J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
N a v i g a t i o n
● M a i n p a g e
● C o n t e n t s
● C u r r e n t e v e n t s
● R a n d o m a r t i c l e
● A b o u t W i k i p e d i a
● C o n t a c t u s
● D o n a t e
C o n t r i b u t e
● H e l p
● L e a r n t o e d i t
● C o m m u n i t y p o r t a l
● R e c e n t c h a n g e s
● U p l o a d f i l e
S e a r c h
Search
A p p e a r a n c e
● C r e a t e a c c o u n t
● L o g i n
P e r s o n a l t o o l s
● C r e a t e a c c o u n t
● L o g i n
P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e
● C o n t r i b u t i o n s
● T a l k
( T o p )
1
I d e n t i t i e s i n v o l v i n g h a r m o n i c n u m b e r s
T o g g l e I d e n t i t i e s i n v o l v i n g h a r m o n i c n u m b e r s s u b s e c t i o n
1 . 1
I d e n t i t i e s i n v o l v i n g π
2
C a l c u l a t i o n
3
G e n e r a t i n g f u n c t i o n s
4
A r i t h m e t i c p r o p e r t i e s
5
A p p l i c a t i o n s
6
G e n e r a l i z a t i o n s
T o g g l e G e n e r a l i z a t i o n s s u b s e c t i o n
6 . 1
G e n e r a l i z e d h a r m o n i c n u m b e r s
6 . 2
M u l t i p l i c a t i o n f o r m u l a s
6 . 3
H y p e r h a r m o n i c n u m b e r s
6 . 4
R o m a n H a r m o n i c n u m b e r s
7
H a r m o n i c n u m b e r s f o r r e a l a n d c o m p l e x v a l u e s
T o g g l e H a r m o n i c n u m b e r s f o r r e a l a n d c o m p l e x v a l u e s s u b s e c t i o n
7 . 1
A l t e r n a t i v e , a s y m p t o t i c f o r m u l a t i o n
7 . 2
S p e c i a l v a l u e s f o r f r a c t i o n a l a r g u m e n t s
7 . 3
R e l a t i o n t o t h e R i e m a n n z e t a f u n c t i o n
8
S e e a l s o
9
N o t e s
10
R e f e r e n c e s
11
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
H a r m o n i c n u m b e r
1 9 l a n g u a g e s
● ا ل ع ر ب ي ة
● C a t a l à
● Č e š t i n a
● E s p a ñ o l
● F r a n ç a i s
● 한 국 어
● ह ि न ् द ी
● I t a l i a n o
● M a g y a r
● N e d e r l a n d s
● 日 本 語
● N o r s k b o k m å l
● P o l s k i
● Р у с с к и й
● S l o v e n č i n a
● S v e n s k a
● த ம ி ழ ்
● У к р а ї н с ь к а
● 中 文
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
I n o t h e r p r o j e c t s
● W i k i m e d i a C o m m o n s
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
The harmonic number
H
n
{\displaystyle H_{n}}
with
n
=
⌊
x
⌋
{\displaystyle n=\lfloor x\rfloor }
(red line) with its asymptotic limit
γ
+
ln
(
x
)
{\displaystyle \gamma +\ln(x )}
(blue line) where
γ
{\displaystyle \gamma }
is the Euler–Mascheroni constant .
In mathematics , the n -th harmonic number is the sum of the reciprocals of the first n natural numbers :
H
n
=
1
+
1
2
+
1
3
+
⋯
+
1
n
=
∑
k
=
1
n
1
k
.
{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.}
Starting from n = 1 , the sequence of harmonic numbers begins:
1
,
3
2
,
11
6
,
25
12
,
137
60
,
…
{\displaystyle 1,{\frac {3}{2}},{\frac {11}{6}},{\frac {25}{12}},{\frac {137}{60}},\dots }
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 numbers have been studied since antiquity and are important in various branches of number theory . They are sometimes loosely termed harmonic series , are closely related to the Riemann zeta function , and appear in the expressions of various special functions .
The harmonic numbers roughly approximate the natural logarithm function [1] : 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers . 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 .
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. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value .
The Bertrand-Chebyshev theorem implies that, except for the case n = 1 , the harmonic numbers are never integers.[2]
The first 40 harmonic numbers
n
Harmonic number, H n
expressed as a fraction
decimal
relative size
1
1
1
1
2
3
/2
1.5
1.5
3
11
/6
~1.83333
1.83333
4
25
/12
~2.08333
2.08333
5
137
/60
~2.28333
2.28333
6
49
/20
2.45
2.45
7
363
/140
~2.59286
2.59286
8
761
/280
~2.71786
2.71786
9
7 129
/2 520
~2.82897
2.82897
10
7 381
/2 520
~2.92897
2.92897
11
83 711
/27 720
~3.01988
3.01988
12
86 021
/27 720
~3.10321
3.10321
13
1 145 993
/360 360
~3.18013
3.18013
14
1 171 733
/360 360
~3.25156
3.25156
15
1 195 757
/360 360
~3.31823
3.31823
16
2 436 559
/720 720
~3.38073
3.38073
17
42 142 223
/12 252 240
~3.43955
3.43955
18
14 274 301
/4 084 080
~3.49511
3.49511
19
275 295 799
/77 597 520
~3.54774
3.54774
20
55 835 135
/15 519 504
~3.59774
3.59774
21
18 858 053
/5 173 168
~3.64536
3.64536
22
19 093 197
/5 173 168
~3.69081
3.69081
23
444 316 699
/118 982 864
~3.73429
3.73429
24
1 347 822 955
/356 948 592
~3.77596
3.77596
25
34 052 522 467
/8 923 714 800
~3.81596
3.81596
26
34 395 742 267
/8 923 714 800
~3.85442
3.85442
27
312 536 252 003
/80 313 433 200
~3.89146
3.89146
28
315 404 588 903
/80 313 433 200
~3.92717
3.92717
29
9 227 046 511 387
/2 329 089 562 800
~3.96165
3.96165
30
9 304 682 830 147
/2 329 089 562 800
~3.99499
3.99499
31
290 774 257 297 357
/72 201 776 446 800
~4.02725
4.02725
32
586 061 125 622 639
/144 403 552 893 600
~4.05850
4.0585
33
53 676 090 078 349
/13 127 595 717 600
~4.08880
4.0888
34
54 062 195 834 749
/13 127 595 717 600
~4.11821
4.11821
35
54 437 269 998 109
/13 127 595 717 600
~4.14678
4.14678
36
54 801 925 434 709
/13 127 595 717 600
~4.17456
4.17456
37
2 040 798 836 801 833
/485 721 041 551 200
~4.20159
4.20159
38
2 053 580 969 474 233
/485 721 041 551 200
~4.22790
4.2279
39
2 066 035 355 155 033
/485 721 041 551 200
~4.25354
4.25354
40
2 078 178 381 193 813
/485 721 041 551 200
~4.27854
4.27854
Identities involving harmonic numbers [ edit ]
By definition, the harmonic numbers satisfy the recurrence relation
H
n
+
1
=
H
n
+
1
n
+
1
.
