Identities for number series and their reciprocals:  
Dirac delta function approach 
S. M. Abrarov 1, R. M. Abrarov 2
April 20, 2007 
Abstract 
Dirac delta function (delta-distribution) approach can be used as efficient method to derive 
identities for number series and their reciprocals. Applying this method, a simple proof for 
identity relating prime counting function (π-function) and logarithmic integral (Li-function) can be 
obtained. 
Keywords: delta function, delta-distribution, prime counting function, distribution of 
primes, Mertens’ formula, harmonic number 
I. Introduction 
Since Paul Dirac introduced the delta function [1], it remains a versatile mathematical tool in 
many applications of the modern sciences and information technologies. Being a physicist P. Dirac 
intuitively believed in the existence of the delta function as a true mathematical object and 
successfully applied it developing the fundamentals of the relativistic quantum mechanics in his 
classic works [1, 2]. 
Nowadays Dirac delta function is widely used in the pure and applied mathematics and 
engineering physics [3-8]. Particularly the sampling methods, based on Dirac delta function, proved 
their reliability and are commonly implemented in the systems of telecommunications and signal 
processing [8]. 
In this work we introduce some useful applications of delta-distribution in the theory of 
numbers. Compared to Stieltjes integration methods [9], Dirac delta function approach is more 
natural and obvious for understanding and can be successfully used in the derivations of the 
identities for the number series and their reciprocals. 
II. Some properties of δ-function 
We will apply two main properties of the δ-function. The first one is the sampling property 
( ) ( ) ( )0 0f x f x x xδ
= −∫ dx ,     (1) 
while the second one relates itself with Heaviside step function [10] 
( ) 00
− = ⎨
via integration 
( ) ( )0 0x x x xχ δ
− = −∫ dx .      (2) 
 It should be noted that integral of the delta function can be determined by infinitely narrow 
area in the vicinity of x0. Therefore (1) and (2) can be rewritten as 
  ( ) ( ) ( )
f x f x x x
= −∫ dx  
  ( ) ( )
x x x x
− = −∫ dx , 
respectively, where ε is infinitely small positive value. 
III. Step function series 
3.1 Number of pulses 
Consider a set consisting of  numbers qN 1, q2, q3, … , qN. Define the series as a sum of 
reciprocals qi as follows 
( ) min
iq x i
     (3) 
where  is the smallest number in this set. According to the sampling property (1) of the delta-
distribution, the series (3) can be rewritten as 
( ) ( )
h x y q dy
= −∑∫ , 
which, in turn, can be expressed in the form 
( ) ( )
ixh x x qδ
′ = −∑ .      (4) 
In compliance with (2), integrating of (4) yields 
( ) ( )
min i
yh y dy x q
′ = −∑∫ i .     (5) 
RHS of (5) counts the number of stepwise pulses, therefore: 
1 1 1 ...
N timesq
yh y dy x
′ = + + + = ⎢ ⎥⎣ ⎦∫ , 
where symbol ⎣ ⎦ denotes the floor operator. Equation (5) contains derivative, which may be 
inconvenient for computing. In order to avoid it, we apply integration by part. This leads to 
( ) ( )
N xh x h y dy= − ∫ ,      (6) 
where  is the number of the stepwise pulses within interval from  up to N minq x . 
3.2 Sum of numbers 
Multiplying x to both parts of (4) and integrating the result, we can find that 
( ) ( )
i i i
y h y q x q
′ = ∑∫ −
.     (7) 
RHS of (7) represents the sum of all numbers up to x. Therefore this equation we can be rewritten as 
q x q
q y h y
′=∑ ∫ .      (8) 
The derivative in (8) can be excluded again. This provides the following relation 
( ) ( )
q x q
q x h x yh y dy
= −∑ ∫ . 
