A very noteworthy HP-15C mention

[Valentin Albillo]

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Hi, all,

I just found a really noteworthy mention of the HP-15C on this interesting PDF document:

Primes, Quantum Chaos, and Computers by Andrew. M. Odlyzko, AT&T Bell Laboratories

Dr. Odlyzko is one of the world's top experts on the Riemann Hypothesis (yes, the #1 Unsolved Problem in Mathematics, one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves it,) and in 1992 conducted large-scale computations to obtain 175,587,726 zeros of the Zeta function of heights around 10²⁰.

Furthermore, quoting from Dr. Odlyzko's Wikipedia page:

"In 1985 he and Herman te Riele disproved the Mertens conjecture. In mathematics, he is probably known best for his work on the Riemann zeta function, which led to the invention of improved algorithms, including the Odlyzko–Schönhage algorithm, and large-scale computations, which stimulated extensive research on connections between the zeta function and random matrix theory. As a direct collaborator of Paul Erdős, he has Erdős number 1."

and in Page 7 of the aforementioned PDF document Dr. Odlyzko writes:

"Around the time of the Titchmarsh computations, Turing decided to investigate the Riemann Hypothesis. After failing to prove it theoretically, he decided to carry out some computations, [...] During and after the war, electronic digital computers were developed, and around 1950 Turing programmed one of them to search for counterexamples to the Riemann Hypothesis.

Although this machine was one of the world’s most powerful at that time, it was extremely weak by modern standards, roughly equivalent to a Hewlett-Packard HP-15C pocket programmable calculator in speed, and far inferior to it in reliability and ease of use."

I don't know about you but for me, finding that such a world-renowned large-scale numeric computations eminence as Dr. Odlyzko does know about the HP-15C and even has some very appreciative comments about it, frankly makes me very proud of this little but awesomest vintage HP calc.   

V.

[EdS2]

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A very nice find! And what a possibility - that we might be able to re-do Turing's analysis on an HP ... he only managed one abortive overnight run.

Rather near by your paper, jointly by the same author, we find
Alan Turing and the Riemann Zeta Function by Dennis A. Hejhal, Andrew M. Odlyzko

Turing made an advance in the technique, although his practical result didn't extend knowledge very far, due to the machine failure.

Quote:The final stage was the verification that the sign changes that have been found account for all the zeros in a given Im(s)- range. Until Turing came out with his method, this step was done by a rather messy, although in principle not very difficult, computation based on the principle of the argument. Turing’s method obviates any need for using the argument principle. It involves only the real-valued function on the critical line

...

Quote:By that time, though, electronic digital computers were becoming available, and Turing was the first one to utilize them to investigate the zeta function. In 1950, he used the Manchester Mark 1 Electronic Computer to extend the Titchmarsh verification of the RH to the first 1104 zeros of the zeta function, the ones with 0 < Im(s) < 1540. This was a very small extension, but it represented a triumph of perseverance over a promising new technology that was still suffering from teething problems. In Turing’s words, “[i]f it had not been for the fact that the computer remained in serviceable condition for an unusually long period from 3 p.m. one afternoon to 8 a.m. the following morning it is probable that the calculations would never have been done at all.”

Notably, Turing suspected that the RH was false. "The calculations were done in an optimistic hope that a zero would be found off the critical line"

Both of Turing's papers as referenced by Odlyzko can be read online.
1943 “A method for the calculation of the zeta-function”
1953 “Some calculations of the Riemann zeta-function”

Perhaps also of interest (these are all things I've found previously and bookmarked)
The Riemann Hypothesis in Computer Science by Yu. V. Matiyasevich

[floppy]

Perhaps a peer review of this will bring a bit light in the tunnel? (far over my skills) https://vixra.org/pdf/2208.0089v1.pdf

[EdS2]

(04-19-2023 08:10 PM)floppy Wrote:  Perhaps a peer review of this ...

Hmm, "THE RIEMANN HYPOTHESIS PROVED" would be such big news, I would wait to see it that way, without reading beyond the title. In other words, it would be safe to suppose that any such claim is a mistake, even though the RH is not yet decided (unlike perpetual motion, squaring the circle, or trisecting the angle.)