{\displaystyle H_{n+1}=H_{n}+{\frac {1}{n+1}}.}
The harmonic numbers are connected to the Stirling numbers of the first kind by the relation
H
n
=
1
n
!
[
n
+
1
2
]
.
{\displaystyle H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].}
The harmonic numbers satisfy the series identities
∑
k
=
1
n
H
k
=
(
n
+
1
)
H
n
−
n
{\displaystyle \sum _{k=1}^{n}H_{k}=(n+1)H_{n}-n}
and
∑
k
=
1
n
H
k
2
=
(
n
+
1
)
H
n
2
−
(
2
n
+
1
)
H
n
+
2
n
.
{\displaystyle \sum _{k=1}^{n}H_{k}^{2}=(n+1)H_{n}^{2}-(2n+1)H_{n}+2n.}
These two results are closely analogous to the corresponding integral results
∫
0
x
log
y
d
y
=
x
log
x
−
x
{\displaystyle \int _{0}^{x}\log y\ dy=x\log x-x}
and
∫
0
x
(
log
y
)
2
d
y
=
x
(
log
x
)
2
−
2
x
log
x
+
2
x
.
{\displaystyle \int _{0}^{x}(\log y)^{2}\ dy=x(\log x)^{2}-2x\log x+2x.}
Identities involving π [ edit ]
There are several infinite summations involving harmonic numbers and powers of π :[3] [better source needed ]
∑
n
=
1
∞
H
n
n
⋅
2
n
=
π
2
12
∑
n
=
1
∞
H
n
2
n
2
=
17
360
π
4
∑
n
=
1
∞
H
n
2
(
n
+
1
)
2
=
11
360
π
4
∑
n
=
1
∞
H
n
n
3
=
π
4
72
{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {H_{n}}{n\cdot 2^{n}}}&={\frac {\pi ^{2}}{12}}\\\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{n^{2}}}&={\frac {17}{360}}\pi ^{4}\\\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{(n+1)^{2}}}&={\frac {11}{360}}\pi ^{4}\\\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{3}}}&={\frac {\pi ^{4}}{72}}\end{aligned}}}
Calculation [ edit ]
An integral representation given by Euler [4] is
H
n
=
∫
0
1
1
−
x
n
1
−
x
d
x
.
{\displaystyle H_{n}=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx.}
The equality above is straightforward by the simple algebraic identity
1
−
x
n
1
−
x
=
1
+
x
+
⋯
+
x
n
−
1
.
{\displaystyle {\frac {1-x^{n}}{1-x}}=1+x+\cdots +x^{n-1}.}
Using the substitution x = 1 − u , another expression for H n is
H
n
=
∫
0
1
1
−
x
n
1
−
x
d
x
=
∫
0
1
1
−
(
1
−
u
)
n
u
d
u
=
∫
0
1
[
∑
k
=
1
n
(
n
k
)
(
−
u
)
k
−
1
]
d
u
=
∑
k
=
1
n
(
n
k
)
∫
0
1
(
−
u
)
k
−
1
d
u
=
∑
k
=
1
n
(
n
k
)
(
−
1
)
k
−
1
k
.
{\displaystyle {\begin{aligned}H_{n}&=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx=\int _{0}^{1}{\frac {1-(1-u)^{n}}{u}}\,du\\[6pt]&=\int _{0}^{1}\left[\sum _{k=1}^{n}{\binom {n}{k}}(-u)^{k-1}\right]\,du=\sum _{k=1}^{n}{\binom {n}{k}}\int _{0}^{1}(-u)^{k-1}\,du\\[6pt]&=\sum _{k=1}^{n}{\binom {n}{k}}{\frac {(-1)^{k-1}}{k}}.\end{aligned}}}
Graph demonstrating a connection between harmonic numbers and the natural logarithm . The harmonic number H n can be interpreted as a Riemann sum of the integral:
∫
1
n
+
1
d
x
x
=
ln
(
n
+
1
)
.
{\displaystyle \int _{1}^{n+1}{\frac {dx}{x}}=\ln(n+1).}
The n th harmonic number is about as large as the natural logarithm of n . The reason is that the sum is approximated by the integral
∫
1
n
1
x
d
x
,
{\displaystyle \int _{1}^{n}{\frac {1}{x}}\,dx,}
whose value is ln n .
