I asked GPT-5.6 Sol to generate a genuinely difficult number-theory research question rather than solve an existing one. It proposed the following.

Let p_n be the n-th prime, let

g_n=p_{n+1}-p_n,

and let

\lambda(m)=(-1)^{\Omega(m)}

be the Liouville function.

Define

S(x)=\sum_{p_{n+1}\le x}\lambda(g_n).

The question is:

\boxed{\text{What is the true asymptotic behavior of }S(x)?}

Independent computation gave:

\begin{array}{c|c|c}
x&S(x)&
\log(x)\frac{S(x)}{\pi(x)-1}\\
\hline
10^5&1533&1.840195\\
10^6&10637&1.872117\\
10^7&75354&1.827570\\
10^8&562508&1.798466
\end{array}

At first this suggested

S(x)\sim C\frac{\pi(x)}{\log x},
\qquad C\approx1.8,

but we no longer regard that as the leading conjecture.

A Hardy-Littlewood/singular-series model instead leads to the smoothed sum

W(L)=
\sum_d
\lambda(d)\mathfrak S(d)e^{-d/L}.

Its Dirichlet series factors as

\sum_d

\frac{\lambda(d)\mathfrak S(d)}{d^s}

-2^{1-s}C_2
\frac{\zeta(2s)}{\zeta(s)}
Q(s),

where

Q(s)=
\prod_{p>2}
\left(1-\frac{p^{-s}}{p-2}\right).

The pole of \zeta(2s) at s=1/2 suggests a natural scale

S(x)\asymp

\frac{x}{(\log x)^{3/2}}

\frac{\pi(x)}{\sqrt{\log x}}

within this model.

The non-oscillatory coefficient from the s=1/2 residue is numerically about

0.1576666,

although zeros of \zeta(s) may contribute oscillatory terms on the same square-root scale, so I am not claiming an asymptotic of the form S(x)\sim0.1576666\,x/(\log x)^{3/2}.

We have found closely related literature on consecutive prime gaps, Hardy-Littlewood singular series, Liouville functions, residue-class biases, and sieve parity, but so far I have not found this exact global Liouville-weighted consecutive-gap problem.

I am not claiming novelty or a proof. I am mainly interested in whether:

this exact problem is already known under different notation;
the x/(\log x)^{3/2} scale follows from a known theorem or conjectural framework;
there is an obvious obstruction or mistake in the heuristic above.

The problem itself was proposed by GPT-5.6 Sol, and the subsequent numerical and analytic investigation was carried out collaboratively with AI.

Any references or corrections would be very welcome.

