Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple of natural numbers $\geq 1$ with the property that the entrys are pairwise coprime. We define $l : M(n) \rightarrow \mathbb{N}_0$ by $$l(m_1,\ldots,m_n) = \max_{(x_1,\ldots,x_n) \in G(m_1,\ldots,m_n)} \min \{ k \in \mathbb{N}_0 | x_i+k \neq 0 \forall 1 \leq i \leq n \},$$ hence $l(m_1,\ldots,m_n)$ may be regarded as the maximal distance of the elements of $G$ from the elements which have no identity in their entrys. It is clear, that $l(m_1,\ldots,m_n)$ is always a natural number.

The question is now: Does there exist a real number $r$, such that $$r \cdot n \cdot \ln (n) \geq \sup_{(m_1,\ldots,m_n) \in M(n)} l(m_1,\ldots,m_n)$$ holds for all $n >> 0$ and is it possible to take $r=2$ ?

  • 2
    $\begingroup$ Could you please clarify the definition of $l(m_i,...)$, there seems to be a typo. $\endgroup$ Mar 21 '11 at 17:10
  • $\begingroup$ Hi, I edited my question. Hopefully, the typos is clear now. $\endgroup$
    – tobias
    Mar 21 '11 at 17:28
  • $\begingroup$ I am afraid not. What is the role of k in the max min? $\endgroup$
    – user9072
    Mar 21 '11 at 17:32
  • $\begingroup$ What do you mean by the role of k ? It should be an absolute constant. $\endgroup$
    – tobias
    Mar 21 '11 at 17:36
  • $\begingroup$ In the displayed formula where a max min appears you say k in N_0 but this k appears nowhere; except later a different(?) k appears. $\endgroup$
    – user9072
    Mar 21 '11 at 17:41

We have $l(m_1,\cdots,m_n)$ is at least equal to the maximum length of a sequence of consecutive integers each of which is divisible by some $m_i$. Let's suppose for a moment that $m_i$ is the $i$th prime. In "On the problem of Jacobsthal", Iwaniec proves that this number is $\ll (n\log n)^2$, while your bound would be much stronger, and if you see the lower bound I mention in this equvalent question ($\sim\frac{n(\log n)^2 \log \log \log n}{(\log \log n)^2}$), it is actually false. In any case the question reduces to finding the best upper bound on Jacobsthal's function (In the comments of the question I linked to above there is an argument for why it suffices to reduce to the $m_i$ prime case).

  • $\begingroup$ Thank you very much Gjergji. But naturally the most important example of this is, as you already said, when $m_i$ is the i-th prime. If we would regard $l(m_1,...,m_n)$ for this case, could we give a good bound for $l(m_1,...,m_n)$ ? $\endgroup$
    – tobias
    Mar 21 '11 at 22:52
  • $\begingroup$ When $m_i$ are prime then $l(m_i)$ is equal to Jacobsthal's function. Except for Iwaniec's bound I mentioned here, and Erdos' bound in the other question, I am not aware of further improvement on lower or upper bounds. $\endgroup$ Mar 21 '11 at 22:58
  • $\begingroup$ I never heard of the Jacobsthal function before. Thas helps me quite a lot. Thank you, Gjergji. $\endgroup$
    – tobias
    Mar 22 '11 at 6:35

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