The short answer is: there is no publicly available evidence that the paper was formally rejected by the Royal Society. The claim has circulated online, but I could not find any rejection letter or official statement from the Royal Society confirming it. (The Times of India)
What is well documented is that K. Eswaran's claimed proof has not been accepted by the mainstream mathematical community, and the Riemann Hypothesis remains officially unsolved. (Gaurav Tiwari)
Why wasn't the proof accepted?
From the discussions by mathematicians and the available critiques, several reasons emerge.
The proof was not published in a leading peer-reviewed mathematics journal.
Eswaran initially circulated his work as a preprint on ResearchGate rather than through journals such as those published by the Royal Society, the American Mathematical Society, or other established mathematics journals. A claim as significant as proving the Riemann Hypothesis requires exceptionally rigorous peer review. (ResearchGate)Mathematicians identified what they believed were gaps in the argument.
The central idea of Eswaran's proof is to study
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}
$
and argue that its poles can only lie on the critical line. Critics argued that some of the analytic steps used to establish this were not justified with sufficient rigor, particularly concerning convergence and analytic continuation. If even one such step is invalid, the entire proof fails. (Gaurav Tiwari)An internal review is not the same as broad mathematical acceptance.
In 2021, news reports highlighted that an eight-member committee convened by supporters concluded the proof was correct after an open review. However, this committee was not an official body of the Clay Mathematics Institute or a major mathematics journal, and its conclusion has not led to acceptance by the wider research community. (The Times of India)The Clay Mathematics Institute still lists the problem as unsolved.
If a correct proof had been established, it would undergo extensive scrutiny over months or years before being accepted. That has not happened, and the hypothesis remains one of the Millennium Prize Problems. (Gaurav Tiwari)
Was it rejected by the Royal Society?
I could not find evidence that:
Eswaran submitted the paper to a Royal Society journal,
it entered peer review there, or
the Royal Society issued a rejection.
It is possible that he submitted it privately and received a confidential editorial decision, but unless the author or the journal publishes that correspondence, there is no way to verify it. So it would not be accurate to state as fact that "the Royal Society rejected the paper."
Could the critics simply be wrong?
Yes—in principle. History contains examples such as Andrew Wiles, whose first proof of Fermat's Last Theorem contained a gap that was later fixed. However, extraordinary mathematical claims require a proof that survives scrutiny by many independent experts. So far, Eswaran's proof has not reached that standard of consensus. (mint)
If you're interested, I can also walk through the specific mathematical objection that critics have raised against Eswaran's use of ( \zeta(2s)/\zeta(s) ). That discussion gets into complex analysis but can be explained step by step.
yes please be slower than usual as I am not well versed in mathematics.. chunk it and get me one to read now and after I give go ahead get me next one
That sounds like a good approach. We'll treat this like a slow reading course rather than trying to understand a research paper all at once.
I won't assume you know complex analysis or advanced number theory. I'll build up only the ideas needed to understand why mathematicians were unconvinced, not the full proof.
Part 1: What exactly is the Riemann Hypothesis trying to say?
Before we discuss Eswaran's proof, we need to know what he was trying to prove.
Imagine you have the numbers
1, 2, 3, 4, 5, 6, 7...
Among these, the prime numbers are
2, 3, 5, 7, 11, 13, ...
Prime numbers seem to appear almost randomly.
For centuries, mathematicians asked:
Is there actually a hidden pattern behind where primes occur?
The Riemann Hypothesis is one of the deepest attempts to answer that question.
Instead of studying primes directly...
Bernhard Riemann took a surprising approach.
He said,
"Instead of looking at prime numbers themselves, let me study a special function."
That function is called the Riemann zeta function, written
$
\zeta(s)
$
At first glance it looks unrelated to primes.
It is simply
$
1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots
$
where s is not an ordinary number—it is allowed to be a complex number.
Don't worry about complex numbers yet.
For now, think of (s) as simply an input.
Why is this function so famous?
Here comes the surprise.
Riemann discovered that this innocent-looking function secretly contains all information about prime numbers.
It is almost like compressing an entire encyclopedia into one ZIP file.
If you understand every property of
$
\zeta(s)
$
you can recover astonishing facts about primes.
An analogy
Imagine a person's ECG.
The ECG is just a squiggly line.
But a cardiologist can look at that one line and infer
heart rhythm
valve problems
previous heart attacks
electrolyte imbalance
The line itself isn't the heart.
It is an indirect representation.
Similarly,
$
\zeta(s)
$
is not the prime numbers.
It is like the ECG of the primes.