3.3 Power series 
More generally, (4) can be extended for any power of x as given by 
( ) ( )1
x h x x x qδ+
′ = −∑ , 
where k is any number. This leads to the relation of the power series 
q x q
q y h y
dy′=∑ ∫ .      (9) 
Excluding the derivative in (9), we have 
( ) ( ) ( )
k k k
q x q
q x h x k y h y d+
= − +∑ ∫ y .    (10) 
3.4 Sum of reciprocals 
Define the following function as 
q x q
By analogy with (4), the superposition of the delta functions is given by 
( ) (
)ix qx
= −∑ .      (11) 
Dividing both parts of (11) by kx  and integrating the result yields 
( ) ( )
i ik k
q x q xi iq
dy x q
′ ⎛ ⎞
= − = ⎜ ⎟
   (12) 
Finally, integrating by part (12) results to 
( ) ( ) ( )
q x i q
g x g y
q x y+≤
= + +⎜ ⎟
∑ ∫ 2k dy+ .    (13) 
IV. Natural and prime numbers 
4.1 Natural numbers 
We will use the relations derived above for natural and prime numbers. Consider the harmonic 
number (further index  will be omitted) i
( ) 1
H x n
⎢ ⎥⎣ ⎦
∑      (14) 
where n = 1, 2, 3 … . Substituting (14) into (10), we can find that 
( ) ( ) ( )1
1 2 3 4 ... 1
k k k k k k k
n x H x k
⎢ ⎥⎣ ⎦
= + + + + = − +∑ ∫ y H y dy
There are two interesting cases. For k = 0¸ we obtain an equation counting the number of 
pulses, similar to that of (6). However, as the gap between two closest integers is always unity, the 
number of pulses must be equal to the largest integer maxn ∈ , i.e.: 
( ) ( )max
n nn xH x H y d= − ∫ y
H y dy
For k = 1, we have the sum of the arithmetic progression of the natural numbers 
( ) ( )2
1 2 3 4 ... 2
n x H x y
⎢ ⎥⎣ ⎦
= + + + + = −∑ ∫ . 
Therefore using the formula , we can write ( )2max max max/ 2 1 2 3 ...n n n+ = + + + +
( ) ( )
2max max
x H x yH y dy
= − ∫ . 
To derive a relation for harmonic number, we use (13) and substitute . This provides the 
following identity 
x x y y
H x dy
+ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦= + ∫ .    (15) 
It should be noted that the algorithm, built on the basis of (15), significantly accelerates the 
computation of the harmonic number, especially for the large values of x . This is possible to 
achieve since RHS of (15) includes the set of integers instead of reciprocals. The computation 
performed directly through (14) involves rapidly decreasing terms as x  increases. As a result of the 
multiple process of division, the calculation becomes problematic due to the decrease of the 
significant digits after floating point. The use of integers effectively resolves this problem. 
4.2 Prime numbers 
Define a sum of the reciprocal primes in the following form  
,     (16) 
where  p = 2, 3, 5, 7 … , i.e. all primes smaller than x. Substituting (16) into (10), we get 
( ) ( ) ( )1
2 3 5 7 ... 1
k k k k k k k
p x H x k y H y dy+
= + + + + = − +∑ ∫ . 
Consider again two interesting cases. For k = 0, we have an equation showing a quantity of all 
primes up to x. In other words, primes counting function can be expressed through the following 
identity 
( ) ( ) ( )
p px xH x H y dyπ = − ∫ .     (17) 
For k =1, we have an equation showing the sum of all primes up to x, i.e.: 
( ) ( )2
2 3 5 7 ... 2
p x H x yH y dy
= + + + + = −∑ ∫ . 
The identity for , expressed through set of primes, can be readily found using (13) and 
substituting : 
( )pH x
( ) 2 3
p x p x
H x p p dy
x y≤ ≤
= +∑ ∑∫ .     (18) 
By analogy to (15), RHS of (18) does not include the set of reciprocals. As a result, the algorithm, 
built on the basis of identity (18), essentially decreases the computation time, especially for the large 
arguments x . 
