Running least common multiples: distinct values and multiplicities

Two sums from running least common multiples

Let $P$ be a finite set of at least two primes, and let ${\mathcal{S}}_P$ consist of $1$ and all positive integers whose prime factors belong to $P$. For $x\ge 1$, put

Every integer in this finite set divides ${\mathrm{H}}_P(x)$, and the set contains the largest power of each prime in $P$ below or equal to $x$. Their product therefore equals the least common multiple: $\mathrm{L}(x)={\mathrm{H}}_P(x)$. The value changes only at a positive prime power. Summing its distinct reciprocals gives

We begin with an irrationality result for three primes.

The proof in Section 2 gives the primes three different roles. Powers of $2$ delimit the blocks; powers of $5$ occur often enough to separate the five affine tail images; powers of $3$ have an irrational block frequency. Separation makes each tail determine the next block and tail. Under rationality there are only finitely many tails, so repetition would force an impossible periodic block sequence.

Erdős asserted the general conclusion for every finite $P$ with $|P|\ge 2$ in his letter of 1 January 1973, without printing a proof [3]. The theorem above is a special case of that assertion. The general argument in Section 5.1 compares nearby tails and finds an isolated interchange of jumps on a torus. We retain it as an ordinary argument awaiting formalisation, with its intermediate claims stated as remarks. The specialised five-map proof is independent of that argument.

The question in Erdős Problem #269 retains multiplicities. If $a_1<a_2<\cdots$ enumerates ${\mathcal{S}}_P$, it asks about

[1][2]; the numbering is Bloom’s [4]. For $P=\{2,3,5\}$, the integers $5$ and $6$ both have running LCM $60$. They contribute $2/60$ to ${\mathcal{R}}_P$ and $1/60$ to ${\mathcal{D}}_P$. Thus Theorem 1.1 concerns the sum over distinct values, whereas the problem concerns the sum with multiplicity. The arguments in this paper leave ${\mathcal{R}}_P$ unresolved when $P$ is finite and $|P|\ge 3$.

For two primes the conclusion is stronger.

Fan’s post of 26 June 2026 gave the finite-prime running-LCM identity, the factorisation of ${\mathcal{R}}_{\{p,q\}}$, and the Hecke–Mahler reduction with its transcendence conclusion [17]. We reproduce that calculation in Section 4. Both values are nonconstant polynomials over $\mathbb{Q}$ in one boundary sum, whose transcendence follows from Loxton and van der Poorten [16], in the formulation of Bugeaud and Laurent [15]. No priority is claimed for this deduction.

Retaining multiplicities preserves the bases but changes the numerators. The repeated tails satisfy $X_{a+1}=b_a{X}_a-m_a$, with $b_a\in \{2,6,10,30\}$, while both $m_a$ and $X_a$ grow quadratically. The preceding finite-tail repetition argument no longer applies. Rationality would still make every sufficiently late $B{X}_a$ integral, where $B$ is the denominator part coprime to $30$. A finite segment excludes such an integer tail when its least positive residue exceeds the quadratic bound. Irrationality is equivalent to finding these segments for every $B$ beyond every prescribed onset. The finite computations do not establish either unbounded quantifier.

The five-map proof comes first. Sections 3 and 4 then supply the running LCM geometry and two-prime formulae. For general distinct sums, Section 5.1 uses comparison of two tails rather than recovery from one; the finite-state, subseries and linear-flow extensions follow as ordinary arguments pending formalisation. Section 7 separates exact kernel rank from approximation. For the repeated target, Sections 8–10 derive the coefficients, bounds and residue criterion; the remaining sections distinguish finite certificates from missing arithmetic inputs. Section 12.5 records formal coverage. The short paper concentrates on the five-map theorem and the repeated-tail criterion.

We use $\mathbb{N}=\{0,1,2,\dots \}$ for exponents and braces for fractional parts. In the finite-geometry section and in the kernel calculation, $P=\{p,q,r\}$ consists of three pairwise distinct primes, $\mathrm{H}={\mathrm{H}}_P$, and

The other sections specify their prime sets separately. For $\{2,3,5\}$, ${\mathcal{S}}_P$ consists of the usual $5$-smooth integers. This fixed support differs from a varying smoothness bound in $\mathrm{\Psi }(x,y)$ [11]. Tijdeman and Meijer [28] and the two-prime study of Languasco, Luca, Moree and Togbé [29] give the relevant lattice-counting context; the estimates below use no smooth-number density asymptotic or gap theorem. Kovač and Tao [13] study other unit-fraction series by elementary methods, without providing an input to these proofs.

Keywords. irrationality; transcendence; least common multiple; smooth numbers; Cantor series; separated rank. MSC 2020. 11J72 (primary); 11A05, 11N25, 68V20 (secondary).