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, Hn
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
Calculation
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
(
−
1
)
k
(
n
k
)
u
k
−
1
]
d
u
=
−
∑
k
=
1
n
(
−
1
)
k
(
n
k
)
∫
0
1
u
k
−
1
d
u
=
−
∑
k
=
1
n
(
−
1
)
k
1
k
(
n
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}(-1)^{k}{\binom {n}{k))u^{k-1}\right]\,du=-\sum _{k=1}^{n}(-1)^{k}{\binom {n}{k))\int _{0}^{1}u^{k-1}\,du\\[6pt]&=-\sum _{k=1}^{n}(-1)^{k}{\frac {1}{k)){\binom {n}{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 .
Arithmetic properties
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}
.
Generalizations
Generalized harmonic numbers
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 kn 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
−
8
G
−
5
6
π
2
{\displaystyle H_((\frac {1}{4)),2}=16-8G-{\tfrac {5}{6))\pi ^{2))
where G is Catalan's constant ,
H
1
2
,
2
=
4
−
π
2
3
{\displaystyle H_((\frac {1}{2)),2}=4-{\tfrac {\pi ^{2)){3))}
H
3
4
,
2
=
8
G
+
16
9
−
5
6
π
2
{\displaystyle H_((\frac {3}{4)),2}=8G+{\tfrac {16}{9))-{\tfrac {5}{6))\pi ^{2))
H
1
4
,
3
=
64
−
27
ζ
(
3
)
−
π
3
{\displaystyle H_((\frac {1}{4)),3}=64-27\zeta (3)-\pi ^{3))
H
1
2
,
3
=
8
−
6
ζ
(
3
)
{\displaystyle H_((\frac {1}{2)),3}=8-6\zeta (3)}
H
3
4
,
3
=
(
4
3
)
3
−
27
ζ
(
3
)
+
π
3
{\displaystyle H_((\frac {3}{4)),3}={({\tfrac {4}{3)))}^{3}-27\zeta (3)+\pi ^{3))
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
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
,
{\displaystyle H_{2x}={\frac {1}{2))\left(H_{x}+H_{x-{\frac {1}{2))}\right)+\ln 2,}
H
3
x
=
1
3
(
H
x
+
H
x
−
1
3
+
H
x
−
2
3
)
+
ln
3
,
{\displaystyle H_{3x}={\frac {1}{3))\left(H_{x}+H_{x-{\frac {1}{3))}+H_{x-{\frac {2}{3))}\right)+\ln 3,}
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
)
)
{\displaystyle 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
3
x
,
2
=
1
9
(
6
ζ
(
2
)
+
H
x
,
2
+
H
x
−
1
3
,
2
+
H
x
−
2
3
,
2
)
,
{\displaystyle 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),}
where
ζ
(
n
)
{\displaystyle \zeta (n)}
is the Riemann zeta function .
Hyperharmonic numbers
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))
.
Harmonic numbers for real and complex values
The formulae given above,
H
x
=
∫
0
1
1
−
t
x
1
−
t
d
t
=
−
∑
k
=
1
∞
(
x
k
)
(
−
1
)
k
k
{\displaystyle H_{x}=\int _{0}^{1}{\frac {1-t^{x)){1-t))\,dt=-\sum _{k=1}^{\infty }{x \choose k}{\frac {(-1)^{k)){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
k
(
x
k
)
H
k
.
{\displaystyle H_{x,2}=-\sum _{k=1}^{\infty }{\frac {(-1)^{k)){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
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
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
{\displaystyle H_{\frac {1}{2))=2-2\ln {2))
H
1
3
=
3
−
π
2
3
−
3
2
ln
3
{\displaystyle H_{\frac {1}{3))=3-{\tfrac {\pi }{2{\sqrt {3))))-{\tfrac {3}{2))\ln {3))
H
2
3
=
3
2
(
1
−
ln
3
)
+
3
π
6
{\displaystyle H_{\frac {2}{3))={\tfrac {3}{2))(1-\ln {3})+{\sqrt {3)){\tfrac {\pi }{6))}
H
1
4
=
4
−
π
2
−
3
ln
2
{\displaystyle H_{\frac {1}{4))=4-{\tfrac {\pi }{2))-3\ln {2))
H
3
4
=
4
3
−
3
ln
2
+
π
2
{\displaystyle H_{\frac {3}{4))={\tfrac {4}{3))-3\ln {2}+{\tfrac {\pi }{2))}
H
1
6
=
6
−
π
2
3
−
2
ln
2
−
3
2
ln
3
{\displaystyle H_{\frac {1}{6))=6-{\tfrac {\pi }{2)){\sqrt {3))-2\ln {2}-{\tfrac {3}{2))\ln {3))
H
1
8
=
8
−
π
2
−
4
ln
2
−
1
2
{
π
+
ln
(
2
+
2
)
−
ln
(
2
−
2
)
}
{\displaystyle H_{\frac {1}{8))=8-{\tfrac {\pi }{2))-4\ln {2}-{\tfrac {1}{\sqrt {2))}\left\{\pi +\ln \left(2+{\sqrt {2))\right)-\ln \left(2-{\sqrt {2))\right)\right\))
H
1
12
=
12
−
3
(
ln
2
+
ln
3
2
)
−
π
(
1
+
3
2
)
+
2
3
ln
(
2
−
3
)
{\displaystyle H_{\frac {1}{12))=12-3\left(\ln {2}+{\tfrac {\ln {3)){2))\right)-\pi \left(1+{\tfrac {\sqrt {3)){2))\right)+2{\sqrt {3))\ln \left({\sqrt {2-{\sqrt {3))))\right)}
For positive integers p and q with p < q , we have:
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
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))}
^ 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 .
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