The computation of  can also be performed though ( )pH x ( )xπ . Rewrite (17) in the following 
form 
( ) ( )
px yH y dy
′= ∫ .      (19) 
Rearranging (19) and applying integration by part, we get  
( ) ( ) ( )2
H x dy
= + ∫ . 
V. Proof of π-function identity 
Dirac delta function approach can also be useful for any kinds of broken-line functions. As an 
example, we represent a simple proof for the identity relating prime counting function and             
Li-function [11] 
( ) ( ) ( ) ( )
x yR y
x Li x dy Li
= + −∫ + ,    (20) 
where ( )R x  is defined as 
( ) log log
= −∑ x .      (21) 
Using the sampling property (1) of Dirac delta function, R(x) can be expressed as  
log log
log log
p x p x
x y p dy x
− = − −∑ ∑∫ .   (22) 
The derivative of (22) yields  
( ) ( )log 1
R x x p
′ = −∑ − , 
( ) (1
log log p x
+ = −∑ .     (23) 
Integrating of (23) results to 
( ) ( ) ( )
Li x dy const x p
+ + =∑∫ − . 
The sum of Heaviside step functions ( )
−∑  is actually π-function counting the number 
of all primes up to x . A constant in LHS can be determined from the initial condition at 2x =  when 
π-function is unity. This provides exact formula for prime counting function and completes the 
proof of (20). 
The proof can be obtained even simpler. Consider Mertens’ formula providing a relation for 
sum consisting of the reciprocal primes [12] 
( ) ( ) ( )
log log 1 log log 2
loglog
R y R
H x x dy
= + − + +∫ .  (24) 
Taking derivative of (24) yields 
( ) ( )1
log logp
x x x
′ = + .      (25) 
Ultimately, substituting (25) into (19) leads to (20). It is proven again. 
VI. Conclusion 
We applied Dirac delta function approach to derive identities for the sums of the numbers and 
their reciprocals. δ-Function is also useful to obtain a simple proof of the identity relating prime 
counting function and Li-function. 
Acknowledgements 
Authors express gratitude to Prof. T. D. Radjabov and Dr. E. N. Tsoy for their constructive 
remarks and suggestions. 
References 
[1] P. A. M. Dirac, Proc. R. Soc., Series A, 113 (1927) 621
[2] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 1930 
[3] R. P. Kanwal, Generalized Functions: Theory and Applications, Birkhäuser, 2004 
[4] P. Antosik, J. Mikusinski, R. Sikorski, Theory of Distributions. The Sequential Approach, 
Elsevier Scientific Publishing Company, 1973 
[5] A. I. Khuri , Advanced Calculus with Applications in Statistics, Wiley, 2003 
[6] A. I. Khuri, Int. J. Math. Educ. Sci. Technol., 35 (2004) 185
[7] C. Guilpin, J. Gacougnolle and Y. Simon, Appl. Num. Math., 48 (2004) 27
[8] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation, 
The Morgan Kaufmann Series in Interactive 3D Technology, pp. 279-367, 2004 
http://links.jstor.org/sici?sici=0950-1207%2819270101%29113%3A765%3C621%3ATPIOTQ%3E2.0.CO%3B2-8
http://dx.doi.org/10.1080/00207390310001638313
http://dx.doi.org/10.1016/S0168-9274(03)00104-1
[9] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, pp. 
152-155, 1988 
[10] We consider an alternative form of Heaviside function when ( )0χ 1≡ . See, e.g.:  
http://en.wikipedia.org/wiki/Heaviside_step_function
[11] R. M. Abrarov, S. M. Abrarov (to be published) 
[12] G. Tenenbaum and M. M. France, The Prime Numbers and Their Distribution, AMS, p. 22, 
2001 
________________________________________________________________________________ 
1 York University, Toronto, Canada abrarov@yorku.ca
Dongguk University, Seoul, South Korea abrarov@dongguk.edu
2 University of Toronto, Canada rabrarov@physics.utoronto.ca
http://en.wikipedia.org/wiki/Heaviside_step_function
mailto:abrarov@yorku.ca
mailto:abrarov@dongguk.edu
mailto:rabrarov@physics.utoronto.ca

  Dirac delta function (delta-distribution) approach can be used as efficient
method to derive identities for number series and their reciprocals. Applying
this method, a simple proof for identity relating prime counting function
(pi-function) and logarithmic integral (Li-function) can be obtained.