The values of the sequence H n − ln n decrease monotonically towards the limit
lim
n
→
∞
(
H
n
−
ln
n
)
=
γ
,
{\displaystyle \lim _{n\to \infty }\left(H_{n}-\ln n\right)=\gamma ,}
where γ ≈ 0.5772156649 is the Euler–Mascheroni constant . The corresponding asymptotic expansion is
H
n
∼
ln
n
+
γ
+
1
2
n
−
∑
k
=
1
∞
B
2
k
2
k
n
2
k
=
ln
n
+
γ
+
1
2
n
−
1
12
n
2
+
1
120
n
4
−
⋯
,
{\displaystyle {\begin{aligned}H_{n}&\sim \ln {n}+\gamma +{\frac {1}{2n}}-\sum _{k=1}^{\infty }{\frac {B_{2k}}{2kn^{2k}}}\\&=\ln {n}+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\cdots ,\end{aligned}}}
where B k are the Bernoulli numbers .
Generating functions [ edit ]
A generating function for the harmonic numbers is
∑
n
=
1
∞
z
n
H
n
=
−
ln
(
1
−
z
)
1
−
z
,
{\displaystyle \sum _{n=1}^{\infty }z^{n}H_{n}={\frac {-\ln(1-z)}{1-z}},}
where ln(z ) is the natural logarithm . An exponential generating function is
∑
n
=
1
∞
z
n
n
!
H
n
=
e
z
∑
k
=
1
∞
(
−
1
)
k
−
1
k
z
k
k
!
=
e
z
Ein
(
z
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}H_{n}=e^{z}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{\frac {z^{k}}{k!}}=e^{z}\operatorname {Ein} (z )}
where Ein(z ) is the entire exponential integral . The exponential integral may also be expressed as
Ein
(
z
)
=
E
1
(
z
)
+
γ
+
ln
z
=
Γ
(
0
,
z
)
+
γ
+
ln
z
{\displaystyle \operatorname {Ein} (z )=\mathrm {E} _{1}(z )+\gamma +\ln z=\Gamma (0,z)+\gamma +\ln z}
where Γ(0, z ) is the incomplete gamma function .
Arithmetic properties [ edit ]
The harmonic numbers have several interesting arithmetic properties. It is well-known that
H
n
{\textstyle H_{n}}
is an integer if and only if
n
=
1
{\textstyle n=1}
, a result often attributed to Taeisinger.[5] Indeed, using 2-adic valuation , it is not difficult to prove that for
n
≥
2
{\textstyle n\geq 2}
the numerator of
H
n
{\textstyle H_{n}}
is an odd number while the denominator of
H
n
{\textstyle H_{n}}
is an even number. More precisely,
H
n
=
1
2
⌊
log
2
(
n
)
⌋
a
n
b
n
{\displaystyle H_{n}={\frac {1}{2^{\lfloor \log _{2}(n )\rfloor }}}{\frac {a_{n}}{b_{n}}}}
with some odd integers
a
n
{\textstyle a_{n}}
and
b
n
{\textstyle b_{n}}
.
As a consequence of Wolstenholme's theorem , for any prime number
p
≥
5
{\displaystyle p\geq 5}
the numerator of
H
p
−
1
{\displaystyle H_{p-1}}
is divisible by
p
2
{\textstyle p^{2}}
. Furthermore, Eisenstein[6] proved that for all odd prime number
p
{\textstyle p}
it holds
H
(
p
−
1
)
/
2
≡
−
2
q
p
(
2
)
(
mod
p
)
{\displaystyle H_{(p-1)/2}\equiv -2q_{p}(2 ){\pmod {p}}}
where
q
p
(
2
)
=
(
2
p
−
1
−
1
)
/
p
{\textstyle q_{p}(2 )=(2^{p-1}-1)/p}
is a Fermat quotient , with the consequence that
p
{\textstyle p}
divides the numerator of
H
(
p
−
1
)
/
2
{\displaystyle H_{(p-1)/2}}
if and only if
p
{\textstyle p}
is a Wieferich prime .
In 1991, Eswarathasan and Levine[7] defined
J
p
{\displaystyle J_{p}}
as the set of all positive integers
n
{\displaystyle n}
such that the numerator of
H
n
{\displaystyle H_{n}}
is divisible by a prime number
p
.
{\displaystyle p.}
They proved that
{
p
−
1
,
p
2
−
p
,
p
2
−
1
}
⊆
J
p
{\displaystyle \{p-1,p^{2}-p,p^{2}-1\}\subseteq J_{p}}
for all prime numbers
p
≥
5
,
{\displaystyle p\geq 5,}
and they defined harmonic primes to be the primes
p
{\textstyle p}
such that
J
p
{\displaystyle J_{p}}
has exactly 3 elements.
Eswarathasan and Levine also conjectured that
J
p
{\displaystyle J_{p}}
is a finite set for all primes
p
,
{\displaystyle p,}
and that there are infinitely many harmonic primes. Boyd[8] verified that
J
p
{\displaystyle J_{p}}
is finite for all prime numbers up to
p
=
547
{\displaystyle p=547}
except 83, 127, and 397; and he gave a heuristic suggesting that the density of the harmonic primes in the set of all primes should be
1
/
e
{\displaystyle 1/e}
. Sanna[9] showed that
J
p
{\displaystyle J_{p}}
has zero asymptotic density , while Bing-Ling Wu and Yong-Gao Chen[10] proved that the number of elements of
J
p
{\displaystyle J_{p}}
not exceeding
x
{\displaystyle x}
is at most
3
x
2
3
+
1
25
log
p
{\displaystyle 3x^{{\frac {2}{3}}+{\frac {1}{25\log p}}}}
, for all
x
≥
1
{\displaystyle x\geq 1}
.
