<s>
In	O
number	O
theory	O
,	O
an	O
arithmetic	O
,	O
arithmetical	O
,	O
or	O
number-theoretic	B-Algorithm
function	I-Algorithm
is	O
for	O
most	O
authors	O
any	O
function	O
f(n )	O
whose	O
domain	O
is	O
the	O
positive	O
integers	O
and	O
whose	O
range	O
is	O
a	O
subset	O
of	O
the	O
complex	O
numbers	O
.	O
</s>
<s>
Hardy	O
&	O
Wright	O
include	O
in	O
their	O
definition	O
the	O
requirement	O
that	O
an	O
arithmetical	B-Algorithm
function	I-Algorithm
"	O
expresses	O
some	O
arithmetical	O
property	O
of	O
n	O
"	O
.	O
</s>
<s>
An	O
example	O
of	O
an	O
arithmetic	B-Algorithm
function	I-Algorithm
is	O
the	O
divisor	O
function	O
whose	O
value	O
at	O
a	O
positive	O
integer	O
n	O
is	O
equal	O
to	O
the	O
number	O
of	O
divisors	O
of	O
n	O
.	O
</s>
<s>
There	O
is	O
a	O
larger	O
class	O
of	O
number-theoretic	B-Algorithm
functions	I-Algorithm
that	O
do	O
not	O
fit	O
the	O
above	O
definition	O
,	O
for	O
example	O
,	O
the	O
prime-counting	O
functions	O
.	O
</s>
<s>
Arithmetic	B-Algorithm
functions	I-Algorithm
are	O
often	O
extremely	O
irregular	O
(	O
see	O
table	O
)	O
,	O
but	O
some	O
of	O
them	O
have	O
series	O
expansions	O
in	O
terms	O
of	O
Ramanujan	O
's	O
sum	O
.	O
</s>
<s>
See	O
Dirichlet	O
convolution	B-Language
,	O
below	O
.	O
</s>
<s>
Although	O
it	O
is	O
hard	O
to	O
say	O
exactly	O
what	O
"	O
arithmetical	O
property	O
of	O
n	O
"	O
it	O
"	O
expresses	O
"	O
,	O
(τ(n )	O
is	O
(	O
2π	O
)	O
−12	O
times	O
the	O
nth	O
Fourier	O
coefficient	O
in	O
the	O
q-expansion	O
of	O
the	O
modular	O
discriminant	O
function	O
)	O
it	O
is	O
included	O
among	O
the	O
arithmetical	B-Algorithm
functions	I-Algorithm
because	O
it	O
is	O
multiplicative	O
and	O
it	O
occurs	O
in	O
identities	O
involving	O
certain	O
σk(n )	O
and	O
rk(n )	O
functions	O
(	O
because	O
these	O
are	O
also	O
coefficients	O
in	O
the	O
expansion	O
of	O
modular	O
forms	O
)	O
.	O
</s>
<s>
These	O
important	O
functions	O
(	O
which	O
are	O
not	O
arithmetic	B-Algorithm
functions	I-Algorithm
)	O
are	O
defined	O
for	O
non-negative	O
real	O
arguments	O
,	O
and	O
are	O
used	O
in	O
the	O
various	O
statements	O
and	O
proofs	O
of	O
the	O
prime	O
number	O
theorem	O
.	O
</s>
<s>
They	O
are	O
summation	O
functions	O
(	O
see	O
the	O
main	O
section	O
just	O
below	O
)	O
of	O
arithmetic	B-Algorithm
functions	I-Algorithm
which	O
are	O
neither	O
multiplicative	O
nor	O
additive	O
.	O
</s>
<s>
It	O
is	O
the	O
summation	O
function	O
of	O
the	O
arithmetic	B-Algorithm
function	I-Algorithm
which	O
takes	O
the	O
value	O
1/k	O
on	O
integers	O
which	O
are	O
the	O
k-th	O
power	O
of	O
some	O
prime	O
number	O
,	O
and	O
the	O
value	O
0	O
on	O
other	O
integers	O
.	O
</s>
<s>
λ(n )	O
,	O
the	O
Carmichael	B-Algorithm
function	I-Algorithm
,	O
is	O
the	O
smallest	O
positive	O