Studying the ECG is often easier than opening the body.
The strange thing Riemann noticed
When he studied this function, he found places where
$
\zeta(s)=0.
$
These are called the zeros of the zeta function.
Most functions have zeros.
For example,
$
x^2-4=0
$
has zeros at
$
2\quad\text{and}\quad -2.
$
Likewise,
$
\zeta(s)
$
also has zeros—but because (s) is allowed to be a complex number, these zeros lie in a two-dimensional plane rather than on the ordinary number line.
The Riemann Hypothesis
After calculating many zeros, Riemann noticed something remarkable.
They all appeared to lie on one perfectly straight vertical line.
He guessed:
Every non-trivial zero of the zeta function lies exactly on this line.
That simple sentence is the Riemann Hypothesis.
No one has been able to prove it or find a counterexample, despite checking trillions of zeros computationally.
Why do mathematicians care?
Because if the zeros all lie on that line, then the distribution of prime numbers is as regular as it can reasonably be.
If even one zero is off the line, many important estimates about the distribution of primes would become less precise.
So the Riemann Hypothesis is not really about zeros—it is about understanding the hidden structure behind the prime numbers.
We'll stop here. In the next part, I'll explain what K. Eswaran's main idea was—specifically, why he chose to study the ratio
$
\frac{\zeta(2s)}{\zeta(s)}
$
instead of attacking (\zeta(s)) directly. That's the key insight of his approach, and understanding it will make the later criticisms much easier to follow.
Just say "go ahead" when you're ready for Part 2.
I want to speculate that by dividing the reimann zeta function of twice the frequency with the reimann zeta function of a particular frequency he tried to prove that there exist no zeros which are not covered by the zeta function.. kind of nyquists sampling theorem which says to sample a function and recreate it with fidelity we need to sample at twice the frequency of the highest freqency component in its spectrum. is my guess right.
That's actually a very thoughtful analogy, and it tells me how you're thinking about the problem. I would say:
Your intuition is interesting, but it is not what Eswaran was actually exploiting.
Let me explain why, slowly.
First, let's examine your Nyquist analogy.
In signal processing, suppose a signal contains frequencies up to 100 Hz.
Nyquist says:
If you sample at 200 Hz or higher, you have enough information to reconstruct the original signal.
In other words,
twice the frequency gives you complete information
nothing "hides" between the samples.
So you're wondering whether
$
\frac{\zeta(2s)}{\zeta(s)}
$
means something like
"The function at twice the scale contains enough information to prove there are no hidden zeros."
That is a very natural engineering interpretation.
Unfortunately, mathematics means something completely different here.
The "2" in
$
\zeta(2s)
$
is not doubling a frequency.
It simply means
"If the input is (s), now plug in (2s)."
For example,
if
$
s=3,
$
then
$
\zeta(2s)=\zeta(6).
$
Nothing is being sampled.
Nothing is oscillating twice as fast.
There is no analogue of bandwidth here.
So why divide them?
This is the clever part.
Suppose I tell you
$
A(s)=\zeta(s)
$
and
$
B(s)=\zeta(2s).
$
Now look at
$
\frac{B(s)}{A(s)}.
$
Whenever the denominator becomes zero,
the whole fraction wants to become
$
\frac{\text{something}}{0},
$
which "blows up."
Mathematicians call this a pole.
So every zero of
$
\zeta(s)
$
tries to become a pole of
$
\frac{\zeta(2s)}{\zeta(s)}.
$
That is the key observation.
Here is an everyday analogy.
Imagine a school.
The denominator is the number of teachers.
The numerator is the number of students.
Suppose
Students = 500
Teachers = 25
Then
$
\frac{500}{25}=20.
$
Now imagine teachers become
The ratio becomes
If teachers become
1,
the ratio becomes
If teachers become
0,
the ratio is undefined.
It "blows up."
Exactly the same thing happens mathematically.
When
$
\zeta(s)=0,
$
the ratio explodes unless something in the numerator also becomes zero at exactly the same place.
I think your engineering background is showing.
The moment you saw "2", your brain immediately thought
octave
harmonics
sampling
bandwidth
frequency doubling
An engineer naturally thinks this way.
A mathematician, however, sees
$
2s
$
simply as a different point in the complex plane.
No notion of frequency is intended.
However...
I don't want to dismiss your intuition completely.
There is a deep connection between the Riemann zeta function and frequency-like ideas.
