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INNOVATIVE ARITHMETICAL FUNCTIONS: INFINITE PRODUCTS
-------Version 2
By
Garimella Rama Murthy,
Associate Professor
ABSTRACT
Various known arithmetical functions are briefly reviewed. Several novel
arithmetical functions are defined. Motivated by the Euler product, several new
infinite products in unknown variable “x” are defined. These infinite products are
related to power series type functions associated with novel arithmetical functions.
Also some important constants associated with the sequence of primes/integers
are defined.
1. Introduction:
Ever since the dawn of civilization, natural numbers
stimulated the curiosity of homo-sapiens. Geometric investigations led to the
discovery of hidden relationships among natural numbers. For instance, the
Pythagorean Theorem led to an interesting Diophantine equation. Books 7,8,9
of Euclid’s elements were devoted to number theoretic investigations [3].
Investigations on relationship between divisors of an integer
and the integer itself resulted in the definition of perfect numbers (A number
which is equal to the sum of its proper divisors e.g 6, 28). Thus
mathematicians defined an interesting function, an arithmetical function: Sum
of proper divisors of an integer. This arithmetical function is related to the
specification of so called amicable numbers. These are a pair of numbers
with the following property. The sum of all proper divisors of the first number
exactly equals the second number while the sum of all proper divisors of the
second like wise equals the first. For instance, it can easily be verified that
220 and 284 are amicable numbers.
Prime numbers were also subjected to
serious investigations from the period of Euclid (who provided an elegant
proof that they are infinite). Efforts to utilize “analysis” techniques in number
theoretic investigations led to the field of analytic number theory. Efforts of
many mathematicians focused on one important arithmetical function: Number
of prime numbers until an integer ‘n’ i.e.  (n). This arithmetical function is
associated with the celebrated prime number theorem. Also
 Euler discovered the famous Totient function: the number of
integers relatively prime to “n”.
 He also studied the partition function p(n): Number of ways in
which an integer “n” can be written as the sum of other integers.
Mathematicians like Hardy, Lambert,
Ramanujan et. al introduced and studied the properties of the following
arithmetical functions (using the techniques of analytic number theory) [1, 2]
  (n) : Number of divisors of an integer “n”

 s (n) : Sum of ‘s’th powers of divisors of an integer “n”.
In this research paper, we define new arithmetical functions. It is
shown how these functions naturally arise in connection with some infinite
products. These results are summarized in Sections 2 and 3. The
technical report concludes in Section 4.
2. Novel Arithmetical Functions:
Consider the “unique” factorization of an
arbitrary integer. Define the following functions related to the “prime” divisors
occurring in the unique factorization.
(a)  (n) : Number of distinct
prime divisors occurring in the unique
factorization of “n”.
This arithmetical function was studied extensively in the
field of probabilistic number theory. For instance Ramanujan studied this
function in his famous paper with Hardy [5] (the normal number of prime
factors of a number n ).
(b)  (n) : Sum of distinct prime divisors occurring in the unique factorization
of “n”.
©  k (n) : Sum of “k” th powers of prime divisors of n.
It should be noted that
 1 ( n)   ( n ) .
(D)  (n) : Product of exponents of primes (distinct) occurring in the unique
factorization of ‘n’.
This arithmetical function is vaguely related to the
n
number of divisors of an integer “N”. Let N = p1n1 p 2n2 .... p l l . It is easy to
see that
 (n) = (1  n1 )(1  n2 )....(1  nl ) whereas
 (n)  n1 n2 ...nl
(E)  (n) : Product of primes occurring in the unique factorization of “n”
(F) Consider the unique factorization of an integer, N i.e.
n
n
n
N = p1 1 p 2 2 .... p l l . ……………………(2.1)
Define a novel arithmetical function, g(N) as
n
n
n
g(N) = p1 1  p 2 2  ....  p l l .
(G) Consider the above unique factorization of integer N,
Define a new arithmetical function,  (N ) as
 ( N )  n1  n2  ...  nl
This function can be interpreted as the total number of divisors
based on distinct prime divisors.
(H) Consider the above unique factorization of an integer, N.
Define a novel arithmetical function, h(N) as
h(N) = n1 p1  n2 p 2  ...  nl pl
Remark:
It is well accepted that Mobius function, Euler Totient function,
Mangoldt function have deeper significance in number theory. Also, as in the
case of  (n) , we expect that detailed investigation of the above arithmetical
functions will be of considerable utility. Thus, we expect large number of
research papers will be written discussing
the properties of
the above
arithmetical functions.
Remark: It should be kept in mind that a large number of arithmetical
functions can be easily defined. For instance, in [4], several novel arithmetical
functions are studied. Some of these functions have deeper significance in
number theory, other mathematical disciplines and even in physics. In the
following, we define some more arithmetical functions.
(I) Consider the primes until an integer “n”. Define
function as the sum of primes until n i.e. say A(n).
a
new arithmetical
(J) Sum of say “s” th power of primes until n i.e. B(n).
(K) Consider the following arithmetical functions: (i) Sum of pairwise products
of primes, (ii) Sum of products of primes (THREE at a time) and so on.
Remark: One of the approaches to define new arithmetical function is to
generalize the basis on which existing arithmetical function is
defined. For
instance the arithmetical functions studied in [4] are generalizations of Euler’s
totient function  (N ). Specifically, we consider