Applications [ edit ]
The harmonic numbers appear in several calculation formulas, such as the digamma function
ψ
(
n
)
=
H
n
−
1
−
γ
.
{\displaystyle \psi (n )=H_{n-1}-\gamma .}
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:
γ
=
lim
n
→
∞
(
H
n
−
ln
(
n
)
)
,
{\displaystyle \gamma =\lim _{n\rightarrow \infty }{\left(H_{n}-\ln(n )\right)},}
although
γ
=
lim
n
→
∞
(
H
n
−
ln
(
n
+
1
2
)
)
{\displaystyle \gamma =\lim _{n\to \infty }{\left(H_{n}-\ln \left(n+{\frac {1}{2}}\right)\right)}}
converges more quickly.
In 2002, Jeffrey Lagarias proved[11] that the Riemann hypothesis is equivalent to the statement that
σ
(
n
)
≤
H
n
+
(
log
H
n
)
e
H
n
,
{\displaystyle \sigma (n )\leq H_{n}+(\log H_{n})e^{H_{n}},}
is true for every integer n ≥ 1 with strict inequality if n >1 ; here σ (n ) denotes the sum of the divisors of n .
The eigenvalues of the nonlocal problem
λ
φ
(
x
)
=
∫
−
1
1
φ
(
x
)
−
φ
(
y
)
|
x
−
y
|
d
y
{\displaystyle \lambda \varphi (x )=\int _{-1}^{1}{\frac {\varphi (x )-\varphi (y )}{|x-y|}}\,dy}
are given by
λ
=
2
H
n
{\displaystyle \lambda =2H_{n}}
, where by convention
H
0
=
0
{\displaystyle H_{0}=0}
, and the corresponding eigenfunctions are given by the Legendre polynomials
φ
(
x
)
=
P
n
(
x
)
{\displaystyle \varphi (x )=P_{n}(x )}
.[12]
Generalizations [ edit ]
Generalized harmonic numbers [ edit ]
The n th generalized harmonic number of order m is given by
H
n
,
m
=
∑
k
=
1
n
1
k
m
.
{\displaystyle H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}.}
(In some sources, this may also be denoted by
H
n
(
m
)
{\textstyle H_{n}^{(m )}}
or
H
m
(
n
)
.
{\textstyle H_{m}(n ).}
)
The special case m = 0 gives
H
n
,
0
=
n
.
{\displaystyle H_{n,0}=n.}
The special case m = 1 reduces to the usual harmonic number:
H
n
,
1
=
H
n
=
∑
k
=
1
n
1
k
.
{\displaystyle H_{n,1}=H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}.}
The limit of
H
n
,
m
{\textstyle H_{n,m}}
as n → ∞ is finite if m >1 , with the generalized harmonic number bounded by and converging to the Riemann zeta function
lim
n
→
∞
H
n
,
m
=
ζ
(
m
)
.
{\displaystyle \lim _{n\rightarrow \infty }H_{n,m}=\zeta (m ).}
The smallest natural number k such that k n 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, ... :
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 A128670 in the OEIS )
The related sum
∑
k
=
1
n
k
m
{\displaystyle \sum _{k=1}^{n}k^{m}}
occurs in the study of Bernoulli numbers ; the harmonic numbers also appear in the study of Stirling numbers .
Some integrals of generalized harmonic numbers are
∫
0
a
H
x
,
2
d
x
=
a
π
2
6
−
H
a
{\displaystyle \int _{0}^{a}H_{x,2}\,dx=a{\frac {\pi ^{2}}{6}}-H_{a}}
and
∫
0
a
H
x
,
3
d
x
=
a
A
−
1
2
H
a
,
2
,
{\displaystyle \int _{0}^{a}H_{x,3}\,dx=aA-{\frac {1}{2}}H_{a,2},}
where A is Apéry's constant ζ (3 ),
and
∑
k
=
1
n
H
k
,
m
=
(
n
+
1
)
H
n
,
m
−
H
n
,
m
−
1
for
m
≥
0.
{\displaystyle \sum _{k=1}^{n}H_{k,m}=(n+1)H_{n,m}-H_{n,m-1}{\text{ for }}m\geq 0.}
Every generalized harmonic number of order m can be written as a function of harmonic numbers of order
m
−
1
{\displaystyle m-1}
using
H
n
,
m
=
∑
k
=
1
n
−
1
H
k
,
m
−
1
k
(
k
+
1
)
+
H
n
,
m
−
1
n
{\displaystyle H_{n,m}=\sum _{k=1}^{n-1}{\frac {H_{k,m-1}}{k(k+1)}}+{\frac {H_{n,m-1}}{n}}}
for example:
H
4
,
3
=
H
1
,
2
1
⋅
2
+
H
2
,
2
2
⋅
3
+
H
3
,
2
3
⋅
4
+
H
4
,
2
4
{\displaystyle H_{4,3}={\frac {H_{1,2}}{1\cdot 2}}+{\frac {H_{2,2}}{2\cdot 3}}+{\frac {H_{3,2}}{3\cdot 4}}+{\frac {H_{4,2}}{4}}}
A generating function for the generalized harmonic numbers is
∑
n
=
1
∞
z
n
H
n
,
m
=
Li
m
(
z
)
1
−
z
,
{\displaystyle \sum _{n=1}^{\infty }z^{n}H_{n,m}={\frac {\operatorname {Li} _{m}(z )}{1-z}},}
where
Li
m
(
z
)
{\displaystyle \operatorname {Li} _{m}(z )}
is the polylogarithm , and |z | <1 . The generating function given above for m = 1 is a special case of this formula.