Introduction 
Since Paul Dirac introduced the delta function [1], it remains a versatile mathematical tool in 
many applications of the modern sciences and information technologies. Being a physicist P. Dirac 
intuitively believed in the existence of the delta function as a true mathematical object and 
successfully applied it developing the fundamentals of the relativistic quantum mechanics in his 
classic works [1, 2]. 
Nowadays Dirac delta function is widely used in the pure and applied mathematics and 
engineering physics [3-8]. Particularly the sampling methods, based on Dirac delta function, proved 
their reliability and are commonly implemented in the systems of telecommunications and signal 
processing [8]. 
In this work we introduce some useful applications of delta-distribution in the theory of 
numbers. Compared to Stieltjes integration methods [9], Dirac delta function approach is more 
natural and obvious for understanding and can be successfully used in the derivations of the 
identities for the number series and their reciprocals. 
II. Some properties of δ-function 
We will apply two main properties of the δ-function. The first one is the sampling property 
( ) ( ) ( )0 0f x f x x xδ
= −∫ dx ,     (1) 
while the second one relates itself with Heaviside step function [10] 
( ) 00
− = ⎨
via integration 
( ) ( )0 0x x x xχ δ
− = −∫ dx .      (2) 
 It should be noted that integral of the delta function can be determined by infinitely narrow 
area in the vicinity of x0. Therefore (1) and (2) can be rewritten as 
  ( ) ( ) ( )
f x f x x x
= −∫ dx  
  ( ) ( )
x x x x
− = −∫ dx , 
respectively, where ε is infinitely small positive value. 
III. Step function series 
3.1 Number of pulses 
Consider a set consisting of  numbers qN 1, q2, q3, … , qN. Define the series as a sum of 
reciprocals qi as follows 
( ) min
iq x i
     (3) 
where  is the smallest number in this set. According to the sampling property (1) of the delta-
distribution, the series (3) can be rewritten as 
( ) ( )
h x y q dy
= −∑∫ , 
which, in turn, can be expressed in the form 
( ) ( )
ixh x x qδ
′ = −∑ .      (4) 
In compliance with (2), integrating of (4) yields 
( ) ( )
min i
yh y dy x q
′ = −∑∫ i .     (5) 
RHS of (5) counts the number of stepwise pulses, therefore: 
1 1 1 ...
N timesq
yh y dy x
′ = + + + = ⎢ ⎥⎣ ⎦∫ , 
where symbol ⎣ ⎦ denotes the floor operator. Equation (5) contains derivative, which may be 
inconvenient for computing. In order to avoid it, we apply integration by part. This leads to 
( ) ( )
N xh x h y dy= − ∫ ,      (6) 
where  is the number of the stepwise pulses within interval from  up to N minq x . 
3.2 Sum of numbers 
Multiplying x to both parts of (4) and integrating the result, we can find that 
( ) ( )
i i i
y h y q x q
′ = ∑∫ −
.     (7) 
RHS of (7) represents the sum of all numbers up to x. Therefore this equation we can be rewritten as 
q x q
q y h y
′=∑ ∫ .      (8) 
The derivative in (8) can be excluded again. This provides the following relation 
( ) ( )
q x q
q x h x yh y dy
= −∑ ∫ . 