Given an integer “N”, the number of integers below N which share
1, 2, …,i,…,m prime divisors with N., say F (i ) ( N ). It is easy to see that
N = (N ) 
L
F
(i )
( N ) ….The sum is a finite sum and denotes
j 1
the number of integers which share atleast one prime divisor with N.

Similarly, given integer N, the number of integers below N which share
1,2,… composite divisors with N.
(L) Given an integer n, the number of integers below “n” which are relatively
prime to each other (note that these integers are “LIKE” prime numbers) leads to
the definition of a new arithmetical function, say J(n).
It is clear that J(n)   (n ) . It should be possible to derive tigher bounds
using the results in [4]. It is easily noted that the celebrated prime number
theorem
says
that
 ( n) 
n
.
log n
Thus
a similar
Theorem on the
asymptotic order of J(n) could be stated and proved. In this
design the following notation:
connection, we
(M) Consider the unique factorization of an integer, N. Define a new arithmetical
function, M(N) as the total number of single prime based divisors (divisor
monomial in primes contains single prime) of an integer. It is easy to see that
M(N) = ( n1  n2  ....nl ) .
In a similar manner, 2,3, ….prime divisor based arithmetical functions can be
defined and studied. The definitions can consider the “number”, “Sum”, “sum of
“s”th powers of divisors (i.e. divisor monomials are based on 2,3,…prime
divisors).
The goal of investigations in this technical report is to associate functions
(power series) related to these arithmetical functions. These functions are
connected with some infinite products. Also the eventual goal is to study the
properties of these novel arithmetical functions. In this direction we briefly
study the multiplicativity of some of the arithmetical functions. We expect that
well developed techniques of analytic number theory are
of considerable
utility.
3. Infinite Products: Novel Arithmetical Functions:
The main motivation for
the investigations in this technical report is the following Euler Product:


n 1
1

nk
1
 (1 p
pP
k

1
 ...) ,
p 2k
where P is the set of all prime
numbers.
Now utilizing the fundamental Theorem of Arithmetic (i.e the unique
factorization of any integer using the prime numbers), the following identities
are derived. It is clear that the identities hold true from the point of view
Of SYMBOLIC ALGEBRA. Using standard arguments from analysis, it can
easily be shown that most of the identities hold true from the point of
view of convergence. Detailed analytic arguments are avoided for the
sake of brevity.
LEMMA A: Consider the following infinite product in unknown variable ‘x’

x
x
x ( n )
( 1  k  2 k  ...) = 1   k , …………………(1)

p
p
n
n 1
pP
where P is the set of prime numbers and  (n) is the number of distinct
prime divisors of “n”.
Proof:
The Left Hand Side (LHS) of the above equation, (1) becomes
L.H.S
=

pP
x / pk
(1
)
1
1 k
p
Using the unique factorization of any integer, it is easy to see
that the above equality holds true from the point of view of symbolic algebra.
Now we consider its truth from the convergence view point.
Typical Convergence Argument:
Consider the power series on the right
hand of equation (1). Upper bound this power series by the following power
series i.e.
(1 


n 1
x ( n )
) < (1 +
nk


x n )………(1.5)
n 1
It is easy to see that the power series defined on RHS of (1.5)
converges absolutely on the interval (-1,1). Thus, invoking standard Theorem from
real analysis, it can be seen that the power series on the right hand side of (1)
converges.
Q.E.D.
LEMMA B. Now let us consider another infinite product in unknown variable
‘x’

(1
pP

xp
xp
x ( n )


