Erdős Problem 290

References:

• erdosproblems.com/290
• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980), p.34.
• [vD24] van Doorn, W., On the non-monotonicity of the denominator of generalized harmonic sums. arXiv:2411.03073 (2024).

noncomputable def Erdos290.harmonicDen (a b : ℕ) : ℕ

The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$.

Equations

• Erdos290.harmonicDen a b = (∑ n ∈ Finset.Icc a b, 1 / ↑n).den

theorem Erdos290.erdos_290 : True ↔ ∀ (a : ℕ), 1 ≤ a → ∃ (b : ℕ), a < b ∧ harmonicDen a (b + 1) < harmonicDen a b

Let $a\geq 1$. Must there exist some $b>a$ such that [\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and } \sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},] with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?

This was resolved in the affirmative by van Doorn [vD24], who proved $b=b(a)$ always exists, and in fact $b(a) \ll a$. Indeed, if $a\in (3^k,3^{k+1}]$ then one can take $b=2\cdot 3^{k+1}-1$. van Doorn also proves that $b(a)>a+(1/2-o(1))\log a$, and considers various generalisations of the original problem.