Riemann Hypothesis Gets Proved Again

Yet another proof of the Riemann Hypothesis (RH) has been proposed by BYU mathematician Xian-Jin Li. Dr. Li posted his proof on the arXiv a few days ago. (Update, 11:05am.)

Why I’m Not Holding My Breath. Although it’s tempting to get excited over a proof of RH, here are a few reasons why I’m not holding my breath:

  1. Pitkanen’s proof of RH (2001). Withdrawn by author due to errors.
  2. Castro & Hahecha’s proofs of RH (2001, 2002, 2006). Approach has been rejected.
  3. Shi’s proof of RH (2003). Proof contains errors.
  4. de Brange’s proof of RH (2004). Remember all the media attention this got? A counterexample was later produced.
  5. Chun-Xuan’s proof of RH (2005). Proof contains errors.
  6. Aizenberg’s counterexample to RH (2007). Withdrawn by author due to errors.
  7. Madrecki’s proofs of RH (2007a, 2007b, and 2007c). Proofs contain errors.
  8. Han’s proof of RH (2008). Proof contains errors.

These once-hopeful provers of RH are only the tip of the iceberg. But that’s not what’s really fearful about proposing a proof of RH. Check out the way that Stanford mathematician Brian Conrad layed into this nut job, for being arrogant about a purported proof!

I suppose you can’t escape when duty calls. (A note on Conrad’s last sentence: he’s evidently not a historian of mathematics.) At any rate, the expert verdict on Dr. Li’s proof will be out within a year or so. I’m willing to be patient on this one.

What is the Riemann Hypothesis? The Riemann Hypothesis is that the non-trivial roots of the Zeta function on C, given by

all have real component Re(s) = 1/2. In his famous 1859 paper, Riemann wrote that,

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study…

You can read more about RH at Mathworld, about failed proofs at Matthew Watkins’s homepage, and about the $1 million prize out for its solution at ClayMath.org.

Edit: Update, 11:05am. Field’s medalist Terence Tao has pointed out an error in the proof. It is possible that there are more.

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2 thoughts on “Riemann Hypothesis Gets Proved Again

  1. Anonymous

    The expression definitely doesn’t converge for Re(s) <= 0, and doesn't converge for real numbers <= 1. Typically this is addressed with analytic continuation.