Corresponding to every zeta function there is a delta series. We make use of this fact to derive an explicit formula for Riemann Prime Counting Function.

1. Zeta and Delta Function Pairs
and an explicit formula for the
Riemann Prime Counting Function

2. Zeta and delta function pairs
We start off with some definitions. Some names here are my
creation to facilitate communication.
Unit step function:
u(t) = 1, t ≥ 0
= 0, t < 0.
Unit delta function:
δ(t)
d
dt
u(t).
Since u(t) has no derivative at t = 0, mathematicians get over the
difficulty by calling δ(t), a generalized function.

3. Laplace transform of f(t):
L{f (t)}
∞
0
f (t)e−st
dt
where it is assumed that the integral is convergent in a right half
plane of s. It is easy to verify that
L{δ(t − a)} = e−as
and
L{u(t − a)} =
e−as
s
where a is a positive contant.

4. Riemann delta function:
Rieδ(t)
∞
n=1
δ(t − log n)
Riemann zeta function:
Rieζ(s) L{Rieδ(t)} =
∞
n=1
e−s log n
=
∞
n=1
n−s
ζ(s)

5. Using the M¨obius function µ(n) we can write the inverse of ζ(s)
explicitly.
Inverse delta function:
Invδ(t)
∞
n=1
µ(n)δ(t − log n)
Inverse zeta function:
Invζ(s) L{Invδ(t)} =
∞
n=1
µ(n)n−s
=
1
ζ(s)

6. Elementary delta function:
Eleδ(t)
∞
m=1
1
m
δ(t − m)
Elementary zeta function:
Eleζ(s) L{Eleδ(t)} =
∞
m=1
1
m
e−sm
= − log(1 − e−s
)

7. Secondary delta function:
Secδ(t; a)
∞
m=1
1
m
δ(t − ma)
where a is a positive constant.
Secondary zeta function:
Secζ(s; a) L{Secδ(t; a)} =
∞
m=1
1
m
e−mas
= − log(1 − e−as
)

8. Prime delta function:
Priδ(t)
∞
k=1
δ(t − log log pk)
where pk is the kth
prime number.
Prime zeta function:
Priζ(s) L{Priδ(t)}
=
∞
k=1
ak
−s
where ak = log pk.

9. Super delta function:
Supδ(t)
∞
k=1
Secδ(t; ak)
Super zeta function:
Supζ(s) L{Supδ(t)} =
∞
k=1
Secζ(s; ak)
log ζ(s) =
∞
k=1
− log(1 − p−s
k ) =
∞
k=1
Secζ(s; ak)
from which it follows that
Supζ(s) = log ζ(s).

10. Hyper delta function:
Hypδ(t) Supδ(et
) =
∞
k=1
∞
m=1
1
m
δ(et
− mak)
=
∞
k=1
∞
m=1
1
m
δ(t − log m − log ak)

11. Hyper zeta function:
Hypζ(s) L{Hypδ(t)}
=
∞
k=1
e−s log ak
∞
m=1
1
m
e−s log m
=
∞
k=1
a−s
k
∞
m=1
m−(s+1)
Using the previously defined symbols, we get
Hypζ(s) = Priζ(s)ζ(s + 1).

12. Derivation of prime counting formula
Rewriting the just derived equation gives us
Priζ(s) = Invζ(s + 1)Hypζ(s)
an explicit form for the prime counting formula. If we take the
inverse Laplace transform, we get
Priδ(t) = e−t
Invδ(t) ∗ Hypδ(t)
where ∗ represents the convolution product.

13. Prime Counting Formula
Priζ(s) = Invζ(s + 1)Hypζ(s)
Priδ(t) = e−t
Invδ(t) ∗ Hypδ(t)