A fractional argument for generalized harmonic numbers can be introduced as follows:
For every
p
,
q
>
0
{\displaystyle p,q>0}
integer, and
m
>
1
{\displaystyle m>1}
integer or not, we have from polygamma functions:
H
q
/
p
,
m
=
ζ
(
m
)
−
p
m
∑
k
=
1
∞
1
(
q
+
p
k
)
m
{\displaystyle H_{q/p,m}=\zeta (m )-p^{m}\sum _{k=1}^{\infty }{\frac {1}{(q+pk)^{m}}}}
where
ζ
(
m
)
{\displaystyle \zeta (m )}
is the Riemann zeta function . The relevant recurrence relation is
H
a
,
m
=
H
a
−
1
,
m
+
1
a
m
.
{\displaystyle H_{a,m}=H_{a-1,m}+{\frac {1}{a^{m}}}.}
Some special values are
H
1
4
,
2
=
16
−
5
6
π
2
−
8
G
H
1
2
,
2
=
4
−
π
2
3
H
3
4
,
2
=
16
9
−
5
6
π
2
+
8
G
H
1
4
,
3
=
64
−
π
3
−
27
ζ
(
3
)
H
1
2
,
3
=
8
−
6
ζ
(
3
)
H
3
4
,
3
=
(
4
3
)
3
+
π
3
−
27
ζ
(
3
)
{\displaystyle {\begin{aligned}H_{{\frac {1}{4}},2}&=16-{\tfrac {5}{6}}\pi ^{2}-8G\\H_{{\frac {1}{2}},2}&=4-{\frac {\pi ^{2}}{3}}\\H_{{\frac {3}{4}},2}&={\frac {16}{9}}-{\frac {5}{6}}\pi ^{2}+8G\\H_{{\frac {1}{4}},3}&=64-\pi ^{3}-27\zeta (3 )\\H_{{\frac {1}{2}},3}&=8-6\zeta (3 )\\H_{{\frac {3}{4}},3}&=\left({\frac {4}{3}}\right)^{3}+\pi ^{3}-27\zeta (3 )\end{aligned}}}
where G is Catalan's constant . In the special case that
p
=
1
{\displaystyle p=1}
, we get
H
n
,
m
=
ζ
(
m
,
1
)
−
ζ
(
m
,
n
+
1
)
,
{\displaystyle H_{n,m}=\zeta (m,1)-\zeta (m,n+1),}
where
ζ
(
m
,
n
)
{\displaystyle \zeta (m,n)}
is the Hurwitz zeta function . This relationship is used to calculate harmonic numbers numerically.
Multiplication formulas [ edit ]
The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain
H
2
x
=
1
2
(
H
x
+
H
x
−
1
2
)
+
ln
2
H
3
x
=
1
3
(
H
x
+
H
x
−
1
3
+
H
x
−
2
3
)
+
ln
3
,
{\displaystyle {\begin{aligned}H_{2x}&={\frac {1}{2}}\left(H_{x}+H_{x-{\frac {1}{2}}}\right)+\ln 2\\H_{3x}&={\frac {1}{3}}\left(H_{x}+H_{x-{\frac {1}{3}}}+H_{x-{\frac {2}{3}}}\right)+\ln 3,\end{aligned}}}
or, more generally,
H
n
x
=
1
n
(
H
x
+
H
x
−
1
n
+
H
x
−
2
n
+
⋯
+
H
x
−
n
−
1
n
)
+
ln
n
.
{\displaystyle H_{nx}={\frac {1}{n}}\left(H_{x}+H_{x-{\frac {1}{n}}}+H_{x-{\frac {2}{n}}}+\cdots +H_{x-{\frac {n-1}{n}}}\right)+\ln n.}
For generalized harmonic numbers, we have
H
2
x
,
2
=
1
2
(
ζ
(
2
)
+
1
2
(
H
x
,
2
+
H
x
−
1
2
,
2
)
)
H
3
x
,
2
=
1
9
(
6
ζ
(
2
)
+
H
x
,
2
+
H
x
−
1
3
,
2
+
H
x
−
2
3
,
2
)
,
{\displaystyle {\begin{aligned}H_{2x,2}&={\frac {1}{2}}\left(\zeta (2 )+{\frac {1}{2}}\left(H_{x,2}+H_{x-{\frac {1}{2}},2}\right)\right)\\H_{3x,2}&={\frac {1}{9}}\left(6\zeta (2 )+H_{x,2}+H_{x-{\frac {1}{3}},2}+H_{x-{\frac {2}{3}},2}\right),\end{aligned}}}
where
ζ
(
n
)
{\displaystyle \zeta (n )}
is the Riemann zeta function .
Hyperharmonic numbers [ edit ]
The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers .[1] : 258 Let
H
n
(
0
)
=
1
n
.
{\displaystyle H_{n}^{(0)}={\frac {1}{n}}.}
Then the nth hyperharmonic number of order r (r>0 ) is defined recursively as
H
n
(
r
)
=
∑
k
=
1
n
H
k
(
r
−
1
)
.
{\displaystyle H_{n}^{(r )}=\sum _{k=1}^{n}H_{k}^{(r-1)}.}
In particular,
H
n
(
1
)
{\displaystyle H_{n}^{(1 )}}
is the ordinary harmonic number
H
n
{\displaystyle H_{n}}
.
Roman Harmonic numbers [ edit ]
The Roman Harmonic numbers ,[13] named after Steven Roman , were introduced by Daniel Loeb and Gian-Carlo Rota in the context of a generalization of umbral calculus with logarithms.[14] There are many possible definitions, but one of them, for
n
,
k
≥
0
{\displaystyle n,k\geq 0}
, is
c
n
(
0
)
=
1
,
{\displaystyle c_{n}^{(0)}=1,}
and
c
n
(
k
+
1
)
=
∑
i
=
1
n
c
i
(
k
)
i
.