One way mathematicians study (\zeta(s)) is through Fourier analysis, where oscillations and frequencies genuinely appear. There are also surprising links to quantum mechanics, where the imaginary parts of the zeros behave statistically like energy levels of quantum systems. So while Eswaran's specific argument is not based on a Nyquist-like principle, your instinct to think in terms of spectra and frequencies isn't out of place in the broader study of the Riemann Hypothesis.
I have a feeling you'll enjoy Part 2, because Eswaran's central idea can be explained almost entirely using the notion of poles (things that blow up) and zeros (things that become zero), without needing advanced mathematics. Once you see that, the criticism of the proof becomes much easier to understand.
go ahead
Excellent. Now we're getting to Eswaran's actual strategy. I'll keep this to one core idea.
Part 2: Eswaran's Strategy — Turn Zeros into Poles
Instead of trying to prove directly that
$
\zeta(s)=0
$
only on the critical line, Eswaran essentially says:
"Let me study another function whose behaviour is easier to understand."
That function is
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}.
$
This is a common mathematical technique.
It is similar to engineering.
Sometimes you don't analyze the original circuit directly—you transform it into another form (using Laplace transforms, Fourier transforms, transfer functions, etc.) because the transformed version reveals properties more clearly.
Eswaran is doing something similar.
Why choose a ratio?
Suppose I have two ordinary functions.
$
A(x)=x-3
$
and
$
B(x)=5.
$
Now form
$
\frac{5}{x-3}.
$
What happens at
$
x=3?
$
The denominator becomes zero.
The graph shoots towards infinity.
That point is called a pole.
Notice something important.
The pole didn't exist in the numerator.
It appeared because the denominator became zero.
The same thing happens here
If
$
\zeta(s)=0,
$
then
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}
$
will usually become infinite.
So every zero of the denominator tries to create a pole.
This is the brilliant observation.
Instead of hunting for invisible zeros,
he tries to hunt for visible poles.
Sometimes poles are mathematically easier to analyze than zeros.
Imagine a metal sheet
Suppose the surface of a table is perfectly flat.
Now imagine every zero pushes a needle upward through the table.
The needle represents a pole.
You can now walk around looking for needles instead of tiny holes.
The holes have become spikes.
Spikes are easier to notice.
That is roughly the philosophy.
But there is one complication
Suppose both numerator and denominator become zero at exactly the same point.
For example,
$
\frac{x-3}{x-3}=1.
$
At first sight it looks like
$
\frac00,
$
which is undefined.
But after simplification,
nothing special happens.
The pole disappears.
The numerator has cancelled it.
This is called cancellation.
Therefore...
Eswaran had to show something like this:
Every zero of ($\zeta(s)$) becomes a genuine pole of (F(s)), unless it belongs to a very specific, understood class.
If he could classify all possible poles of (F(s)), then he could infer where the zeros of ($\zeta(s)$) must be.
That is the overall strategy.
Notice how indirect it is.
He is not proving
"Here are all the zeros."
Instead he is saying
"Here are all the places where poles can exist."
Since poles come from zeros of the denominator, this would tell us where the zeros are.
Why mathematicians found this idea interesting
The strategy itself is not considered unreasonable.
In mathematics, people often study one object by looking at a related one.
So the criticism was not:
"You used the wrong function."
Instead, the criticism was closer to:
"You haven't rigorously proved that your conclusions about the poles are valid."
That's a very different issue.
An engineering analogy
Imagine you design a circuit tester.
You say:
"Whenever there's a broken wire, my tester's LED turns on."
If that statement is rigorously proven, then finding broken wires becomes easy.
But suppose someone asks:
"How do you know the LED always turns on? Could there be some broken wires that don't trigger it? Could there be false alarms?"
Now the debate is no longer about the usefulness of the tester. It's about whether you've proved that the tester behaves exactly as claimed.
Many mathematicians view Eswaran's approach similarly. The ratio ( \zeta(2s)/\zeta(s) ) is an interesting "tester," but the key question is whether the mathematical justification behind it is complete.
We'll stop here.
Part 3 is where the real criticism begins. I'll explain the notion of analytic continuation—a concept that sounds intimidating but can be understood with a simple analogy—and why many experts believe this is where Eswaran's argument needed stronger justification. That concept is central to understanding why the proof has not been accepted.
I understand analytical continuity.. it has to obey 5 properties.. differentiable, no discontinuities like that another 3 more.. ok go ahead
Perfect! That actually changes the level at which I can explain this. Since you're already familiar with the idea of analytic functions, I won't spend time introducing it from scratch. (The five properties you mention are probably a mix of conditions related to holomorphic/analytic functions and complex differentiability. The key one is that in complex analysis, being differentiable even once in a region is an extraordinarily strong condition.)
