Running least common multiples: distinct values and multiplicities

Hypotheses for auxiliary integer recurrences

For ordered primes $p_1<\cdots <p_s$, the interval $(p_1^a,p_1^{a+1})$ contains at most one power of each $p_i>p_1$. The radix including the final $p_1$-jump consequently belongs to

This finite alphabet determines neither the frequency of the radices nor the error terms in the weighted block counts. Estimates for those quantities, or for specified shifted kernels, would be additional arithmetic inputs. The conditional statements about integer recurrences below keep such inputs explicit.

Some auxiliary results explain why bounded integer approximations alone are insufficient. Here are their hypotheses in explicit form.

Let $j(n)\in \{2,3,5\}$ label a sequence of jumps, and suppose each label occurs. A perturbation ${\epsilon }_n$ in an additive abelian group has sum zero on every interval $[a,b)$ whose endpoints have the same label exactly when

for three potential values $u(2),u(3),u(5)$. One direction follows by telescoping. Conversely, the zero-sum assumption makes $\sum _{k<n}{\epsilon }_k$ the same at any two indices with the same label. Assign that common value to $u(j(n))$; every label occurs, so all three values are defined. Consecutive partial sums then give the displayed identity. In the usual terminology, the perturbation is a coboundary of the function $u$ on the labels: it is the difference of successive potential values. This is a telescoping condition, not a bound on the size of the perturbation. It does not by itself force the perturbation to vanish: unequal potential values give nonzero differences when the label changes. If the perturbation also vanishes on a genuine $2\to 3$ transition and on a genuine $2\to 5$ transition, all three potential values agree, so every perturbation is zero.

For an integer sequence satisfying $c_{n+1}=b_n{c}_n-D{m}_n$, compare it with an integer sequence $z_n$. Define the error $e_n=D{z}_n-c_n$ and the perturbation ${\epsilon }_n=b_n{z}_n-z_{n+1}-m_n$. Then

Under the preceding zero-sum and transition hypotheses, ${\epsilon }_n=0$ and hence $e_N=(\prod _{n<N}{b}_n)e_0$. If every $b_n\ge 2$ and the integer $e_0$ is nonzero, then $|e_N|\ge 2^N$. This contradicts even one bound $|e_N|<2^N$, and therefore also excludes a uniform bound for all $N$. The nonzero initial error is essential: the identically zero error causes no contradiction.

A calculation at four consecutive indices gives a related obstruction. If $0<T_i<1$, $z_i\in \mathbb{Z}$ and $|z_i-T_i|<1$, then $z_i$ is $0$ or $1$. For bases $2,3,2,5$, the equalities $2z_0-z_1-1=0$ and $2z_2-z_3-1=0$ force all four integers to be $1$. The sum of the first two perturbations is then

not zero. Thus unit accuracy, the two exact equalities and a zero sum on that block cannot all hold. Unit accuracy alone does not give either exact equality.

No integer approximation to the tails of $S$ is constructed here that satisfies all these conditions. Rationality supplies integer carries of at most quadratic growth, but not the additional zero-sum and transition identities. The carries obtained from $S$ satisfy a weighted block identity instead. Applying the auxiliary results would require that identity to imply their hypotheses, or would require a different construction of the integer approximations.

There is also a conditional criterion for a sum supported on the powers of $2$. Its elementary ingredients can be stated without introducing another formal vocabulary. From $y^{\prime }=by-1$ with $b>0$ and $0<y^{\prime }<1$ one gets $1/b<y<2/b$. For integers $a>0$ and $p\ge 3$, if $u>1/a$ and $v<2/(pa)$, then $u-v>1/(3a)$. Finally, a real number $x$ is irrational if, for every positive integer $d$, there are integers $n,k$ with $0<|nx-k|<1/d$.

To use these observations for a series, the conditional result requires exact identities $y_M=H_{M}x-z_M$, with $H_M,z_M\in \mathbb{Z}$, and, for every positive $d$, two indices with different $H_M$, different $y_M$, and $|y_M-y_{M^{\prime }}|<1/d$. Their difference gives the required nonzero linear form. Equal tail values would give zero and would not suffice; a merely bounded gap would not suffice either. Indeed, for a rational $x$ of denominator $d$, every nonzero difference of this form has size at least $1/d$. No concrete series in this record is shown to satisfy all these clearing and arbitrarily small nonzero-gap conditions.