{\displaystyle c_{n}^{(k+1)}=\sum _{i=1}^{n}{\frac {c_{i}^{(k )}}{i}}.}
Of course,
c
n
(
1
)
=
H
n
.
{\displaystyle c_{n}^{(1 )}=H_{n}.}
If
n
≠
0
{\displaystyle n\neq 0}
, they satisfy
c
n
(
k
+
1
)
−
c
n
(
k
)
n
=
c
n
−
1
(
k
+
1
)
.
{\displaystyle c_{n}^{(k+1)}-{\frac {c_{n}^{(k )}}{n}}=c_{n-1}^{(k+1)}.}
Closed form formulas are
c
n
(
k
)
=
n
!
(
−
1
)
k
s
(
−
n
,
k
)
,
{\displaystyle c_{n}^{(k )}=n!(-1)^{k}s(-n,k),}
where
s
(
−
n
,
k
)
{\displaystyle s(-n,k)}
is Stirling numbers of the first kind generalized to negative first argument, and
c
n
(
k
)
=
∑
j
=
1
n
(
n
j
)
(
−
1
)
j
−
1
j
k
,
{\displaystyle c_{n}^{(k )}=\sum _{j=1}^{n}{\binom {n}{j}}{\frac {(-1)^{j-1}}{j^{k}}},}
which was found by Donald Knuth .
In fact, these numbers were defined in a more general manner using Roman numbers and Roman factorials , that include negative values for
n
{\displaystyle n}
. This generalization was useful in their study to define Harmonic logarithms .
Harmonic numbers for real and complex values [ edit ]
The formulae given above,
H
x
=
∫
0
1
1
−
t
x
1
−
t
d
t
=
∑
k
=
1
∞
(
x
k
)
(
−
1
)
k
−
1
k
{\displaystyle H_{x}=\int _{0}^{1}{\frac {1-t^{x}}{1-t}}\,dt=\sum _{k=1}^{\infty }{x \choose k}{\frac {(-1)^{k-1}}{k}}}
are an integral and a series representation for a function that interpolates the harmonic numbers and, via analytic continuation , extends the definition to the complex plane other than the negative integers x . The interpolating function is in fact closely related to the digamma function
H
x
=
ψ
(
x
+
1
)
+
γ
,
{\displaystyle H_{x}=\psi (x+1)+\gamma ,}
where ψ (x ) is the digamma function, and γ is the Euler–Mascheroni constant . The integration process may be repeated to obtain
H
x
,
2
=
∑
k
=
1
∞
(
−
1
)
k
−
1
k
(
x
k
)
H
k
.
{\displaystyle H_{x,2}=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{x \choose k}H_{k}.}
The Taylor series for the harmonic numbers is
H
x
=
∑
k
=
2
∞
(
−
1
)
k
ζ
(
k
)
x
k
−
1
for
|
x
|
<
1
{\displaystyle H_{x}=\sum _{k=2}^{\infty }(-1)^{k}\zeta (k )\;x^{k-1}\quad {\text{ for }}|x|<1}
which comes from the Taylor series for the digamma function (
ζ
{\displaystyle \zeta }
is the Riemann zeta function ).
Alternative, asymptotic formulation [ edit ]
When seeking to approximate H x for a complex number x , it is effective to first compute H m for some large integer m . Use that as an approximation for the value of H m +x . Then use the recursion relation H n = H n −1 + 1/n backwards m times, to unwind it to an approximation for H x . Furthermore, this approximation is exact in the limit as m goes to infinity.
Specifically, for a fixed integer n , it is the case that
lim
m
→
∞
[
H
m
+
n
−
H
m
]
=
0.
{\displaystyle \lim _{m\rightarrow \infty }\left[H_{m+n}-H_{m}\right]=0.}
If n is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined harmonic numbers for non-integers. However, we do get a unique extension of the harmonic numbers to the non-integers by insisting that this equation continue to hold when the arbitrary integer n is replaced by an arbitrary complex number x ,
lim
m
→
∞
[
H
m
+
x
−
H
m
]
=
0
.
{\displaystyle \lim _{m\rightarrow \infty }\left[H_{m+x}-H_{m}\right]=0\,.}
Swapping the order of the two sides of this equation and then subtracting them from H x gives
H
x
=
lim
m
→
∞
[
H
m
−
(
H
m
+
x
−
H
x
)
]
=
lim
m
→
∞
[
(
∑
k
=
1
m
1
k
)
−
(
∑
k
=
1
m
1
x
+
k
)
]
=
lim
m
→
∞
∑
k
=
1
m
(
1
k
−
1
x
+
k
)
=
x
∑
k
=
1
∞
1
k
(
x
+
k
)
.
{\displaystyle {\begin{aligned}H_{x}&=\lim _{m\rightarrow \infty }\left[H_{m}-(H_{m+x}-H_{x})\right]\\[6pt]&=\lim _{m\rightarrow \infty }\left[\left(\sum _{k=1}^{m}{\frac {1}{k}}\right)-\left(\sum _{k=1}^{m}{\frac {1}{x+k}}\right)\right]\\[6pt]&=\lim _{m\rightarrow \infty }\sum _{k=1}^{m}\left({\frac {1}{k}}-{\frac {1}{x+k}}\right)=x\sum _{k=1}^{\infty }{\frac {1}{k(x+k)}}\,.\end{aligned}}}
This infinite series converges for all complex numbers x except the negative integers, which fail because trying to use the recursion relation H n = H n −1 + 1/n backwards through the value n = 0 involves a division by zero. By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1 ) H 0 = 0 , (2 ) H x = H x −1 + 1/x for all complex numbers x except the non-positive integers, and (3 ) limm →+∞ (H m +x − H m ) = 0 for all complex values x .