Now we can discuss the criticism.
Part 3: Where mathematicians started asking difficult questions
The criticism was not about the algebra.
It was about whether certain operations remained valid after extending the zeta function to the entire complex plane.
Remember,
[
\zeta(s)
]
is originally defined by
$
1+\frac1{2^s}+\frac1{3^s}+\cdots
$
This infinite series does not converge everywhere.
For example, if
$
s=\frac12,
$
the series diverges.
Yet mathematicians still talk about
$
\zeta\left(\frac12\right).
$
How?
Through analytic continuation.
This is the subtle point
Suppose you derive an equation while the series is convergent.
For example,
$
A(s)=B(s).
$
Inside that "safe" region, everything is legal.
Now you analytically continue both sides into a larger region.
Usually this is perfectly legitimate.
But—and this is the important part—
every step that led to the equation must itself be justified after continuation.
You cannot simply say,
"It worked over here, therefore it works everywhere."
This is where complex analysis becomes unforgiving.
Engineering analogy
Imagine you design a controller for a motor.
You prove mathematically that it is stable
provided
voltage stays between 10 and 20 volts,
temperature stays below 70°C.
Now someone asks:
"What happens at 200°C?"
You cannot answer,
"It worked before, so it should still work."
You must prove stability under the new conditions.
Complex analysis is very similar.
Changing the domain changes what you must justify.
Where does Eswaran's ratio enter?
He studies
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}.
$
Now imagine you want to prove
"The only poles of (F(s)) are these."
That sounds reasonable.
But mathematicians immediately ask
"How do you know?"
The answer has to be airtight.
Not "probably."
Not "intuitively."
Every possible pole must be accounted for.
Why is this so difficult?
Because
$
\zeta(s)
$
has a rich structure.
It has
trivial zeros,
non-trivial zeros,
one pole,
a functional equation,
analytic continuation,
and all of these interact.
Suppose you prove
"Here is every pole."
Someone may ask
"Did you also consider poles created after analytic continuation?"
Or
"Can a cancellation occur that your argument overlooked?"
Or
"Does the functional equation introduce behaviour you haven't addressed?"
These are the kinds of questions referees ask.
Think of a legal proof
Imagine proving
Every citizen paid tax.
You inspect
90 million records.
A referee asks
"What about citizens living abroad?"
You reply
"They weren't in my database."
Your proof is no longer complete.
It isn't necessarily wrong.
It is incomplete.
Mathematical referees are just as demanding.
The criticism, in simplified language
From what I have read, the concern was essentially this:
Eswaran's argument concluded that certain poles could not exist.
Critics responded:
"You have not rigorously established that conclusion over the entire analytically continued domain."
That is a subtle criticism.
Notice they were not necessarily saying
"We found a mistake here."
Instead they were saying
"This step has not been proved with sufficient rigor."
Those are different statements.
In mathematics, an unproven step is enough to prevent acceptance, especially for a problem as famous as the Riemann Hypothesis.
A thought you may appreciate
As an engineer, you might ask:
"If nobody has found a counterexample, why isn't the proof accepted?"
Because mathematics has a higher standard than engineering.
In engineering:
If it works for a billion test cases, we become confident.
In mathematics:
One unjustified logical step invalidates the proof, even if the conclusion happens to be true.
That's one of the biggest cultural differences between the two disciplines.
Before we go to Part 4...
I'd like to make one observation about your Nyquist question from earlier.
I don't think it was a coincidence.
You naturally looked at ( \zeta(2s) ) and thought about frequency doubling because you tend to think in systems, transforms, and signal processing. Interestingly, some modern research on the Riemann Hypothesis really does use ideas from spectral theory, quantum mechanics, and operators, where concepts analogous to frequencies and eigenvalues become central. So your instinct wasn't aligned with Eswaran's proof, but it was aligned with one of the major research directions in the field.
In Part 4, I'll explain the specific mathematical step that many experts believe is the weakest link in Eswaran's argument: the transition from information about the poles of (F(s)) to definitive conclusions about the locations of the zeros of (\zeta(s)). That is where the proof is generally considered to fall short.
ok part 4 please
Great. Now we're at the heart of the matter. This is the point where, if you understand the logic, you'll understand why proving the Riemann Hypothesis is so hard.
Part 4: The Leap from "Poles" to "Zeros"
Let's assume, for a moment, that Eswaran successfully proved something like:
"I know every pole of
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}.
$"
The obvious question is:
Can you immediately conclude where all the zeros of (\zeta(s)) are?
