Topology vis a vis arithmetic
Ignacio Añón
ianon at latahona.com.uy
Wed Feb 16 10:11:09 EST 2022
Apologies for the delayed response.
Caballero wrote:
*"In my own research [2], I have developed a q-analog of Jacobi's formula
for the number of representations of an integer as the sum of two squares,
denoted r_2(n, q). I suspect that the following problem is undecidable: *
*Given a list of non-zero integers a_1,...,a_k, if there is n such that
a_1,...,a_k are the non-zero coefficients of r_2(n, q), return true,
otherwise, return false*
*If this problem is decidable, I would appreciate seeing the algorithm. If
it is undecidable, the function that I used to define r_2(n, q) may give
some clue to prove undecidability in areas of complex analysis with a
strong connection to number theory. This function is a variation (change of
sign) of the Jordan-Kronecker function [3]."*
This is a beautifully succinct conjecture, that goes quite deep. But
profound elementary problems like these, require, at the "FOM" level,
subtler ideas than "consistency", "independence", "decidability", or the
arsenal of techniques used in set classification.
Since few Mathematicians would deny the mathematical importance, and
naturality, of your conjecture, its independence, would call for a delicate
axiom system that acts as the "minimal" model, where this conjecture can be
decided. This calls for a complete reconfiguration in FOM...
There is a whole sphere of elementary number theoretic problems, similar to
yours, that need this reconfiguration. At the end of this post, I list 3 of
them: the style of these problems goes back to Fermat and Lagrange, whose
ideas were afterwards popularized by Guass and Landau, but with little
sophistication, and are still today, after a few centuries, unsurpassed...
Quadratic forms were a shallow use of Lagrange's beautiful technique to
write any square integer as the descending difference between an even
integer and three squares. Using this technique there is infinite potential
to simplify modern modular forms, group theory, and analysis...
Even a genius like Gauss, got quite blinded by certain elementary issues.
For instance: he was convinced that the number of prime numbers less than
x, was always less than the logarithmic integral of x. Today you could use
powerful techniques in proof assistants, and computational halting, to
attempt to solve this issue, achieving millions of concrete examples where
the Gauss conjecture holds; and yet Hardy and Littlewood, using direct
analytic methods, showed that Gauss was deluded...
This is a cautionary tale for the use of digital technology in analytic
number theory: I'm a passionate defender of software in mathematics, but am
also convinced that microprocessors will only be a powerful tool, on our
quest to simplify analysis, by enhancing our capacity to visualize in
simple images, complex structures(multiresolutional analysis).
Binary languages as those used in software, are trivial intelectually: we
often underestimate the infinite power that technology has, to simplify and
to clarify, rather than to solve, or to discover, intellectual problems. A
rudimentary calculator shows this...
In 1000 years, we will have a proof of the four color theorem, that is much
more beautifull, and simpler, than the thing we have today: creative
technological literacy, will be, by then, something as incorporated into
Man's mind, as speaking a language is today...
Godel [1] was interested in minimal models of the sort mentioned above: he
devised a plausible one, where CH holds. The absoluteness of Hausdorf gaps
makes the consistency of these axioms difficult to prove, but he evidently
thought that set theory and analysis would move in this direction in the
future. To him the Hausdorff "pantachie" problem was much deeper than any
issue of decidability or incompleteness...
The function which you used to define r_2(n, q), is at the level of the
best functional equations in analytic number theory; in his original paper,
while attacking a similar problem, Riemann discovered his analytic
expression for the infinite series of prime powers, by observing that if
you split the complex variable of the Zeta function, dividing it into
infinitesimally short halves, and you then apply a Jacobi style elliptic
inversion to it, and use certain simple properties of the complex
factorial(gamma function), then you can find an even function of the zeta
function's imaginary part: this even function is infinitely differentiable,
and of first order, so it illuminates the zeta function's topological, and
analytic qualities, directly...
Riemann evidently thought that this property of the zeta function, revealed
certain aspects of the prime continuum, which his so called "hypothesis"
didn't. This hypothesis has a shallow prestige surrounding it: any
specialist in the area knows quite well that the problem of Zero
multiplicity, or the Erdos' conjecture on prime differences, are much
deeper questions...
By sophisticating the topological analysis of infinitesimally short
intervals, you can understand why the Landau "mollifier", popularized by
Selberg, is a "dead end", and gives no new insight into concrete
arithmetic...
My response is already too long: you also mentioned, in your post, group
theoretic results by Connes: it seems unlikely that such abstract group
theoretic ideas, will illuminate concrete number theory any time soon,
until they're given a Gromov style, topological, spatial, simple, more
natural, form. May be we can discuss this in future emails...
Best regards,
Ignacio
(1) Godel collected Works Volume III.
Here's a summary of simple arithmetic problems that call for a
reconfiguration in FOM:
*Problem I.*
Consider the simple diophantine equation:
D(y) = y - 1 = 2^2^n
Here n runs through the natural numbers, and y is an unknown integer.
Fermat thought, incorrectly, that y is always a prime. Hardy conjectured
that y is not a prime just finitely many times.
*Problem II. *
p runs through all the primes.
We define a function g(p), such that 2g(p) - 1 = p.
Consider the infinite series:
P(y) = Sigma (-1)^g(p) e^py.
P(y) tends to infinity, as y tends to 0. Sigma is infinite summation, e the
natural logarithm base. The conjecture was first conceived by Lagrange and
Chebyshev.
*Problem III. *
g_n is the difference between consecutive primes p_n such that:
g_n = p_n - p_(n - 1).
For all p <= x, the following holds:
Sigma (g_n)^2 <= O(xlogx).
This idea was first stated by Erdos. Sigma stands for infinite summation, O
is Landau's notation for bounds with constants.
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