This last formula can be used to show that
∫
0
1
H
x
d
x
=
γ
,
{\displaystyle \int _{0}^{1}H_{x}\,dx=\gamma ,}
where γ is the Euler–Mascheroni constant or, more generally, for every n we have:
∫
0
n
H
x
d
x
=
n
γ
+
ln
(
n
!
)
.
{\displaystyle \int _{0}^{n}H_{x}\,dx=n\gamma +\ln(n!).}
Special values for fractional arguments [ edit ]
There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
H
α
=
∫
0
1
1
−
x
α
1
−
x
d
x
.
{\displaystyle H_{\alpha }=\int _{0}^{1}{\frac {1-x^{\alpha }}{1-x}}\,dx\,.}
More values may be generated from the recurrence relation
H
α
=
H
α
−
1
+
1
α
,
{\displaystyle H_{\alpha }=H_{\alpha -1}+{\frac {1}{\alpha }}\,,}
or from the reflection relation
H
1
−
α
−
H
α
=
π
cot
(
π
α
)
−
1
α
+
1
1
−
α
.
{\displaystyle H_{1-\alpha }-H_{\alpha }=\pi \cot {(\pi \alpha )}-{\frac {1}{\alpha }}+{\frac {1}{1-\alpha }}\,.}
For example:
H
1
2
=
2
−
2
ln
2
H
1
3
=
3
−
π
2
3
−
3
2
ln
3
H
2
3
=
3
2
+
π
2
3
−
3
2
ln
3
H
1
4
=
4
−
π
2
−
3
ln
2
H
3
4
=
4
3
+
π
2
−
3
ln
2
H
1
6
=
6
−
3
2
π
−
2
ln
2
−
3
2
ln
3
H
1
8
=
8
−
1
+
2
2
π
−
4
ln
2
−
1
2
(
ln
(
2
+
2
)
−
ln
(
2
−
2
)
)
H
1
12
=
12
−
(
1
+
3
2
)
π
−
3
ln
2
−
3
2
ln
3
+
3
ln
(
2
−
3
)
{\displaystyle {\begin{aligned}H_{\frac {1}{2}}&=2-2\ln 2\\H_{\frac {1}{3}}&=3-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3\\H_{\frac {2}{3}}&={\frac {3}{2}}+{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3\\H_{\frac {1}{4}}&=4-{\frac {\pi }{2}}-3\ln 2\\H_{\frac {3}{4}}&={\frac {4}{3}}+{\frac {\pi }{2}}-3\ln 2\\H_{\frac {1}{6}}&=6-{\frac {\sqrt {3}}{2}}\pi -2\ln 2-{\frac {3}{2}}\ln 3\\H_{\frac {1}{8}}&=8-{\frac {1+{\sqrt {2}}}{2}}\pi -4\ln {2}-{\frac {1}{\sqrt {2}}}\left(\ln \left(2+{\sqrt {2}}\right)-\ln \left(2-{\sqrt {2}}\right)\right)\\H_{\frac {1}{12}}&=12-\left(1+{\frac {\sqrt {3}}{2}}\right)\pi -3\ln {2}-{\frac {3}{2}}\ln {3}+{\sqrt {3}}\ln \left(2-{\sqrt {3}}\right)\end{aligned}}}
Which are computed via Gauss's digamma theorem , which essentially states that for positive integers p and q with p < q
H
p
q
=
q
p
+
2
∑
k
=
1
⌊
q
−
1
2
⌋
cos
(
2
π
p
k
q
)
ln
(
sin
(
π
k
q
)
)
−
π
2
cot
(
π
p
q
)
−
ln
(
2
q
)
{\displaystyle H_{\frac {p}{q}}={\frac {q}{p}}+2\sum _{k=1}^{\lfloor {\frac {q-1}{2}}\rfloor }\cos \left({\frac {2\pi pk}{q}}\right)\ln \left({\sin \left({\frac {\pi k}{q}}\right)}\right)-{\frac {\pi }{2}}\cot \left({\frac {\pi p}{q}}\right)-\ln \left(2q\right)}
Relation to the Riemann zeta function [ edit ]
Some derivatives of fractional harmonic numbers are given by
d
n
H
x
d
x
n
=
(
−
1
)
n
+
1
n
!
[
ζ
(
n
+
1
)
−
H
x
,
n
+
1
]
d
n
H
x
,
2
d
x
n
=
(
−
1
)
n
+
1
(
n
+
1
)
!
[
ζ
(
n
+
2
)
−
H
x
,
n
+
2
]
d
n
H
x
,
3
d
x
n
=
(
−
1
)
n
+
1
1
2
(
n
+
2
)
!
[
ζ
(
n
+
3
)
−
H
x
,
n
+
3
]
.
{\displaystyle {\begin{aligned}{\frac {d^{n}H_{x}}{dx^{n}}}&=(-1)^{n+1}n!\left[\zeta (n+1)-H_{x,n+1}\right]\\[6pt]{\frac {d^{n}H_{x,2}}{dx^{n}}}&=(-1)^{n+1}(n+1)!\left[\zeta (n+2)-H_{x,n+2}\right]\\[6pt]{\frac {d^{n}H_{x,3}}{dx^{n}}}&=(-1)^{n+1}{\frac {1}{2}}(n+2)!\left[\zeta (n+3)-H_{x,n+3}\right].\end{aligned}}}
And using Maclaurin series , we have for x < 1 that
H
x
=
∑
n
=
1
∞
(
−
1
)
n
+
1
x
n
ζ
(
n
+
1
)
H
x
,
2
=
∑
n
=
1
∞
(
−
1
)
n
+
1
(
n
+
1
)
x
n
ζ
(
n
+
2
)
H
x
,
3
=
1
2
∑
n
=
1
∞
(
−
1
)
n
+
1
(
n
+
1
)
(
n
+
2
)
x
n
ζ
(
n
+
3
)
.