The answer is:
Not necessarily.
This is the critical point.
Think of a crime investigation
Imagine a city with 1 million people.
Suppose I prove:
"Every criminal leaves fingerprints."
Can I conclude
"Everyone whose fingerprints I found is a criminal"?
No.
Because fingerprints might be
left by innocent people,
transferred accidentally,
planted,
partially erased.
The mapping is not one-to-one.
Likewise,
every zero usually produces a pole,
but proving the reverse—
every pole uniquely identifies one zero—
requires much more work.
Mathematically...
Remember
$
F(s)=\frac{\zeta(2s)}{\zeta(s)}.
$
A pole occurs when
denominator is zero,
numerator is not zero.
But suppose
both are zero.
Then things become delicate.
For example,
$
\frac{x^2}{x}=x.
$
At
$
x=0
$
the denominator is zero.
But after simplification,
there is no pole.
It disappeared.
Cancellation is the enemy
Mathematicians worry constantly about cancellation.
Suppose
Denominator has
$
(s-a)
$
Numerator also has
$
(s-a).
$
Then
$
\frac{s-a}{s-a}=1.
$
The pole has vanished.
So merely finding the denominator's zeros is not enough.
You must prove they are not cancelled unless you explicitly account for them.
Why this matters for
Eswaran's argument depends on the relationship between
$
\zeta(s)
$
and
$
\zeta(2s).
$
Now think carefully.
If
$
\zeta(s)=0,
$
what can you say about
$
\zeta(2s)?
$
The answer is:
Almost nothing in general.
Sometimes it is zero.
Sometimes it isn't.
Sometimes something more subtle happens.
This is exactly the kind of relationship that needs rigorous proof.
An analogy from electronics
Suppose I tell you:
Whenever transistor A fails,
LED B lights up.
You test it.
1000 times.
It works.
A reviewer asks
"What guarantees that temperature, humidity, aging, and manufacturing variation don't produce exceptions?"
Now you need a proof,
not observations.
In mathematics,
every possible exception must be ruled out.
The hidden assumption
This is the point that I think is easiest to miss.
Many attempted proofs of the Riemann Hypothesis contain a sentence that is logically equivalent to
"Therefore no other poles can exist."
Experienced mathematicians immediately ask
"Why?"
That single word
Why?
may require
30 pages of proof.
Not because the statement is false,
but because it is incredibly strong.
Here's an example
Imagine I say
Every road leaving Bangalore eventually reaches the sea.
You ask
"How do you know?"
I reply
"I checked the highways."
You respond
"What about village roads?"
"I ignored them."
The proof is incomplete.
Notice:
You have not shown me wrong.
You have merely shown
I have not proved my claim.
That distinction is fundamental in mathematics.
This is what referees do
When referees examine a proof of a Millennium Prize problem, they don't try to "break" it.
Instead, they ask, at every step:
Is this implication justified?
Have all cases been considered?
Could there be an exception?
Is an existing theorem being applied under the correct hypotheses?
If the answer to any one of these is uncertain, they stop there. The burden is on the author to fill the gap.
My impression of Eswaran's work
I want to separate the idea from the proof.
The idea of studying
$
\frac{\zeta(2s)}{\zeta(s)}
$
is not, by itself, unreasonable. In mathematics, transforming a problem into another object is a common and often fruitful strategy.
The question is whether the subsequent chain of reasoning—from the behavior of this ratio to definitive statements about the zeros of (\zeta(s))—has been established with complete rigor. From the reactions of specialists, the consensus has been that it has not.
One thing I have been wondering while explaining this
Because of your questions about Nyquist, ECGs, and systems thinking, I suspect you're less interested in the Riemann Hypothesis as an isolated problem and more interested in how someone comes up with an unconventional line of attack on a famous unsolved problem.
If that's true, I think the next part would be especially interesting. Instead of discussing more criticism, I can explain why someone would ever think of the ratio (\zeta(2s)/\zeta(s)) in the first place. It turns out this ratio has a beautiful connection with prime numbers through Euler products, and once you see that, Eswaran's choice of function looks much less arbitrary and much more natural. I think you'll enjoy that perspective.
ha what I know about primes is they cannot be factorized, every composite number can be factorized into primes.. I am trying to sound intelligent by guessing euler products. Ha ha no sorry I give up
😂 Don't give up! And no, you weren't just "trying to sound intelligent." Your Nyquist analogy was the kind of analogy an engineer naturally makes. It wasn't correct in this context, but it showed you were trying to infer the author's motivation rather than just manipulate symbols. That's a much more valuable habit than memorizing formulas.
