In this presentation, I will demonstrate an alternative novel proof of the prime-counting function, by utilizing the Heaviside function, the Laplace transform, and the residue theorem.

1. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
The s-Domain Prime-Counting Function Π(s)
Prof. Fayez A. Alhargan, PhD. BEng.
21 July 2021

2. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Background

3. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Figure 1: Bernhard Riemann (1826-1866).
In his landmark paper Riemann [1] stated
"The known approximate expression F (x) = Li(x) is
therefore valid up to quantities of the order x
1
2 and gives
somewhat too large a value; because the non-periodic
terms in the expression for F (x) are, apart from quantities
that do not grow infinite with x:
Figure 2: Bernhard Riemann Paper
Published 1859.
Li(x) − 1
2 Li(x
1
2 ) − 1
3 Li(x
1
3 ) − 1
5 Li(x
1
5 ) + 1
6 Li(x
1
6 ) − 1
7 Li(x
1
7 ) + · · ·
"
This is the final equation in Riemann’s paper for the approximation of the
prime-counting function, see Figure (2).

4. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Introduction
Riemann [1] landmark paper for the prime-counting function is the
foundation for the modern prime numbers analysis.
An accessible reference is Edwards book [2], where he examines and amplifies
Riemann’s paper [1], also von Mangoldt [3] gives alternative formula for
prime-counting function via Chebyshev function.
In this presentation, I will demonstrate an alternative novel proof of the
prime-counting function, by utilizing the Heaviside function, the Laplace
transform, and the residue theorem.
The proof consists of the following main techniques:
• Formulate the prime-count function π(x) using the Heaviside function.
• Laplace transform π(x) to the s-domain prime-counting function Π(s).
• Identify the relation between ζ(s) and Π(s).
• Inverse Laplace transform Π(s) using the residue theorem to obtain π(x).
Such techniques provide concise and elegant solutions both in the x-domain
and s-domain, revealing the profound connections between the Heaviside
prime-counting function, and the zeta function.
Detailed theoretical analysis is available in [4].

5. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Formulation

6. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Graph
The prime-counting function has not been formally defined in the literature,
we can visualize the function, as shown in Figure 3, below.
π(x)
x
2 3 5 7
1
2
3
4
Figure 3: Prime Counting Heaviside Step Function.
Here, we observe the staircase Heaviside step function.

7. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Definition
The number of primes less than a given magnitude x can be formulated in
the x-domain on a fundamental building block, using the staircase Heaviside
step function H(ln x − ln p) as a base for the prime-counting function π(x), see
Figure 3.
Therefore, the real function π(x) can be formally defined as
π(x) :=
X
p
H(ln x − ln p), (1)
where the sum is over the set of all prime numbers p ∈ {2, 3, 5, · · · , pm, · · · }.
Equation (1), seems to be elementary, and probably one would think it is
very simplistic and very likely impractical.
In the following slides, you will be astonished as to how such an elementary
equation leads to a fundamental shift on simplifying the prime-counting
function derivation, as it results in far fewer steps in the proof of the
prime-counting function.
In fact, the proof can be summarized in one slide – this will be shown on the
last slide of this presentation.

8. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Differentiating the Heaviside function in Equation (1); gives the Dirac delta
function δ(x), see Figure 4, thus we have
π0
(x) =
X
p
1
x
δ(ln x − ln p). (2)
x π0
(x)
x
2 3 5 7
1
2
3
4
Figure 4: Prime Dirac Delta Function δ(ln x − ln p).

9. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
The s-domain

10. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Laplace Transform
Now, we can take the Laplace transform of the prime-counting function in
Equation (1), to transform π(x) to Π(s), giving
Π(s) = L {π(x)} =
X
p
L {H(ln x − ln p)} =
X
p
e−s ln p
s
, (3)
Also, we note that
sΠ(s) = L
π0
(x)
=
X
p
L
n
1
x
δ(ln x − ln p)
o
=
X
p
e−s ln p
. (4)
It is important to note the newly defined form of the prime-counting function in
the s-domain, denoted by the symbol Π(s).

11. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
The s-Domain Prime-Counting Function Π(s)
Thus, we can formally define the s-domain prime-counting complex function
Π(s), in the half-plane (s) 1, by the sum over the prime numbers of the
following absolutely convergent series
Π(s) :=
1
s
X
p
1
ps
, ( s 1), (5)
and in the whole complex plane by analytic continuation.
Here, the symbol Π(s) is a newly defined function and should not be
confused with the same symbol used in the literature for different purposes.

12. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
ζ(s) and Π(s) Relationship
Now, we recall the log expansion of the Euler product of Riemann zeta
function, which is given by
ln ζ(s) = −
X
p
ln(1 − e−s ln p
) =
X
k∈N
X
p
1
k
e−ks ln p
. (6)
Therefore, from Equations (5) and (6), we discover that in the s-domain, the
relationship between the prime-counting function Π(s) and the zeta function
ζ(s), is simply given by
ln ζ(s) =
X
k∈N
X
p
1
k
e−ks ln p
= s
X
k∈N
Π(ks). (7)
Here, we observe the power of employing the Heaviside function and the
s-domain analysis, which immediately demonstrate the profound relationship
between ζ(s) and the prime counting function, for Equation (7) reveals that
ln ζ(s) is the sum of all the harmonics of the prime counting function Π(s) in
the s-domain.

13. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Now, the reverse of Equation (7), is given by
sΠ(s) =
X
k∈N
µ(k)
k
ln ζ(ks), (8)
where µ(k) is the Möbius function.
Equation (5) for the s-domain prime-counting function Π(s), exhibits
elegance as well as deceptive simplicity.
However, its complexity is revealed by Equation (8). The function Π(s) has
poles at s = 0, s = 1, and at the zeros of ζ(s).
Figure (5) shows the real part of sΠ(s); along the critical line s = 1
2
, we also
observe the poles at s = 1
2
+ itm.
Figure 5: {sΠ(s)} along the critical line s = 1
2 + it.

14. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Inverse Laplace Transform

15. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Now, the inverse Laplace of Π(s) gives directly the prime-counting function
π(x), i.e.
π(x) = L−1
{Π(s)} . (9)
From the Laplace transform properties, multiplication by s results in
differentiation of π(x), and multiplication by −x results in differentiation of
sΠ(s), thus we have
−x π0
(x) ln x = L−1
[sΠ(s)]0
; (10)
i.e.
x π0
(x) ln x = L−1
(
−
X
k∈N
µ(k)
k
ζ0(ks)
ζ(ks)
)
. (11)
Here, the inverse Laplace can be evaluated using the residue theorem; i.e.
x π0
(x) ln x = −
X
all poles
Res
X
k∈N
µ(k)
k
ζ0(ks)es ln x
ζ(ks)
#
. (12)

16. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
The Residue Evaluation
Now consider the following expression
ζ
0
(ks)
ζ(ks)
, (13)
which has simple poles at the zeros of ζ(ks), i.e. s = sm
k
, s = s̄m
k
and
s = −2m
k
, with m = 1, 2, 3, as well as at s = 1
k
.
The residues at the simple poles are obtained as follows:
R(s=sm/k) = lim
s→
sm
k
nζ
0
(ks)(s − sm
k
)
ζ(ks)
o
=ζ
0
(sm) lim
s→
sm
k
n(s − sm
k
)
ζ(s)
o
=
ζ
0
(sm
ζ0
(sm)
= 1
(14)
similarily for s = −2m/k
R(s=−2m/k) = lim
s→−2m
n
ζ
0
(s)(s + 2m)
ζ(s)
o
= 1 (15)
In fact, for any function f(z) with simple zm zeros, all the residues of
f0
(z)
f(z)
at
the poles z = zm are always 1, as demonstrated above.

17. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
To evaluate the residue at the pole s = 1, we note that
lim
s→1
[ζ(s)(s − 1)] = 1
differentiation, we have
lim
s→1
[ζ
0
(s)(s − 1) + ζ(s)] = 0
rearranging, we have
R(s=1) = lim
s→1
n
ζ
0
(s)(s − 1)
ζ(s)
o
→ −1 (16)
Therefore, the residues of the simple poles at the zeros of ζ(ks), i.e.
s = sm
k
, s = s̄m
k
and s = −2m
k
, with m = 1, 2, 3, as well as at s = 1
k
, for
ζ
0
(ks)
ζ(ks)
es ln x, are respectively given by
e
sm
k
ln x
, e
s̄m
k
ln x
, e− 2m
k
ln x
, and − e
1
k
ln x
.
Simplifying, we have
x
sm
k , x
s̄m
k , x− 2m
k , and − x
1
k .

18. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Therefore, the sum of the residues in Equation (12), gives
xπ0
(x) ln x =
X
k∈N
µ(k)
k
x
1
k −
∞
X
m=1
X
k∈N
µ(k)
k
[x
sm
k + x
s̄m
k + x− 2m
k ]. (17)
Integrating, we have
π(x) =
X
k∈N
µ(k)
k
Li(x
1
k ) −
X
k∈N
µ(k)
k
∞
X
m=1
Li(x
sm
k ) + Li(x
s̄m
k ) + Li(x−
2m
k ).
(18)
Here, we come back a full circle to the same result as given by Riemann [1].
However, this approach gives much better clarity and coherence with far fewer
steps.
The elegant s-domain forms present a new perspective in the relation
between the ζ(s) function and the prime-counting function Π(s). The
behaviour of Π(s) needs further investigation that might reveal new insights
into the computations of the primes.

19. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Finally, with the new perspective, consider the natural number-counting
function φ(x) based on Heaviside function, which we can define as
φ(x) :=
X
n
H(ln x − ln n), (19)
and
φ0
(x) =
X
n
1
x
δ(ln x − ln n). (20)
Taking the Laplace transform of the above equations, we have
L {φ(x)} =
X
n
L {H(ln x − ln n)} =
∞
X
n=1
e−s ln n
s
= 1
s
ζ(s). (21)
and
L
φ0
(x)
=
X
n
L
n
1
x
δ(ln x − ln n)
o
=
∞
X
n=1
e−s ln n
= ζ(s), (22)
Therefore, the inverse Laplace transform of 1
s
ζ(s) to the x-domain, manifests
itself as the natural number counting function φ(x). Also, we observe from
Equation (22) an interesting x-domain representation of the inverse Laplace
transform of ζ(s), as a decreasing Dirac impulse function. Exploring the
features of these equations is an interesting topic for further research

20. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
References

21. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
References
Riemann, Bernhard (1859). Über die Anzahl der Primzahlen unter einer
gegebenen Grösse. Monatsberichte der Berliner Akademie.. In
Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New
York (1953).
Edwards, H. M. (1974). Riemann’s Zeta Function, Academic Press.
Mangoldt, H. von (1895), Zu Riemanns Abhandlung Ueber die Anzahl
der Primzahlen unter einer gegebenen Grösse. Journal für die reine und
angewandte Mathematik 1895, Vol. 114, pp. 255-305, http://gdz.sub.
uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002163144.
Alhargan, Fayez (2021), A Concise Proof of the Riemann Hypothesis via
Hadamard Product,
https://hal.archives-ouvertes.fr/hal-03294415/document.

22. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
Summary

23. Prime-Counting
Function Π(s)
Fayez A. Alhargan
Background
Formulation
The s-domain
Inverse Laplace
Transform
References
Summary
Background Formulation The s-domain Inverse Laplace Transform References Summary
The Prime-Counting Function Proof on One Slide
π(x) :=
X
p
H(ln x − ln p), (23)
Π(s) = L {π(x)} =
X
p
L {H(ln x − ln p)} =
X
p
e−s ln p
s
, (24)
ln ζ(s) = s
X
k∈N
Π(ks), (25)
sΠ(s) =
X
k∈N
µ(k)
k
ln ζ(ks), (26)
π(x) = L
−1
{Π(s)} , (27)
xπ
0
(x) ln x =
X
k∈N
µ(k)
k x
1
k −
∞
X
m=1
X
k∈N
µ(k)
k x
sm
k + x
s̄m
k + x
−
2m
k , (28)
π(x) =
X
k∈N
µ(k)
k Li(x
1
k ) −
X
k∈N
µ(k)
k
∞
X
m=1
Li(x
sm
k ) + Li(x
s̄m
k ) + Li(x
−
2m
k ).
Q.E.D.
Fayez A. Alhargan