Has Euler's Constant $\gamma$ been proven to be irrational?

Question

I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$.

I know people have been trying to prove that $\gamma$ is irrational or not for hundreds of years. I could not find very much on this paper online. Does this paper in fact have a conclusive proof that $\gamma$ is irrational?

Answer 1

No, the paper is incorrect, and in many places. A quick way to check would be to see if it were published anywhere (it isn't). This isn't conclusive, but such a result would almost certainly be sent to some peer-reviewed journal.

I should also note some of the incredible things Kaida Shi has 'proven' in addition to Hilbert's 7th problem:

• Here's a link to his combined proofs of the Goldbach Conjecture, the Twin Primes Conjecture, and some parts of the vastly stronger Schinzel's hypothesis (which he doesn't mention in the abstract, but it nonetheless there).
• Here's a link to a geometric proof of the Riemann Hypothesis. But don't get too excited, because...
• Here's a link to a geometric proof of the generalized Riemann Hypothesis.

Answer 2

@Argon. Shi starts out fine - looking at the 1/x curve, which is the obvious thing to try first. Before delving into the detail there notice the section heading of 3.2 "The proof ... ". Read that. He claims to have established various relationships between alpha, beta, gamma etc. It is obviously true that if S_1 = 1/2 - alpha, gamma=1-S_1, and alpha is irrational, then gamma is irrational. His argument is that alpha, beta are both rational or both irrational, but they cannot both be rational because their difference is 3/4 - ln 2, known to be irrational. That is all fine.

Indeed, the worrying part is that it is totally trite. It is spelt out at far too much length. But still. Why does he think alpha and beta are both rational or both irrational? That is spelt out in 3.1. He gives them as limits. The two expressions are set out at the bottom of page 5, top of page 6.

Beware \Lambda is not some strange constant, it is a coding glitch and seems to mean ... [ie dots]. He has a divergent series of terms which are roughly 1/n and is subtracting ln n to get something which tends to a finite limit. The series for alpha and beta differ only in that the alpha one has an extra term at the start and the beta one an extra term at the end so if you subtract 3/4 from alpha they line up. Similarly, the ln terms differ essentially by ln 2. So it looks as though he is right about their difference.

But the main point is why does he think they are both rational or both irrational? He has several pages deriving his limit expressions (which I have not bothered to do more than skim). But only a few sentences on how they help him:

"the series part of both series [sic] alpha_n and beta_n differ only a rational fraction, and the logarithm parts of both series [sic] alpha_n and beta_n have the identical attributes too. Therefore alpha_n and beta_n have the identical attributes too. Therefore, alpha_n and beta_n have identical attributes. Because alpha=lim alpha_n [expression given] and beta = lim beta_n [expression given] therefore both both series [sic] alpha and beta have identical attribute too. Namely, they are identically rational numbers or irrational numbers."

The bad English does not help, but does not seem to be the problem. Suppose for example we slightly garbled his expressions and made them alpha = lim(3/4 + 1/1+...+1/n - ln n) and beta = lim(1/1+...+1/n - ln n + ln 2). All his comments would still apply, but all he would have is alpha=gamma, beta=gamma+3/4-ln 2. Now why does he think that gamma is irrational iff gamma - ln 2 is irrational?

That is a ludicrous statement without further explanation. His argument boils down to:

1. gamma is irrational iff gamma-ln 2 is irrational.
2. gamma and gamma-ln 2 cannot both be rational because then ln 2 would be rational, whereas it is known to be irrational
3. Therefore gamma is irrational.

This would prove every number irrational. Without more the fact that X-Y and Y are irrational tells you nothing about whether X is irrational. Indeed it is always the case (as he uses in 2.) that if X is rational and Y is irrational then X-Y is irrational.

We all do stupid things but this paper is so incredibly stupid that part of me wonders whether it is a magnificent hoax. According to Wikipedia, Zhejiang Ocean University exists, so if it is a hoax he has covered the obvious bases. But these days many university maths departments are useless. Sadly there are plenty of people churning out junk, although most are sensible enough not to claim to have solved notorious long-standing problems.