Equivalent form of the Riemann hypothesis

The famous Riemann hypothesis is equivalent to the following not-so-famous conjecture:

For every N ≥ 100, | log( lcm(1, 2, …, N) ) – N | ≤ 2 log(N) √N.

Here “lcm” stands for “least common multiple” and “log” means natural log.

Source: Andrew Granville’s chapter on analytic number theory in Princeton Companion to Mathematics.

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8 comments on “Equivalent form of the Riemann hypothesis
  1. Colin says:

    For clarification’s sake, is the log N on the RHS included inside the square root?

  2. Here’s another one. Let H_n be the nth harmonic number (sum of 1/k for k=1 to n). That the sum of the divisors of n is less than or equal to H_n + exp(H_n)·ln(H_n) for any n is equivalent to RH according to this paper.

    It almost makes the problem seem approachable! But experts say there is NO WAY anyone’s going to get to RH via this route.

  3. John says:

    No, the rhs is 2 log(N) √N. I’ll edit the post to make that clearer. Thanks.

  4. Tim Hopper says:

    The proof is left to the reader.

  5. John John Deragon says:

    By the way, in Princeton book, the statement is cited as

    |log(lcm[1, 2, . . . , N]) − N|  < √N (log N)^2.

  6. David Feuer says:

    Of course, the experts also have no clue how to solve the conjecture by other means…

  7. Dave Tate says:

    Slow to comment here, but let me put in a plug for a book I enjoyed tremendously. It’s called Prime Obsession, and it’s by John Derbyshire. Chapters alternate between (a) historical and biographical discussion of Riemann, and (b) a mathematical development of the Riemann Hypothesis, its antecedents, and some insight into why it’s so interesting and important. Derbyshire’s goal was to present the Riemann Hypothesis in a way that anyone with basic high school algebra skills could follow it. He almost succeeded; there’s a tiny bit of calculus at the end. For readers of this blog, though, all of the math is elementary.

  8. susumu says:

    Please, look at Mochizuki’s recently updated paper,
    ” Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations ” .
    He added “Remark 2.2.1.” pp.46 .
    It seems to me that, perhaps, he is going to make a “double-play”, ABC conj and RH.