3 3.89146Harmonic number_cell_0_28_4
28Harmonic number_cell_0_29_0 315 404 588 903Harmonic number_cell_0_29_1 /80 313 433 200Harmonic number_cell_0_29_2 ~3.92717Harmonic number_cell_0_29_3 3.92717Harmonic number_cell_0_29_4
29Harmonic number_cell_0_30_0 9 227 046 511 387Harmonic number_cell_0_30_1 /2 329 089 562 800Harmonic number_cell_0_30_2 ~3.96165Harmonic number_cell_0_30_3 3.96165Harmonic number_cell_0_30_4
30Harmonic number_cell_0_31_0 9 304 682 830 147Harmonic number_cell_0_31_1 /2 329 089 562 800Harmonic number_cell_0_31_2 ~3.99499Harmonic number_cell_0_31_3 3.99499Harmonic number_cell_0_31_4
31Harmonic number_cell_0_32_0 290 774 257 297 357Harmonic number_cell_0_32_1 /72 201 776 446 800Harmonic number_cell_0_32_2 ~4.02725Harmonic number_cell_0_32_3 4.02725Harmonic number_cell_0_32_4
32Harmonic number_cell_0_33_0 586 061 125 622 639Harmonic number_cell_0_33_1 /144 403 552 893 600Harmonic number_cell_0_33_2 ~4.05850Harmonic number_cell_0_33_3 4.0585Harmonic number_cell_0_33_4
33Harmonic number_cell_0_34_0 53 676 090 078 349Harmonic number_cell_0_34_1 /13 127 595 717 600Harmonic number_cell_0_34_2 ~4.08880Harmonic number_cell_0_34_3 4.0888Harmonic number_cell_0_34_4
34Harmonic number_cell_0_35_0 54 062 195 834 749Harmonic number_cell_0_35_1 /13 127 595 717 600Harmonic number_cell_0_35_2 ~4.11821Harmonic number_cell_0_35_3 4.11821Harmonic number_cell_0_35_4
35Harmonic number_cell_0_36_0 54 437 269 998 109Harmonic number_cell_0_36_1 /13 127 595 717 600Harmonic number_cell_0_36_2 ~4.14678Harmonic number_cell_0_36_3 4.14678Harmonic number_cell_0_36_4
36Harmonic number_cell_0_37_0 54 801 925 434 709Harmonic number_cell_0_37_1 /13 127 595 717 600Harmonic number_cell_0_37_2 ~4.17456Harmonic number_cell_0_37_3 4.17456Harmonic number_cell_0_37_4
37Harmonic number_cell_0_38_0 2 040 798 836 801 833Harmonic number_cell_0_38_1 /485 721 041 551 200Harmonic number_cell_0_38_2 ~4.20159Harmonic number_cell_0_38_3 4.20159Harmonic number_cell_0_38_4
38Harmonic number_cell_0_39_0 2 053 580 969 474 233Harmonic number_cell_0_39_1 /485 721 041 551 200Harmonic number_cell_0_39_2 ~4.22790Harmonic number_cell_0_39_3 4.2279Harmonic number_cell_0_39_4
39Harmonic number_cell_0_40_0 2 066 035 355 155 033Harmonic number_cell_0_40_1 /485 721 041 551 200Harmonic number_cell_0_40_2 ~4.25354Harmonic number_cell_0_40_3 4.25354Harmonic number_cell_0_40_4
40Harmonic number_cell_0_41_0 2 078 178 381 193 813Harmonic number_cell_0_41_1 /485 721 041 551 200Harmonic number_cell_0_41_2 ~4.27854Harmonic number_cell_0_41_3 4.27854Harmonic number_cell_0_41_4

## Identities involving harmonic numbers Harmonic number_section_0

By definition, the harmonic numbers satisfy the recurrence relation Harmonic number_sentence_11