...)
=
,
1


pk
p 2k
nk
n 1
………………..(2)
where P is the set of prime numbers and  (n ) is the sum of prime
divisors on ‘n’.
The Left Hand Side (LHS) of the above equation, (2)
becomes
L.H.S
=

(1
pP
x p / pk
)
1
1 k
p
Typical Convergence Argument:
Consider the power series on the right
hand of equation (2). Upper bound this power series by the following power
series i.e.
(1 


n 1
x  ( n)
) < (1 +
nk


2
x n )………(2.5)
n 1
It is easy to see that the power series defined on RHS of (2.5)
converges absolutely on the interval (-1,1). It represents a theta function. Thus,
invoking standard Theorem from real analysis, it can be seen that the power
series on the right hand side of (2) converges.
Q.E.D.
LEMMA C: Generalizing the above logical approach, we have the following
infinite product in unknown variable ‘x’.

pP

x (n)
xp
xp
,
( 1  k  2 k  ...) = 1  
nk
p
p
n 1
l
l
l
Where P is the set of prime numbers and  l (n)
of prime divisors of “n”.
……..(3)
is the sum of “l”th power
Note: Equation (2) is a special case of equation (3). The
(LHS) of the above equation, (3) becomes
L.H.S
=

pP
Left Hand Side
l
x p / pk
(1
)
1
1 k
p
LEMMA D: Another interesting infinite product in unknown “x” is given by

pP

x
x2
x ( n )
( 1  k  2 k  ...) = 1  
,………..(4)
p
p
nk
n 1
l
Where
n  p1n1 p2n2 ......... pl l k and
 (n)  n1  n2  .....  nl i.e sum of exponents occurring in unique
n
factorization.
The Left Hand Side (LHS) of the above equation,(4) becomes
L.H.S
=

(1
pP
x / pk
)
x
1 k
p
Remark: From symbolic algebra point of
view (based on the
factorization of an integer), the equations (1), (2), (3), (4) hold true.
unique
LEMMA E. The most general form of identity is given by

(1
pP

x h( p) x h( p)
x ( n )


...)
=
,………….(5)
1


pk
p 2k
nk
n 1
Where  (n)  h( p1 )  h( p 2 )  .....  h( pl ) . h(.) is an arbitrary function
With
n  p1n1 p2n2 ......... pl
The Left Hand Side (LHS) of the above equation, (5) becomes
nl
L.H.S
=

(1
pP
x h( p) / p k
).
1
1 k
p
LEMMA F. Consider the following infinite product in unknown variable ‘x’

 (n) x ( n )
x
2x
3x
=
, …………………(6)


.
..)
1



p k p 2k
p 3k
nk
n 1
pP
where P is the set of prime numbers,  (n) is the product of exponents of
primes occurring in the unique factorization of ‘n’ and  (n) is the number of
(1
distinct prime divisors of “n”.
The most general term
becomes
(n1 x)( n 2 x)...( nl x)
p1n1 p 2n2 k ... p lnl l
=
in
the
infinite product on the LHS of (6)
 ( n) x ( n )
nk
Thus the identity in equation (6) is
readily obtained.
LEMMA G. Consider the following infinite product in unknown variable ‘x

 (n) x  ( n )
x p 2x p 3 x p
( 1  k  2 k  3k  ......) = 1  
…………(7)

p
p
p
nk
pP
n 1
Where the arithmetical functions  (n) ,  (n) are as defined previously.
The
most
general term in the infinite product on the LHS of (7) becomes
(n1 x p1 )( n2 x p2 )...( nl x pl )
p1n1k p 2n2 k ... plnl k
 ( n) x  ( n )
=
nk
Thus the identity in equation (7) is readily obtained.
LEMMA H. Using similar argument as in (7), the following identity follows:

pP

3xp
 ( n) x 
xp
2x p
( 1  k  2 k  3k  ...) = 1  
nk
p
p
p
n 1
l
l
l
l
(n)
The arithmetical functions arising in the above identity are as defined
previously.
LEMMA I. Consider the following infinite product in unknown variable ‘x’:

pP

 ( n) x 
x
2 x 2 3x 3
( 1  k  2 k  3k ...) = 1  
p
p
p
nk
n 1
l
(n)
.
By considering the general term on the left handside, it can be easily
Seen that the above identity follows.
LEMMA J. Consider the following infinite product in unknown variable ‘x’