3.3 Power series 
More generally, (4) can be extended for any power of x as given by 
( ) ( )1
x h x x x qδ+
′ = −∑ , 
where k is any number. This leads to the relation of the power series 
q x q
q y h y
dy′=∑ ∫ .      (9) 
Excluding the derivative in (9), we have 
( ) ( ) ( )
k k k
q x q
q x h x k y h y d+
= − +∑ ∫ y .    (10) 
3.4 Sum of reciprocals 
Define the following function as 
q x q
By analogy with (4), the superposition of the delta functions is given by 
( ) (
)ix qx
= −∑ .      (11) 
Dividing both parts of (11) by kx  and integrating the result yields 
( ) ( )
i ik k
q x q xi iq
dy x q
′ ⎛ ⎞
= − = ⎜ ⎟
   (12) 
Finally, integrating by part (12) results to 
( ) ( ) ( )
q x i q
g x g y
q x y+≤
= + +⎜ ⎟
∑ ∫ 2k dy+ .    (13) 
IV. Natural and prime numbers 
4.1 Natural numbers 
We will use the relations derived above for natural and prime numbers. Consider the harmonic 
number (further index  will be omitted) i
( ) 1
H x n
⎢ ⎥⎣ ⎦
∑      (14) 
where n = 1, 2, 3 … . Substituting (14) into (10), we can find that 
( ) ( ) ( )1
1 2 3 4 ... 1
k k k k k k k
n x H x k
⎢ ⎥⎣ ⎦
= + + + + = − +∑ ∫ y H y dy
There are two interesting cases. For k = 0¸ we obtain an equation counting the number of 
pulses, similar to that of (6). However, as the gap between two closest integers is always unity, the 
number of pulses must be equal to the largest integer maxn ∈ , i.e.: 
( ) ( )max
n nn xH x H y d= − ∫ y
H y dy
For k = 1, we have the sum of the arithmetic progression of the natural numbers 
( ) ( )2
1 2 3 4 ... 2
n x H x y
⎢ ⎥⎣ ⎦
= + + + + = −∑ ∫ . 
Therefore using the formula , we can write ( )2max max max/ 2 1 2 3 ...n n n+ = + + + +
( ) ( )
2max max
x H x yH y dy
= − ∫ . 
To derive a relation for harmonic number, we use (13) and substitute . This provides the 
following identity 
x x y y
H x dy
+ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦= + ∫ .    (15) 
It should be noted that the algorithm, built on the basis of (15), significantly accelerates the 
computation of the harmonic number, especially for the large values of x . This is possible to 
achieve since RHS of (15) includes the set of integers instead of reciprocals. The computation 
performed directly through (14) involves rapidly decreasing terms as x  increases. As a result of the 
multiple process of division, the calculation becomes problematic due to the decrease of the 
significant digits after floating point. The use of integers effectively resolves this problem. 
4.2 Prime numbers 
Define a sum of the reciprocal primes in the following form  
,     (16) 
where  p = 2, 3, 5, 7 … , i.e. all primes smaller than x. Substituting (16) into (10), we get 
( ) ( ) ( )1
2 3 5 7 ... 1
k k k k k k k
p x H x k y H y dy+
= + + + + = − +∑ ∫ . 
Consider again two interesting cases. For k = 0, we have an equation showing a quantity of all 
primes up to x. In other words, primes counting function can be expressed through the following 
identity 
( ) ( ) ( )
p px xH x H y dyπ = − ∫ .     (17) 
For k =1, we have an equation showing the sum of all primes up to x, i.e.: 
( ) ( )2
2 3 5 7 ... 2
p x H x yH y dy
= + + + + = −∑ ∫ . 
The identity for , expressed through set of primes, can be readily found using (13) and 
substituting : 
( )pH x
( ) 2 3
p x p x
H x p p dy
x y≤ ≤
= +∑ ∑∫ .     (18) 
By analogy to (15), RHS of (18) does not include the set of reciprocals. As a result, the algorithm, 
built on the basis of identity (18), essentially decreases the computation time, especially for the large 
arguments x . 
The computation of  can also be performed though ( )pH x ( )xπ . Rewrite (17) in the following 
form 
( ) ( )
px yH y dy
′= ∫ .      (19) 
Rearranging (19) and applying integration by part, we get  
( ) ( ) ( )2
H x dy
= + ∫ . 