{\displaystyle {\begin{aligned}H_{x}&=\sum _{n=1}^{\infty }(-1)^{n+1}x^{n}\zeta (n+1)\\[5pt]H_{x,2}&=\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)x^{n}\zeta (n+2)\\[5pt]H_{x,3}&={\frac {1}{2}}\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)(n+2)x^{n}\zeta (n+3).\end{aligned}}}
For fractional arguments between 0 and 1 and for a > 1,
H
1
/
a
=
1
a
(
ζ
(
2
)
−
1
a
ζ
(
3
)
+
1
a
2
ζ
(
4
)
−
1
a
3
ζ
(
5
)
+
⋯
)
H
1
/
a
,
2
=
1
a
(
2
ζ
(
3
)
−
3
a
ζ
(
4
)
+
4
a
2
ζ
(
5
)
−
5
a
3
ζ
(
6
)
+
⋯
)
H
1
/
a
,
3
=
1
2
a
(
2
⋅
3
ζ
(
4
)
−
3
⋅
4
a
ζ
(
5
)
+
4
⋅
5
a
2
ζ
(
6
)
−
5
⋅
6
a
3
ζ
(
7
)
+
⋯
)
.
{\displaystyle {\begin{aligned}H_{1/a}&={\frac {1}{a}}\left(\zeta (2 )-{\frac {1}{a}}\zeta (3 )+{\frac {1}{a^{2}}}\zeta (4 )-{\frac {1}{a^{3}}}\zeta (5 )+\cdots \right)\\[6pt]H_{1/a,\,2}&={\frac {1}{a}}\left(2\zeta (3 )-{\frac {3}{a}}\zeta (4 )+{\frac {4}{a^{2}}}\zeta (5 )-{\frac {5}{a^{3}}}\zeta (6 )+\cdots \right)\\[6pt]H_{1/a,\,3}&={\frac {1}{2a}}\left(2\cdot 3\zeta (4 )-{\frac {3\cdot 4}{a}}\zeta (5 )+{\frac {4\cdot 5}{a^{2}}}\zeta (6 )-{\frac {5\cdot 6}{a^{3}}}\zeta (7 )+\cdots \right).\end{aligned}}}
See also [ edit ]
^ a b
John H., Conway; Richard K., Guy (1995). The book of numbers . Copernicus.
^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics . Addison-Wesley.
^ Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html
^ Sandifer, C. Edward (2007), How Euler Did It , MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638 .
^ Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics . Boca Raton, FL: Chapman & Hall/CRC. p. 3115. ISBN 978-1-58488-347-0 .
^ Eisenstein, Ferdinand Gotthold Max (1850). "Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen ahhängen und durch gewisse lineare Funktional-Gleichungen definirt werden". Berichte Königl. Preuβ. Akad. Wiss. Berlin . 15 : 36–42.
^ Eswarathasan, Arulappah; Levine, Eugene (1991). "p-integral harmonic sums" . Discrete Mathematics . 91 (3 ): 249–257. doi :10.1016/0012-365X(90 )90234-9 .
^ Boyd, David W. (1994). "A p-adic study of the partial sums of the harmonic series" . Experimental Mathematics . 3 (4 ): 287–302. CiteSeerX 10.1.1.56.7026 . doi :10.1080/10586458.1994.10504298 .
^ Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers" (PDF) . Journal of Number Theory . 166 : 41–46. doi :10.1016/j.jnt.2016.02.020 . hdl :2318/1622121 .
^ Chen, Yong-Gao; Wu, Bing-Ling (2017). "On certain properties of harmonic numbers". Journal of Number Theory . 175 : 66–86. doi :10.1016/j.jnt.2016.11.027 .
^ Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly . 109 (6 ): 534–543. arXiv :math.NT/0008177 . doi :10.2307/2695443 . JSTOR 2695443 .
^ E.O. Tuck (1964). "Some methods for flows past blunt slender bodies". J. Fluid Mech . 18 (4 ): 619–635. Bibcode :1964JFM....18..619T . doi :10.1017/S0022112064000453 . S2CID 123120978 .
^ Sesma, J. (2017). "The Roman harmonic numbers revisited" . Journal of Number Theory . 180 : 544–565. arXiv :1702.03718 . doi :10.1016/j.jnt.2017.05.009 . ISSN 0022-314X .
^ Loeb, Daniel E; Rota, Gian-Carlo (1989). "Formal power series of logarithmic type" . Advances in Mathematics . 75 (1 ): 1–118. doi :10.1016/0001-8708(89 )90079-0 . ISSN 0001-8708 .
References [ edit ]
External links [ edit ]
This article incorporates material from Harmonic number on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Harmonic_number&oldid=1230558894 "
C a t e g o r y :
● N u m b e r t h e o r y
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● U s e A m e r i c a n E n g l i s h f r o m M a r c h 2 0 1 9
● A l l W i k i p e d i a a r t i c l e s w r i t t e n i n A m e r i c a n E n g l i s h
● A l l a r t i c l e s l a c k i n g r e l i a b l e r e f e r e n c e s
● A r t i c l e s l a c k i n g r e l i a b l e r e f e r e n c e s f r o m F e b r u a r y 2 0 2 2
● A r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s f r o m M a y 2 0 1 9
● A l l a r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s
● C S 1 : l o n g v o l u m e v a l u e
● W e b a r c h i v e t e m p l a t e w a y b a c k l i n k s
● W i k i p e d i a a r t i c l e s i n c o r p o r a t i n g t e x t f r o m P l a n e t M a t h
● T h i s p a g e w a s l a s t e d i t e d o n 2 3 J u n e 2 0 2 4 , a t 1 2 : 1 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w