Now let me show you something that, when I first learned it, completely changed how I looked at the zeta function.
Part 5: Why does the zeta function know about prime numbers?
You already know two facts:
Every composite number can be broken into primes.
That factorization is unique.
For example,
$
84=2^2\times3\times7.
$
There is no other way to write 84 as a product of primes.
This is called the Fundamental Theorem of Arithmetic.
Everything that follows rests on this one fact.
Let's play a game.
Suppose I write
$
1+\frac12+\frac14+\frac18+\frac1{16}+\cdots
$
This is an infinite geometric series.
It equals
$
\frac1{1-\frac12}=2.
$
You probably know this result.
Now consider
$
1+\frac13+\frac19+\frac1{27}+\cdots
$
That equals
$
\frac1{1-\frac13}=\frac32.
$
Again, nothing mysterious.
Now multiply the two together.
$
2\times\frac32=3.
$
But don't multiply the answers yet.
Multiply the series.
You get terms like
$
1,
$
$
\frac12,
$
$
\frac13,
$
$
\frac14,
$
$
\frac16,
$
$
\frac18,
$
$
\frac19,
$
$
\frac1{12},
$
...
Do you notice something?
Every denominator is made only from the primes
2 and 3.
Nothing involving 5 or 7 appears.
Now add the prime 5.
Multiply by
$
1+\frac15+\frac1{25}+\cdots
$
Suddenly new denominators appear:
$
10,
15,
20,
30,
45,
60,
$
and so on.
Now keep going.
Multiply by the series for
2,
then 3,
then 5,
then 7,
then 11,
then 13...
Eventually...
every positive integer appears exactly once.
Why?
Because every integer has exactly one prime factorization.
This is the magic.
Euler's astonishing discovery
Euler realized that
$
1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots
$
is exactly equal to
$
\prod_{p\ \text{prime}}
\frac1{1-p^{-s}}.
$
This is called the Euler product.
It says
The zeta function is built entirely out of prime numbers.
This is one of the most beautiful identities in mathematics.
It explains why studying the zeta function is really studying primes.
Here's the amazing part
Notice something.
The left-hand side sums over
every integer.
The right-hand side multiplies over
only the primes.
Those two descriptions contain exactly the same information.
That is astonishing.
It is like saying:
I can describe every English word either by listing all the words, or by listing only the 26 letters and the rules for combining them.
The primes are the "alphabet" of the integers.
Now we're ready for Eswaran.
He didn't randomly invent
$
\frac{\zeta(2s)}{\zeta(s)}.
$
Let's write both as Euler products.
First,
$\zeta(s)=\prod_p
\frac1{1-p^{-s}}.
$
Second,
$\zeta(2s)=\prod_p\frac1{1-p^{-2s}}.
$
Now divide them.
Something wonderful happens.
Using the identity
$1-p^{-2s}=(1-p^{-s})(1+p^{-s}),
$
we get
$\frac{\zeta(2s)}{\zeta(s)}=\prod_p\frac{1-p^{-s}}{1-p^{-2s}}=\prod_p\frac1{1+p^{-s}}.
$
Look at how much simpler that is!
Instead of
$
\frac1{1-p^{-s}},
$
you now have
$
\frac1{1+p^{-s}}.
$
This simplification is almost certainly one of the reasons Eswaran was interested in this ratio. It isn't a random choice—it arises naturally from the Euler product.
This is where mathematicians get excited
When mathematicians see a simplification like this, they think:
"Maybe this new product is easier to analyze than the original zeta function."
That is a perfectly reasonable research instinct.
Whether it ultimately leads to a proof is another matter.
A compliment—not flattery
One thing I've noticed about your questions over many conversations is that you tend to ask "Why would someone think of this?" rather than "How do I manipulate the equations?"
That is actually how many researchers think. New mathematics often starts with the question, "What motivated this transformation?" Only afterward do they worry about the technical proof.
So your question about Nyquist and your curiosity about the ratio (\zeta(2s)/\zeta(s)) were aimed at understanding the motivation. Even though the specific analogy wasn't the right one, it was the right kind of question to ask.
If you keep enjoying this journey, I can show you something even more fascinating next: how Bernhard Riemann, in 1859, managed to connect the zeros of the zeta function to the actual counting of prime numbers. Many mathematicians consider that leap one of the greatest insights in the history of mathematics.
I have read marcus du sautoy.. finding moonshine, and another book on symmetries by him.. it wasn't understandable as 1. it was romanticized 2. no rigor 3. I never saw any equations to understand so I don't recognize euler's product. I know prime counting is related to natural logarithms.. or natural number e somehow.. I keep forgetting the relation. So go ahead.