V. Proof of π-function identity 
Dirac delta function approach can also be useful for any kinds of broken-line functions. As an 
example, we represent a simple proof for the identity relating prime counting function and             
Li-function [11] 
( ) ( ) ( ) ( )
x yR y
x Li x dy Li
= + −∫ + ,    (20) 
where ( )R x  is defined as 
( ) log log
= −∑ x .      (21) 
Using the sampling property (1) of Dirac delta function, R(x) can be expressed as  
log log
log log
p x p x
x y p dy x
− = − −∑ ∑∫ .   (22) 
The derivative of (22) yields  
( ) ( )log 1
R x x p
′ = −∑ − , 
( ) (1
log log p x
+ = −∑ .     (23) 
Integrating of (23) results to 
( ) ( ) ( )
Li x dy const x p
+ + =∑∫ − . 
The sum of Heaviside step functions ( )
−∑  is actually π-function counting the number 
of all primes up to x . A constant in LHS can be determined from the initial condition at 2x =  when 
π-function is unity. This provides exact formula for prime counting function and completes the 
proof of (20). 
The proof can be obtained even simpler. Consider Mertens’ formula providing a relation for 
sum consisting of the reciprocal primes [12] 
( ) ( ) ( )
log log 1 log log 2
loglog
R y R
H x x dy
= + − + +∫ .  (24) 
Taking derivative of (24) yields 
( ) ( )1
log logp
x x x
′ = + .      (25) 
Ultimately, substituting (25) into (19) leads to (20). It is proven again. 
VI. Conclusion 
We applied Dirac delta function approach to derive identities for the sums of the numbers and 
their reciprocals. δ-Function is also useful to obtain a simple proof of the identity relating prime 
counting function and Li-function. 
Acknowledgements 
Authors express gratitude to Prof. T. D. Radjabov and Dr. E. N. Tsoy for their constructive 
remarks and suggestions. 
References 
[1] P. A. M. Dirac, Proc. R. Soc., Series A, 113 (1927) 621
[2] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 1930 
[3] R. P. Kanwal, Generalized Functions: Theory and Applications, Birkhäuser, 2004 
[4] P. Antosik, J. Mikusinski, R. Sikorski, Theory of Distributions. The Sequential Approach, 
Elsevier Scientific Publishing Company, 1973 
[5] A. I. Khuri , Advanced Calculus with Applications in Statistics, Wiley, 2003 
[6] A. I. Khuri, Int. J. Math. Educ. Sci. Technol., 35 (2004) 185
[7] C. Guilpin, J. Gacougnolle and Y. Simon, Appl. Num. Math., 48 (2004) 27
[8] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation, 
The Morgan Kaufmann Series in Interactive 3D Technology, pp. 279-367, 2004 
http://links.jstor.org/sici?sici=0950-1207%2819270101%29113%3A765%3C621%3ATPIOTQ%3E2.0.CO%3B2-8
http://dx.doi.org/10.1080/00207390310001638313
http://dx.doi.org/10.1016/S0168-9274(03)00104-1
[9] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, pp. 
152-155, 1988 
[10] We consider an alternative form of Heaviside function when ( )0χ 1≡ . See, e.g.:  
http://en.wikipedia.org/wiki/Heaviside_step_function
[11] R. M. Abrarov, S. M. Abrarov (to be published) 
[12] G. Tenenbaum and M. M. France, The Prime Numbers and Their Distribution, AMS, p. 22, 
2001 
________________________________________________________________________________ 
1 York University, Toronto, Canada abrarov@yorku.ca
Dongguk University, Seoul, South Korea abrarov@dongguk.edu
2 University of Toronto, Canada rabrarov@physics.utoronto.ca
http://en.wikipedia.org/wiki/Heaviside_step_function
mailto:abrarov@yorku.ca
mailto:abrarov@dongguk.edu
mailto:rabrarov@physics.utoronto.ca
