Let $H_{n}$ be the $n$th harmonic number. In Lagarias's paper "An Elementary Problem Equivalent to the Riemann Hypothesis," he shows that the statement $$\sum_{d\mid n}d\leq H_{n}+\exp(H_{n})\log(H_{n})\tag{1.1}$$ for every positive integer $n$, with equality if and only if $n=1$, is equivalent to the Riemann hypothesis.

In the last paragraph of section $2$, he says,

One can prove unconditionally that inequality $(1.1)$ holds for nearly all integers. Even if the Riemann hypothesis is false, the exceptions to $(1.1)$ will form a very sparse set. Furthermore, if there exists any counterexample to $(1.1)$ the value of $n$ will be very large.

So far as I can see, none of these statements is justified, or even suggested, by anything in the paper, though it may just be too subtle for me, of course.

Can anyone give me a reference justifying these statements, and quantifying the terms "nearly all integers", "very sparse set", "very large"? I assume that the first two mean "density $1$" and "density $0$" for some definition of density, but the last seems to indicate that the statement is known to be true below a specific bound, and it seems odd that the bound isn't mentioned.

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    $\begingroup$ Hint : the LHS is $$n\prod_{p^k \| n} \frac{1-p^{-k-1}}{1-p^{-1}}$$ $\endgroup$ – reuns May 26 at 0:26

In the proof of Theorem 1.1, he states that one verifies it "directly by computer" for $1\leq n\leq5040$, and it's possible that the calculations were verified beyond 5040. I'm not sure how Lagarias classifies "very large $n$," but searching yielded no numbers beyond 5040.

As for the "nearly all integers" and "very sparse set," a quick search came up with a paper by Marek Wójtowicz, titled "Robin’s inequality and the Riemann hypothesis," which proves that the set of integers in $\mathbb{N}$ which satisfy Robin's inequality (and, consequently, Lagarias's inequality, as per Lemma 3.1 of Lagarias's paper) has natural/asymptotic density 1 (and, hence, the set of counterexamples will have density 0). This paper is from 2007, after Lagarias's, but I'm sure some form of the result was known by those in the field when Lagarias wrote his paper. It's possible that it could be found in one of the references but wasn't labeled in the paper explicitly.

  • $\begingroup$ Thanks for the references. I'm sure that $5,040$ is not what Lagarias means by "very large," however. $\endgroup$ – saulspatz May 25 at 20:31
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    $\begingroup$ I programmed a C++ "verification" for the inequality. Running for a few minutes gave me a verification for $n$ up into the millions. It's not rigorously vetted, so it shouldn't be taken as such. One may be able to verify the inequality up to, and beyond, similar values of $n$ in a reasonable amount of time using proven-correct code, so it's possible that "very large" means beyond what can be verified in a reasonable amount of time on the hardware available at the time. gitlab.com/mathnerd/lagarias-inequality $\endgroup$ – Brian May 27 at 21:01

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