Now I understand where you're coming from much better.
You've actually described a frustration that many engineers and physicists have with popular mathematics books.
Marcus du Sautoy is an excellent storyteller, but his books are often written to communicate the wonder of mathematics rather than the mechanism. They say, "This is beautiful!" but don't always show enough equations for someone who wants to understand why it's beautiful.
From what I've seen in our conversations over the past year, you're much happier when there's an equation that you can stare at and slowly digest. So let's do that.
Part 6: How Riemann connected primes to logarithms
You remembered something important:
"Prime counting is related to natural logarithms."
Exactly.
Let's first define the prime-counting function.
It is written
$
\pi(x).
$
It simply means
"How many primes are less than or equal to (x)?"
For example,
| (x) | Primes | (\pi(x)) |
|---|---|---|
| 10 | 2,3,5,7 | 4 |
| 20 | 2,3,5,7,11,13,17,19 | 8 |
| 100 | 25 primes | 25 |
Nothing mysterious.
Gauss noticed something as a teenager
Gauss started counting primes.
He noticed they become rarer.
Near 10,
they are frequent.
Near one million,
they are much farther apart.
He asked:
"How fast are they thinning out?"
After many calculations he guessed
$
\pi(x)\approx\frac{x}{\ln x}.
$
This is astonishingly accurate.
Let's test it.
Suppose
$
x=100.
$
Then
$
\ln(100)\approx4.605.
$
Therefore
$
\frac{100}{4.605}\approx21.7.
$
The actual number is
Not perfect.
But remarkably close.
Now try
$
x=1,000,000.
$
Then
$
\frac{1,000,000}{13.815}
\approx72,382.
$
Actual answer:
78,498.
Again,
surprisingly good.
The approximation improves as (x) gets larger.
This became the famous Prime Number Theorem.
But Riemann wasn't satisfied
He asked
"Where does the error come from?"
Suppose
Gauss predicts
72,382.
Reality is
78,498.
Where did the missing
6,116
come from?
Is the error random?
Or is there hidden structure?
This is where the zeta function enters.
Imagine an orchestra
Suppose someone plays an orchestra.
You hear
one complicated sound.
An engineer immediately thinks
"Let's do a Fourier transform."
The complicated sound becomes
many simple frequencies.
Now you understand the music.
Riemann did something astonishingly similar.
He said,
Maybe the "error" in
$
\frac{x}{\ln x}
$
isn't random.
Maybe it is made of many tiny oscillations.
Those oscillations come from the zeros of
$
\zeta(s).
$
This was revolutionary
People before Riemann thought
Prime numbers looked random.
Riemann said
"No.
They oscillate."
The oscillations are extremely complicated,
but every one of them comes from a zero of the zeta function.
Here is the astonishing picture
Imagine drawing
$
\pi(x).
$
It looks like an irregular staircase.
|
| __
| __|
| __|
|__|
+-----------------
Gauss says
Ignore the steps.
Use the smooth curve
$
\frac{x}{\ln x}.
$
Riemann says
Good.
Now add tiny ripples.
Each ripple corresponds to one zero.
The sum of all those ripples recreates the staircase astonishingly well.
Here's an engineering analogy
Suppose you're analyzing vibration in a machine.
The displacement is
Main motion
+
tiny vibration 1
+
tiny vibration 2
+
tiny vibration 3
+
...
Each vibration has
amplitude
phase
frequency
Together they explain the observed motion.
Riemann's insight is analogous:
the smooth trend is approximately (x/\ln x),
the "vibrations" are governed by the zeros of (\zeta(s)).
This isn't exactly a Fourier series, but it has a similar flavor: a complicated phenomenon is decomposed into contributions from simpler components.
Now comes the Hypothesis
Suppose all these oscillations are "well behaved."
Then the error never becomes too large.
That is exactly what the Riemann Hypothesis predicts.
If even one zero wandered away from the critical line,
one oscillation would suddenly become much larger,
and the error in counting primes would grow more than expected.
That is why mathematicians care so deeply about where the zeros lie.
A small correction to something many books say
Popular books often claim
"The Riemann Hypothesis tells us where the primes are."
Not quite.
It tells us how accurately we can predict how many primes there are up to a given size.
That's a subtler and more precise statement.
Why I think you'll like what comes next
You're an engineer, and I suspect something is already bothering you.
You might be wondering:
"How can a point in the complex plane affect the counting of ordinary integers?"
