论文标题
算术进展的最小常见倍数的非平凡上限
Nontrivial upper bounds for the least common multiple of an arithmetic progression
论文作者
论文摘要
在本文中，我们为有限算术进展的连续术语中最不常见的倍数建立了一些非平凡且有效的上限。确切地说，我们证明，对于任何两个codrime积极整数$ a $ a $ a $ a $ a $ a和$ b $，带有$ b \ geq 2 $，我们有\ [\ mathrm {lcm} \ left（a，a+b，dots，a+b，dots，a+nb \ nb \ right） \ frac {a} {b} \ right \ rfloor} ~~~~（\ forall n \ geq b+1），\]，其中$ c_1 = 41.30142 $。如果另外$ b $是质量数量和$ a <b $，那么我们证明，对于任何$ n \ geq b+1 $ $ C_2 = 12.30641 $。最后，我们应用这些不等式来估计由$ m（n）定义的算术函数$ m $：= \ frac {1} {1} {φ（n）} \ sum _ {\ sum {\ ordack {1 \ leq \ leq \ leq \ ell \ ell \ ell \ lel \ leq n \\ ell \ ell \ ell \ ell \ ell \ ell \ ell \ eld \ wedge n = 1}} \ geq 1 $），以及广义chebyshev函数的某些值$θ（x; k，\ ell）$。
In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with $b\geq 2$, we have \[\mathrm{lcm}\left(a,a+b,\dots,a+nb\right) \leq \left(c_1\cdot b\log b\right)^{n+\left\lfloor \frac{a}{b}\right\rfloor}~~~~(\forall n\geq b+1),\] where $c_1=41.30142$. If in addition $b$ is a prime number and $a<b$, then we prove that for any $n\geq b+1$, we have $\mathrm{lcm}\left(a,a+b,\dots,a+nb\right) \leq \left(c_2\cdot b^{\frac{b}{b-1}}\right)^n$, where $c_2=12.30641$. Finally, we apply those inequalities to estimate the arithmetic function $M$ defined by $M(n):=\frac{1}{φ(n)}\sum_{\substack{1\leq\ell\leq n \\ \ell \wedge n=1}}\frac{1}{\ell}$ ($\forall n \geq 1$), as well as some values of the generalized Chebyshev function $θ(x;k,\ell)$.
