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I have essentially no real knowledge about this topic, but I suppose some progress has been made by tightening the bounds for this constant?

Can somebody with more info chime in--- is this a huge, groundbreaking amount of progress?

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Reading https://en.wikipedia.org/wiki/De_Bruijn–Newman_constant, it moves the bound from −1.1×10−12 (probably an approximation) to zero.

So, not huge in absolute terms, but on the other hand, huge, as it removes all remaining negative numbers.

I would presume that entirely different methods would be needed to prove that the constant can't be positive. There are many types analytical proofs in math that break down to "negative", "zero" and "positive" cases and often a couple of those are relatively easy whereas others are extremely difficult. The Calabi conjecture is an example.

So really it's hard to say. You might even say this proof made things harder. It guarantees that you have to prove the constant is 0 if you want to prove the conjecture. Before you might have hoped prove negativity.

Right, near the bottom of the paper he writes “...this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis”

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