An Elementary Problem Equivalent to the Riemann Hypothesis

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M. Kaneko notes that these results together with a small computation can be used to show that the Riemann hypothesis is equivalent to the assertion that sigma(n) < exp(H_n) log(H_n) for n > 60.

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Lemma 3.1. For n >= 3, exp(H_n) log(H_n) >= e^gamma n log log n . (3.3)

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Lemma 3.2. For n >= 3, H_n + exp(H_n) log(H_n) <= e^gamma n log log n + (4n)/(log n) . (3.7)

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Problem E. Let H_n = sum_{j=1}^n 1/j. Show that, for each n >= 1, sum_{d|n} d <= H_n + exp(H_n) log(H_n), (1.1) with equality only for n = 1.

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Proof of Theorem 1.1. (<=) Suppose the Riemann hypothesis is true. Then Theorem 3.1 and Lemma 3.1 together give, for n >= 5041, sum_{d|n} d <= e^gamma n log log n < H_n + exp(H_n) log H_n. For 1 <= n <= 5040 one verifies (1.1) directly by computer, the only case of equality being n = 1. (=>) Suppose (1.1) holds for all n. We argue by contradiction, assuming that the Riemann hypothesis is false. Then Theorem 3.2 applies. However its lower bound (which is valid for infinitely many n) contradicts the upper bound of Lemma 3.2 (which holds for all sufficiently large n). We conclude that the Riemann hypothesis must be true.

Referencing norms: verified-direct-1, verified-direct-5040, verified-direct-6, verified-direct-60, verified_by_direct_computation

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The Riemann hypothesis, stated by Riemann [21] in 1859, concerns the complex zeros of the Riemann zeta function. The Riemann zeta function zeta(s) is defined in the half-plane Re(s) > 1 by the convergent Dirichlet series zeta(s) = sum_{n=1}^infinity n^(-s) and is extended by analytic continuation to the complex plane, where it has one singularity; namely, a simple pole with residue 1 at s = 1. The Riemann hypothesis states that the nonreal zeros of the Riemann zeta function zeta(s) all lie on the line Re(s) = 1/2.

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Robin's criterion states that the Riemann hypothesis is true if and only if sigma(n) < e^gamma n log log n for all n >= 5041, (1.2) where gamma ~ 0.57721 is Euler's constant. This criterion is related to the density of primes, as explained in Section 2.

Referencing norms: RobinCriterionAsPresentedByLagarias, attributed_to, lagarias-presents-robin, robin-attribution, robin-iff-rh

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Theorem 1.1 Problem E is equivalent to the Riemann hypothesis.

Referencing norms: problem-e-iff-rh

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Theorem 3.1 (Robin). If the Riemann hypothesis is true, then for each n >= 5041, sum_{d|n} d <= e^gamma n log log n , (3.1) where gamma is Euler's constant.

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Theorem 3.2 (Robin). If the Riemann hypothesis is false, then there exist constants 0 < beta < 1/2 and C > 0 such that sum_{d|n} d >= e^gamma n log log n + (C n log log n)/((log n)^beta) (3.2) holds for infinitely many n.

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An Elementary Problem Equivalent to the Riemann Hypothesis Jeffrey C. Lagarias (July 29, 2001 version)

Referencing norms: LagariasSource