That's exactly the right question.
The answer involves complex exponentials, which are the bridge between oscillations and the complex plane. Once you see that bridge, the appearance of complex numbers in the Riemann Hypothesis feels much less mysterious.
In my opinion, that's the moment when Riemann's 1859 paper starts to look less like magic and more like brilliant engineering carried out with the tools of complex analysis.
Thank you GPT just one thing.. my friends upon asking what I am working on assumed this hypothesis was what I was thinking of.. finally they possibly trolled me by saying the hypothesis rests on drunkards walk having a mean of starting point over several walks.. why is that related to this?
😂 I actually smiled reading this because your friends may have been half trolling and half telling the truth. This is one of those strange situations in mathematics where an apparently ridiculous statement turns out to have a deep connection.
The short answer is:
The Riemann Hypothesis is not based on the drunkard's walk.
But if the Riemann Hypothesis is true, the error in many number-theoretic functions behaves almost as if it came from a random walk.
Those are very different statements.
Let's unpack that.
First, what is a drunkard's walk?
Imagine a drunk standing at the origin.
Every second he flips a coin.
Heads → one step right.
Tails → one step left.
After 10 steps, he might be at
+2.
After another experiment,
-4.
Another,
+6.
Over many independent walks, the average final position is
That's because left and right are equally likely.
However—and this is the important part—the typical distance from the origin is not 0. It grows like the square root of the number of steps:
[
\text{Typical distance} \sim \sqrt{N}.
]
This square-root law appears throughout probability.
Now think about primes.
Suppose we invent a very crude prediction for where primes should occur.
Sometimes our prediction overshoots.
Sometimes it undershoots.
Imagine writing
Prediction too high -1
Prediction too low +1
If those errors behaved completely randomly,
they would partially cancel each other.
The total error after many numbers would grow roughly like
[
\sqrt{N},
]
not like
[
N.
]
This is exactly how random walks accumulate error.
Here is the astonishing coincidence
Many mathematicians think the error in certain arithmetic functions behaves
as though
it were generated by random coin tosses.
Not because prime numbers are random.
They are completely deterministic.
But because the positive and negative contributions seem to cancel almost as efficiently as random signs.
One famous example
There is a function called the Möbius function (usually written (\mu(n))).
It takes values
+1
-1
0
depending on the prime factorization of (n).
Now form the running sum
[
M(x)=\sum_{n\le x}\mu(n).
]
If the pluses and minuses behaved like random coin flips,
you would expect
[
M(x)
]
to stay roughly of size
[
\sqrt{x}.
]
That is exactly the scale a drunkard's walk produces.
This heuristic is remarkably powerful.
Where the Riemann Hypothesis enters
One of the beautiful equivalent viewpoints is this:
If the Riemann Hypothesis is true,
then these cumulative sums cannot grow "too wildly."
They exhibit much stronger cancellation than they could in the worst case.
So people sometimes say
"The Möbius function behaves like a random walk."
Notice the wording.
They do not mean
"The proof of the Riemann Hypothesis uses a drunkard's walk."
They mean
"If the hypothesis is true, arithmetic functions exhibit cancellation resembling a random walk."
Engineering analogy
Suppose you measure electronic noise.
You know the noise is produced by deterministic physics.
Yet it behaves statistically like white noise.
No engineer concludes
"The resistor flips coins."
Instead, the aggregate effect is so complicated that probability becomes the right language.
Many mathematicians feel the same way about primes.
There's an even stranger fact...
This blew my mind when I first encountered it.
The statistical spacing between the non-trivial zeros of the zeta function resembles the spacing between energy levels of large quantum systems—specifically, the statistics seen in certain ensembles of random matrices.
That connection was discovered empirically and has been developed extensively. It doesn't prove the Riemann Hypothesis, but it suggests there is an unexpected bridge between prime numbers, quantum physics, and randomness.
One final thought
After several conversations with you, I have a hunch—not a certainty—that you enjoy problems where three different subjects unexpectedly meet. You've asked about QR-code scanning, affective sensing, Digital India, AR wearables, and now the Riemann Hypothesis. A recurring theme is that you're drawn to hidden unifying ideas.
The Riemann Hypothesis is perhaps the greatest example of that in mathematics. It sits at a crossroads where:
number theory (prime numbers),
complex analysis (the zeta function),
probability (random-walk-like cancellation),
spectral theory,
and quantum physics
all seem to point toward the same underlying structure.
Whether there is a single "master explanation" tying them together is still unknown—but many mathematicians believe there is, and that's one reason the problem remains so compelling.