(1
pP

 ( n )  ( n) x  ( n )
px 2 px 3 px
=
. Where  (n) is


...)
1


pk
p 2k
p 3k
nk
n 1
the
product of primes occurring in the unique factorization of “n”.
By considering the general term on the LHS, it can easily be seen that
the identity follows.
LEMMA K. Consider the following infinite product in unknown variable “x”

pP
px
px
px
( 1  k  2 k  3k ..)
p
p
p

 (n) x ( n )
n 1
nk
1 
=
,
Once again by considering the general term on the LHS, it can easily be
seen that the identity follows.
It is easily seen that along the lines of derivation
of identities (B), (C); identities associated with infinite sums,

 ( n) x  ( n )
n 1
nk
1 
, 1


n 1
 ( n) x 
l
(n)
nk
are easily derived. Details are avoided for brevity.
LEMMA L. Consider the following infinite product in unknown variable “x”

pP

px
p x2 p x3
 ( n) x  ( n )
( 1  k  2 k  3k ...) = 1  
.
p
p
p
nk
n 1
By considering the general
identity follows.
term on the LHS, it can easily be
seen that the
LEMMA M: Consider the following infinite product in unknown variable “x”
2

pP
3

xp
xp
xp
x g (n)
.
( 1  k  2 k  3k ...) = 1  
p
p
p
nk
n 1
Once again, by considering the general term on the LHS, the identity follows.
LEMMA N: Consider the following infinite product in unknown varaiable “x”

(1
pP

xp
x2 p
x3 p
x h( n)


...)
=
1


pk
p 2k
p 3k
nk
n 1
By considering an arbitrary term on the LHS, it can be easily seen that
the identity follows.
It is seen that all the above identities (leading to novel
arithmetical functions) are effectively derived based on the unique factorization
in the integer ring. By abstracting the essential idea (in all the above
identities), the following most general identity is derived.
THEOREM O: Consider the following infinite product
 (1
 ( p,1) x K ( p , 1)
pk
pP

 ( p, 2) x K ( p , 2)
p 2k


=
1 +
 (  ( p , n ) ….  ( p , n )
n 1
1
1
l
l
 ( p, 3 ) x K ( p , 3 )
p 3k
...)
x K ( p1 ,n1 ) ......  K ( pl ,nl ) ) / n k ,
Where the integer “n” has the following unique factorization
n  p1n1 p 2n2 ..... p lnl
As in the previous identities, by considering an arbitrary term on LHS, It
can be seen that the identity holds true.
REMARK: Utilizing the logical procedure adopted in this identity (and thus all
previous identities), various new arithmetical identities can be defined and
associated with infinite products (in variable x). Detailed enumeration of all
possible identities are avoided for brevity.

Multiplicativity of Novel Arithmetical Functions:
In existing discussions on
arithmetical functions, various properties of the function (under consideration)
are studied. One of the very basic properties studied is the multiplicative
nature of the function [6].
Definition: An arithmetical function f is called multiplicative if
identically zero and if
f(mn) = f(m) f(n) whenever (m,n) =1 i.e
f
is not
when integers “m” and “n” are relatively prime.
A multiplicative function f is
called completely multiplicative if we also have
f(mn) = f(m) f(n) for all m, n.
 Now let us consider the arithmetical function:  (n) i.e number of
distinct prime divisors of “n”. It is easy to see that if integers “m”, “n”
are relatively prime, then
 (m n) =  (m)  (n)
i.e  (.) is multiplicative, but not completely multiplicative The counter example
follows. Let
and N 2  p 2 p3 and N 1 N 2  p1 p 22 p 3 .
N1  p1 p2
It is clear that  ( N1 )  2 ,  ( N 2 )  2 . But  ( N1 N 2 )  3   ( N1 ) ( N 2 ) .
 Now
for
the sake
of convenience, consider
the arithmetical
function,  (n ).
It is easily seen that this arithmetical function is not multiplicative. For
instance, let
N1  p1 p2 , N 2  p 3 p 4 (
N1 , N 2
N 1 N 2  p1 p 2 p 3 p 4 . It is clear that
are
relatively
prime)
and
 ( N 1 )  ( p1  p 2 ) ,  ( N 2 )  ( p 3  p 4 ) . But  ( N1 N 2 )   ( N1 )  ( N 2 ) .


For the sake of completeness, let us investigate the completeness of
arithemetical function  (N ).
Consider integers m, n. For the sake of
concreteness (logic is easily generalized), let
m  p12 p 23 ; n  p 32 p 43 . Clearly (m, n ) =1 i.e relatively prime.
 (m)  p1 p 2 ,  (n)  p 3 p 4 and  (mn)   (m)  (n).
Thus  (.) is multiplicative. By considering ( m, n)  1 , it is easy to
see that  (.) is not completely multiplicative.
By using the same procedure, it is easy to see that  (.) is multiplicative
but not completely multiplicative.
In the spirit of the above efforts,
multiplicativity of other arithmetical functions defined above can easily be
determined. Details are avoided for brevity.

