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km-git-acc / dbn_upper_bound

Computational effort to upper bound the de Bruijn-Newman constant as part of a Polymath project
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Typo in definition of γ? #75

Closed WilCrofter closed 6 years ago

WilCrofter commented 6 years ago

There may be sign error in your (excellent) paper's definition of γ. Currently definition (16) reads

(16) γ(x+iy) := Mₜ(t,(1-y+i*x)/2)/Mₜ(t,(1+y-i*x)/2)

It should, perhaps, read

(16) γ(x+iy) := Mₜ(t,(1-y+i*x)/2)/Mₜ(t,(1+y+i*x)/2)

IOW, that ix/2 should have the same sign in numerator and denominator. The mean value theorem as used in first line of the proof of Proposition 6.6 (estimates) implies, as far as can tell that log|Mₜ(t,(σ+i*x)/2)| should be a function of σ only which it would not be if the signs of ix were different in numerator and denominator.

Apologies if I'm missing something obvious. As far as I've checked computationally, the inequality seems to be true in either case.

km-git-acc commented 6 years ago

@WilCrofter Is the question addressed to Prof. Tao. It's not clear if Github sends the comment to the mail unless participants are actually marked. You could try doing that or asking on the blog.

WilCrofter commented 6 years ago

@km-git-acc thanks for clarifying. I'll close the issue and repost to the blog. It's a trivial matter, but it does look like a sign error.