The harmonic numbers are connected to the Stirling numbers of the first kind by the relation Harmonic number_sentence_12

The functions Harmonic number_sentence_13

satisfy the property Harmonic number_sentence_14

In particular Harmonic number_sentence_15

is an integral of the logarithmic function. Harmonic number_sentence_16

The harmonic numbers satisfy the series identities Harmonic number_sentence_17

these two results are closely analogous to the corresponding integral results Harmonic number_sentence_18

### Identities involving π Harmonic number_section_1

There are several infinite summations involving harmonic numbers and powers of π: Harmonic number_sentence_19

## Calculation Harmonic number_section_2

An integral representation given by Euler is Harmonic number_sentence_20

The equality above is straightforward by the simple algebraic identity Harmonic number_sentence_21

Using the substitution x = 1 − u, another expression for Hn is Harmonic number_sentence_22

The nth harmonic number is about as large as the natural logarithm of n. The reason is that the sum is approximated by the integral Harmonic number_sentence_23

whose value is ln n. Harmonic number_sentence_24

The values of the sequence Hn − ln n decrease monotonically towards the limit Harmonic number_sentence_25

where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. Harmonic number_sentence_26

The corresponding asymptotic expansion is Harmonic number_sentence_27

where Bk are the Bernoulli numbers. Harmonic number_sentence_28

## Generating functions Harmonic number_section_3

A generating function for the harmonic numbers is Harmonic number_sentence_29

where ln(z) is the natural logarithm. Harmonic number_sentence_30

An exponential generating function is Harmonic number_sentence_31

where Ein(z) is the entire exponential integral. Harmonic number_sentence_32

Note that Harmonic number_sentence_33

where Γ(0, z) is the incomplete gamma function. Harmonic number_sentence_34

## Applications Harmonic number_section_5

The harmonic numbers appear in several calculation formulas, such as the digamma function Harmonic number_sentence_35

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 earlier: Harmonic number_sentence_36

although Harmonic number_sentence_37

converges more quickly. Harmonic number_sentence_38

In 2002, Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to the statement that Harmonic number_sentence_39

is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n. Harmonic number_sentence_40

The eigenvalues of the nonlocal problem Harmonic number_sentence_41

## Generalizations Harmonic number_section_6

### Generalized harmonic numbers Harmonic number_section_7

The generalized harmonic number of order m of n is given by Harmonic number_sentence_42

Other notations occasionally used include Harmonic number_sentence_43

The limit as n → ∞ is finite if m > 1, with the generalized harmonic number bounded by and converging to the Riemann zeta function Harmonic number_sentence_44

The smallest natural number k such that k does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H′(k, n) is, for n=1, 2, ... : Harmonic number_sentence_45

Harmonic number_description_list_0

• 77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... (sequence in the OEIS)Harmonic number_item_0_0

Some integrals of generalized harmonic numbers are Harmonic number_sentence_46

and Harmonic number_sentence_47

and Harmonic number_sentence_48

Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using: Harmonic number_sentence_49

A generating function for the generalized harmonic numbers is Harmonic number_sentence_50

A fractional argument for generalized harmonic numbers can be introduced as follows: Harmonic number_sentence_51

Some special values are: Harmonic number_sentence_52

### Multiplication formulas Harmonic number_section_8

The multiplication theorem applies to harmonic numbers. Harmonic number_sentence_53

Using polygamma functions, we obtain Harmonic number_sentence_54

or, more generally, Harmonic number_sentence_55

For generalized harmonic numbers, we have Harmonic number_sentence_56

### Hyperharmonic numbers Harmonic number_section_9

The next generalization was discussed by J. Harmonic number_sentence_57 H. Conway and R. Harmonic number_sentence_58 K. Guy in their 1995 book The Book of Numbers. Harmonic number_sentence_59

Let Harmonic number_sentence_60

Then the nth hyperharmonic number of order r (r>0) is defined recursively as Harmonic number_sentence_61