Some Important Constants Associated with Primes/Integers:
In the following
numbers. For the sake
Integers. It should be
defined. But some of
sciences.
we list some interesting constants associated with prime
of completeness, we also list constants associated with
kept in mind that whole set of such constants could be
them will have applications in physics and other natural
Infinite Sequence based Constants:
1
1
1
1
1
 3  5  7  ...  p  ...... = C1
2
2
3
5
7
p
1
1
1
1
1
(2) 1 + 2  2  2  2  .....  2  ..... = C2
2
3
5
7
p
(1) 1 +
In the same spirit of (2), we have
(3) 1 +
1
1
1
1
1
 k  k  k  .....  k  ..... = C 3 (k ) for any arbitrary integer ‘k’.
k
2
3
5
7
p
As in the case of Euler sums, it will be nice if these constants can be
specified. In the spirit of Zeta function, we can define a function of complex
variable “s”,  (s ).
 (s)  1 +

p K
1
, where “K” is the set of
ps
prime
numbers. Study of properties of this function may lead to new results in analytic
number theory.
(4) 1 +
1
1
1
 4  5  ... ………………. = C4
3
2
3
4
(5) 1 +
1
1
1
 3  4  ............. = C5 .
2
2
3
4
4.
Arithmetical Functions of multiple variables:
Traditionally, mathematicians have
defined arithmetical functions of
single integer variable e.g Euler’s totient
function. A question naturally arises as to why we cannot define arithmetical
functions of
multiple
independent
variables. In research literature, various
arithmetical functions of multiple independent variable are defined and studied.
We add to that list in the following:
Consider
Definition 1:
a
function
of
“k”
integer
variables
i.e.
say
 ( n1 , n2 ,...., nk ) . Let the function be the number of primes common/shared in
the unique factorization of the integers { ni }li 1 .
It should be noted that if the integers
l
{ ni }i 1 are either primes or they are relatively prime, then  (.,.,....,.)  0. Also, it
easily seen
Specifically
that  (.,.,....,.) is a symmetric function in the arguments/variables.
 ( n1 , n2 )   ( n2 , n1 ) .
In this case consider “ n1 ” and its prime divisors
 ( n1 , n2 ) 
 ( p ,n
i
2
)
where the sum extends over the divisors of n1 .
pi
In this spirit some trivial arithmetical functions can easily be defined i.e
(i) Number of primes between
n1 , n2 , say the function be K( n1 , n2 ).
(ii)
Sum of “s”th powers of primes between n1 , n2 .
(iii)
Number of relatively prime integers between n1 , n2
(iV) Sum of “l”th degree
composite divisors between n1 , n2 .
Definition 2: Given integers, { ni }li 1 , let a new arithmetical function be the
composite divisors shared by them.
5. Interesting Investigation:
The above discussion leads to the definition and
study of functions of the following form:


n 1
f1 (n) x f 2 ( n ) ………………………………(6)
Where f1 (n) , f 2 (n) are arithmetical functions like  (n),  s (n) ….etc.
6.Conclusions:
Several well known arithmetical functions are summarized. Various
novel arithmetical functions are defined. Motivated by the Euler product,
several infinite products in unknown variable “x” are defined. These infinite
products are related to “power series type functions” associated with the novel
arithmetical functions.
REFERENCES:
[1] S. Ramanujan, “On Certain Arithmetical Functions, “ Transactions
Cambridge Philosophical Society, “XXII, No. 9, 1916, Pages 159-184
of the
[2] R. Kanigel, “The Man Who Knew Infinity: A Life of the Genius Ramanujan, “
Washington Square Press, Published by Pocket Books
[3] W.Dunham, “Journey through Genius, The Great Theorems of Mathematics ,”
Penguin Books, 1990
[4] A. Sangameshwar and G. Rama Murthy, “Novel Arithmetical
Technical Report: IIIT/TR/2004/18
Functions, “ IIIT
[5] G. H. Hardy and E.M.Wright, “An Introduction to the Theory of Numbers, “,
Oxford (at the Clarendon Press )
[6] Tom M. Apostol, “Introduction to Analytic Number Theory, “Narosa Publishing
House, New Delhi, 1989.