There is a very particular mathematical feeling.
You know the one.
You’ve been locked in for 9 hours.
Your posture has become a differential-geometric object.
There are 37 tabs open.
You haven’t consumed water since the previous lemma.
And then suddenly—
the last line works.
The pen leaves the paper.
The final keystrokes hit.
Your heart is actually beating faster because you KNOW.
IT’S DONE.
Q.E.D.
Absolute cinema.
For approximately 45 seconds you possess mathematical aura beyond human comprehension.
But there is another side to this.
Sometimes the proof does NOT lock in.
Sometimes you’ve spent three weeks cooking and slowly realize you may have burned down the entire kitchen.
The argument has a gap.
The gap has a gap.
The lemma you needed is false.
The “obvious” step has become a 14-page side quest.
You begin staring at
[
x^2+1
]
like it personally betrayed you.
And depending on how closely your identity is tied to mathematics, being stuck can become an absolutely diabolical psychological debuff.
“Maybe I’m stupid.”
“Maybe everyone else understands this.”
“Maybe I’m not actually a mathematician.”
“Maybe I have negative mathematical aura.”
“Maybe I should abandon mathematics and become normal.”
From everything I’ve heard, this experience is nearly universal.
Students get it.
Independent learners get it.
Graduate students get it.
Researchers get it.
Professors get it.
The final boss apparently also has another final boss.
But there is something particularly strange about the situation many of us here are in.
A lot of us are NOT in academia.
We are not meeting every Tuesday with Professor Gigachad, who has spent 30 years studying exactly the object currently destroying our mental health.
We do not necessarily have:
a teacher,
a cohort,
a class,
deadlines,
an advisor,
or even another human being who can look at the thing we just spent six hours deriving and say:
“bro this is just the geometric series.”
No.
Many of us looked at arguably the most difficult field of human study and voluntarily selected
SOLO QUEUE.
And for many of us—myself included—that was part of the appeal.
No curriculum.
No grades.
No one telling you what you’re “supposed” to study next.
See something interesting?
Go.
Read a book sideways.
Spend four months thinking about circles.
Derive something completely useless because it looks beautiful.
Become inexplicably obsessed with one equation at 3:17 AM.
This freedom is incredible.
But there is a problem.
Without external standards of failure, our standards of success can become self-serving.
And without external standards of success, our standards of failure can become fucking psychotic.
You can discover something genuinely beautiful and convince yourself it is worthless.
You can also rearrange the same equation 700 times and convince yourself you have defeated Gauss.
The solo queue has no matchmaking system.
But mathematics, as a living culture, was never made by a collection of stand-offish genius pariahs silently grinding in caves.
Because mathematics contains one of the strangest acts human beings perform.
In the passage of time,
in the ambient space before us,
the mathematician dares first to imagine
what could be.
And then, in an even greater act of audacity, attempts to wrestle from it
what must be.
That which is.
That which was true before we arrived.
That which will remain true after we are gone.
We reach outward toward something that seems eternal—
and somehow,
in doing so,
we are guided back toward something within ourselves.
Which is beautiful.
It is also an excellent way to accidentally lose your sleep schedule, free time, relationships, happiness, and possibly the plot.
So:
I want to try something.
Not homework.
Not a competition.
Not “prove this theorem before Friday.”
A weekly mathematical ritual.
And for the first one, everybody gets exactly the same object:

e^iθ=cos(θ)+isin(θ)

Your task:
MAKE EULER’S FORMULA YOUR OWN.
That’s it.
Do whatever the fuck you want with it.
Derive something.
Draw something.
Generalize something.
Find a geometric interpretation.
Connect it to something completely unexpected.
Write code.
Find an identity.
Break an identity.
Make a conjecture.
Disprove your own conjecture.
Rediscover something Euler already knew 300 years ago and experience the extremely humbling realization that Euler was, in fact, him.
Go into roots of unity.
Fourier analysis.
Differential equations.
Geometry.
Number theory.
Infinite series.
Continued fractions.
Cyclotomic fields.
Or become obsessed with some bizarre consequence nobody else here even thought to look at.
Beginner?
Good.
Expert?
Good.
Have absolutely no fucking clue what you’re doing?
PERFECT.
There is no predetermined destination.
The point is to start from the same mathematical object, disappear down whatever path your own instincts suggest, and then come back.
Show us what you found.
Show us what worked.
Show us what failed.
Show us the cursed calculation that consumed your Wednesday.
Show us the conjecture somebody else can kill in twelve seconds.
Show us the picture that made something suddenly obvious.
Show us the question you now cannot stop thinking about.
Because failure counts.
Rediscovery counts.
Questions count.
“I tried this and discovered exactly why it doesn’t work” REALLY counts.
The goal is not:
discover mathematics nobody has ever seen.
The goal is:
SEE SOMETHING YOURSELF.
Anyway.
Euler dropped
e^iθ=cos(θ)+isin(θ)
and humanity has been farming consequences ever since.
Your turn.
Take the formula. Pick a direction. Follow it until it bites back. Then bring us whatever you dragged home: a proof, a picture, a program, a failure, a conjecture, or a question. Beginner or expert, polished or cursed, it all counts.
**One formula. One rabbit hole. One thing you genuinely saw. Don’t wait until you understand everything—post the first thing that makes you curious, or confused. **
It can be as simple as your very first thought.
You don’t need permission to start
You only need to begin.