In the previous episodes, we
developed the analytic machinery needed to connect prime numbers with the
complex zeros of the Riemann zeta function. In this episode, we bring those
ideas to life. Our focus is the explicit formula for the Chebyshev function
ψ(t), a central object in analytic number theory that encodes the distribution
of prime powers.
This episode marks a shift from
purely theoretical derivations to concrete computation and visualization.
We evaluate ψ(t) directly from its arithmetic definition and then reconstruct
it using the explicit formula, where primes emerge as a superposition of
oscillatory contributions arising from the nontrivial zeros of ζ(s). Under the
assumption of the Riemann Hypothesis, these oscillations take a particularly
transparent and wave-like form.
By comparing the exact Chebyshev
function with truncated versions of the explicit formula, we gain intuition for
how much information is carried by finitely many zeta zeros and how their
collective interference shapes the observed irregularities in prime
distribution. This perspective reveals the primes not as random objects, but as
the result of a precise spectral decomposition.
The goal of this episode is twofold:
to make the explicit formula computable, and to make its meaning visible.
In doing so, we bridge analysis, computation, and intuition—setting the stage
for deeper investigations into the zeros themselves in the episodes that
follow.
Previously, we established the
approximate explicit formula
where
is the Chebyshev 
-(psi)-function, defined as the
cumulative sum of the von Mangoldt function 
(see
Season 2, Episode 4).
Assuming that the nontrivial zeros of
the Riemann zeta function lie on the critical line, each zero can be written as
Pairing complex conjugate zeros
yields a real-valued expression,
This is the explicit formula for the
Chebyshev function. It expresses a prime-counting object in terms of a smooth
main term 
and a correction built from the zeros of the
zeta function.
The shocking part: Primes are being
reconstructed from the zeros of a complex function.
It is a wave decomposition. Each term
is a wave in 
. So the formula says: 
linear growth 
interference pattern of infinitely many waves.
This is like a sound wave being
decomposed into frequencies. A signal being reconstructed from Fourier modes.
Here, frequencies are the imaginary parts of zeta zeros and the signal is the
distribution of primes.
To gain insight into this formula, we
compute 
directly from its definition on the left-hand
side and compare it with a truncated version of the right-hand side, obtained
by summing over finitely many zeta zeros. The resulting approximations can then
be visualized through suitable plots.
This C++ code implements two
classical objects from analytic number theory:
The Von Mangoldt Function Λ(n)
Λ(n) is defined as:
- log(p) if n = pᵏ for some prime p and
integer k ≥ 1
- 0 otherwise
So it "fires" only at prime
powers, and its value is always the log of the underlying prime.
The Chebyshev ψ Function
ψ(n) is simply the cumulative
sum of Λ:
ψ(n) = Λ(1) + Λ(2) + ... + Λ(n)
It's a key tool in the proof of the
Prime Number Theorem — ψ(x) ~ x is essentially equivalent to PNT.
How the Code Works
The implementation has three main
pieces:
- mangoldt_cfg — A configuration object
that takes an input x and computes a sieve bound L
~ x^(2/3). Values up to L are precomputed; values
beyond that are handled on-the-fly.
- compute_lambda_psi — The core sieve. It
uses a linear sieve (each composite is visited exactly
once via its smallest prime factor) to fill a lambda[] array up
to L, then does a prefix sum to build psi[]. This runs
in O(L) time.
- chebyshev_function — The public-facing
class. It preloads the sieve on construction, then answers queries:
- For n ≤ L: direct array lookup —
O(1).
- For n > L: falls back
to compute_lambda_direct(n), which trial-divides n to
check if it's a prime power — O(√n).
- It also caches the running tail sum in tail_psi_ to
avoid recomputing from scratch on sequential large queries.
Key Design Choice
The x^(2/3) sieve bound is
a classic analytic number theory trade-off: sieving further costs more memory
and setup time, but reduces how often the slower O(√n) fallback is needed. For
most use cases, this hits a sweet spot between precomputation cost and query
speed.
How to use
Create a main.cpp file and add:
Then in main
Therefore, the values of the
Chebyshev function are stored in chebychev.dat. You can then use your
preferred plotting software to generate the corresponding graphs.
Using the data obtained from the code
above we obtain:
We can also visualize its evolution
through animation.
Using the truncated explicit formula
and retaining the first 60 nontrivial
zeros of the Riemann zeta function, we obtain the following approximation:
Epilogue (Season 2, Episode 5)
What we have done in this episode is
to bring the explicit formula out of abstraction and into direct contact with
computation. The Chebyshev function ψ(t), defined purely in terms of prime
powers, has been evaluated numerically and then reconstructed using only the
zeros of the Riemann zeta function. The agreement between these two viewpoints
is not a coincidence—it is the analytic heart of the Prime Number Theorem and
the deepest reason the Riemann Hypothesis matters.
Seen through this lens, the explicit
formula is not merely a theoretical statement. It is a spectral
decomposition of the primes. Each nontrivial zero contributes an
oscillatory mode, and the observed irregularities in the distribution of primes
arise from the interference of infinitely many such waves. When we truncate the
sum over zeros, we do not destroy the structure—we simply smooth it, much like
filtering high frequencies from a signal.
The numerical experiments performed
here illustrate a crucial point: primes are not random, nor are they simple.
Their structure is encoded globally in the analytic behavior of ζ(s). The zeros
are not auxiliary objects—they are the carriers of arithmetic information.
Computing ψ(t) directly and comparing it with its reconstruction from zeta
zeros makes this relationship tangible.
In the next episode, we will move
beyond visualization and return to analysis. The tools developed here will be
used to study the density and distribution of zeros themselves, closing the
loop between primes, complex analysis, and asymptotic control. What began as an
abstract contour integral has now become an explicit bridge between computation
and theory—and that bridge is the central theme of analytic number theory.
Readers who wish to build a solid
foundation in analytic number theory and Fourier analysis, and to understand
these ideas from first principles, are encouraged to consult my books The Riemann Hypothesis Revealed and The Essential Transform Toolkit.
Both are written with the explicit aim of carefully reconstructing the analytic
framework, filling in omitted steps, and making the underlying techniques fully
transparent.
PS. The first 100 roots of zeta here: https://www.plouffe.fr/simon/constants/zeta100.html
Comments
Post a Comment