number	O
such	O
that	O
for	O
all	O
a	O
coprime	O
to	O
n	O
.	O
Equivalently	O
,	O
it	O
is	O
the	O
least	O
common	O
multiple	O
of	O
the	O
orders	O
of	O
the	O
elements	O
of	O
the	O
multiplicative	O
group	O
of	O
integers	O
modulo	O
n	O
.	O
</s>
<s>
Since	O
such	O
functions	O
are	O
often	O
represented	O
by	O
series	O
and	O
integrals	O
,	O
to	O
achieve	O
pointwise	O
convergence	B-Algorithm
it	O
is	O
usual	O
to	O
define	O
the	O
value	O
at	O
the	O
discontinuities	O
as	O
the	O
average	O
of	O
the	O
values	O
to	O
the	O
left	O
and	O
right	O
:	O
</s>
<s>
Individual	O
values	O
of	O
arithmetic	B-Algorithm
functions	I-Algorithm
may	O
fluctuate	O
wildly	O
–	O
as	O
in	O
most	O
of	O
the	O
above	O
examples	O
.	O
</s>
<s>
An	O
average	O
order	O
of	O
an	O
arithmetic	B-Algorithm
function	I-Algorithm
is	O
some	O
simpler	O
or	O
better-understood	O
function	O
which	O
has	O
the	O
same	O
summation	O
function	O
asymptotically	O
,	O
and	O
hence	O
takes	O
the	O
same	O
values	O
"	O
on	O
average	O
"	O
.	O
</s>
<s>
Given	O
an	O
arithmetic	B-Algorithm
function	I-Algorithm
a(n )	O
,	O
let	O
Fa(s )	O
,	O
for	O
complex	O
s	O
,	O
be	O
the	O
function	O
defined	O
by	O
the	O
corresponding	O
Dirichlet	O
series	O
(	O
where	O
it	O
converges	B-Algorithm
)	O
:	O
</s>
<s>
Consider	O
two	O
arithmetic	B-Algorithm
functions	I-Algorithm
a	O
and	O
b	O
and	O
their	O
respective	O
generating	O
functions	O
Fa(s )	O
and	O
Fb(s )	O
.	O
</s>
<s>
This	O
function	O
c	O
is	O
called	O
the	O
Dirichlet	O
convolution	B-Language
of	O
a	O
and	O
b	O
,	O
and	O
is	O
denoted	O
by	O
.	O
</s>
<s>
A	O
particularly	O
important	O
case	O
is	O
convolution	B-Language
with	O
the	O
constant	O
function	O
a(n )	O
=	O
1	O
for	O
all	O
n	O
,	O
corresponding	O
to	O
multiplying	O
the	O
generating	O
function	O
by	O
the	O
zeta	O
function	O
:	O
</s>
<s>
There	O
are	O
a	O
great	O
many	O
formulas	O
connecting	O
arithmetical	B-Algorithm
functions	I-Algorithm
with	O
each	O
other	O
and	O
with	O
the	O
functions	O
of	O
analysis	O
,	O
especially	O
powers	O
,	O
roots	O
,	O
and	O
the	O
exponential	O
and	O
log	O
functions	O
.	O
</s>
<s>
The	O
page	O
divisor	O
sum	O
identities	O
contains	O
many	O
more	O
generalized	O
and	O
related	O
examples	O
of	O
identities	O
involving	O
arithmetic	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Here	O
"	O
convolution	B-Language
"	O
does	O
not	O
mean	O
"	O
Dirichlet	O
convolution	B-Language
"	O
but	O
instead	O
refers	O
to	O
the	O
formula	O
for	O
the	O
coefficients	O
of	O
the	O
product	O
of	O
two	O
power	O
series	O
:	O
</s>
<s>
The	O
sequence	O
is	O
called	O
the	O
convolution	B-Language
or	O
the	O
Cauchy	O
product	O
of	O
the	O
sequences	O
an	O
and	O
bn	O
.	O
</s>
<s>
where	O
stands	O
for	O
Dirichlet	O
convolution	B-Language
.	O
</s>