[Valentin Albillo]

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Hi, floppy,

(04-19-2023 08:10 PM)floppy Wrote:  Perhaps a peer review of this will bring a bit light in the tunnel? (far over my skills) https://vixra.org/pdf/2208.0089v1.pdf

There are dozens upon dozens of alleged "proofs" of the Riemann Hypothesis, even in arXiv, and hundreds more in other less reputed sites (like the one you linked from viXra) and the Internet at large. For instance, these are a few of the most recent ones submitted to arXiv:

A Simple Proof of the Riemann Hypothesis

Proof of the Riemann Hypothesis

A Proof of the Riemann Hypothesis Using Bombieri's Equivalence Theorem

All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis

Local geometric proof of Riemann Hypothesis

A Proof of Riemann Hypothesis

A Simple Proof of the Riemann's Hypothesis

A Proof of the Riemann Hypothesis Through the Nicolas Inequality

A Proof of the Riemann Hypothesis

A proof of Riemann Hypothesis

Proof of Riemann hypothesis

A Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers

A proof of the Riemann hypothesis using the remainder term of the Dirichlet eta function

Proof of the Riemann's hypothesis

Modeling the creative process of the mind by prime numbers and a simple proof of the Riemann Hypothesis

A Simple Solution to a Major Problem: Proof of the Riemann Hypothesis

One page proof of the Riemann hypothesis

A short Brownian motion proof of the Riemann hypothesis

etc., etc.

so most mathematicians wouldn't bother to waste their time peer-reviewing any of those unless submitted by authorities on the matter, and even so. Thus, the most advisable thing to do is to wait for the big news to appear in newspapers all over the world before committing any of your time to review them or in fact pay any attention at all. Oh, and don't hold your breath while you wait ...

By the way, this has already happened a number of times, most famously with the Squaring the Circle problem ("constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge",) where prestigious institutions were swarmed with zillions of alleged solutions (which they were eventually forced to reject for lack of resources to review them,) until Ferdinand von Lindemann proved the transcendence of \(\pi\) in 1882, thus showing the impossibility of constructing \(\pi\) within the requirements of the problem.

Same here, but there's also the fact that perhaps the Riemann Hypothesis is false, in which case it would be impossible to try and prove it, no matter who and no matter how many attempts, as happened with the problem of algebraically finding roots of a quintic equation, which resisted all attempts to solve it for longer than 200 years until Niels Henrik Abel attempted (and succeeded !) to prove that perhaps it was actully impossible, against all expectations.

V.
Edit: corrected typos.

[floppy]

Thanks for sharing your insights.
What we see is sometimes valuable to be understood and in my opinion can (sometimes) be explained. Example here https://www.youtube.com/shorts/_MV-TXnsOn4
My thoughts/acting are following (I dont say what others have to do): read / understand / ask a question / feedback if I think I could see a flaw (most of the people accept a friendly question which lead to a self flaw discovery).
Regarding the article I gave the link previously, I have already 2 remarks I will perhaps share with the author (not here). However I dont want to offence a mathematician.
Sharing the link from my side was not an announcement it was done. Just an advice a try was there.
Finally I already met discoveries at an early stage.. and seeing it in the street 25years later is good fun. I could observe how sceptical people were when they saw the prototypes, how others were "sniffing" something interesting like chasing dogs etc. Discovery and Re-Discovery and Acceptance is a long human story. Example https://en.wikipedia.org/wiki/Nicolaus_Copernicus

[EdS2]

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(Edit: oops, this might come across as harsh and personal - it's not intended that way.)

(04-21-2023 07:09 AM)floppy Wrote:  My thoughts/acting are following (I dont say what others have to do): read / understand / ask a question / feedback if I think I could see a flaw (most of the people accept a friendly question which lead to a self flaw discovery).
...
Sharing the link from my side was not an announcement it was done. Just an advice a try was there.

Unfortunately, you found a bad thing in a bad place: mathematicians and scientists need to protect themselves against crackpots, of which there are very many. The time needed to understand and counterargue is time wasted - and time is precious. So the best thing to do is to move on as quickly as you can. The skill that's needed is a form of memetic hygiene - beware of which ideas you spend your precious time on. Being open minded can be good, learning new things is good, but you do have to be careful.

Also unfortunately, we now have a sidetrack below an interesting post, about computing zeros of the Riemann Zeta function, and about early computing compared to HP calculators, and about Alan Turing.

[EdS2]

(04-19-2023 07:42 PM)EdS2 Wrote:  .
A very nice find! And what a possibility - that we might be able to re-do Turing's analysis on an HP ... he only managed one abortive overnight run.

Reading further in this area, I see two interesting papers by RS Lehman, following Turing and correcting some errors too:
1966 Separation of Zeros of the Riemann Zeta-Function
1970 On the Distribution of Zeros of the Riemann Zeta-Function. doi:10.1112/plms/s3-20.2.303 

The first paper includes some ALGOL code, and running on an IBM 7090 separated about 33 zeros per second - 125 minutes of runtime in total, for 250,000 zeros.

The second paper notes and corrects some errors in the first.

Both papers are under 20 pages.

(I note that the ALGOL code extends over a few pages and includes more than one GOTO.)

Edit: this paper from 2000 looks well worth a perusal too
Borwein, J. M., Bradley, D. M., & Crandall, R. E. (2000). Computational strategies for the Riemann zeta function. Journal of Computational and Applied Mathematics, 121(1-2), 247–296. doi:10.1016/s0377-0427(00)00336-8