论文标题
关于贝克问题的扩展
On an extension of a question of Baker
论文作者
论文摘要
这是贝克的一个开放问题，即非平凡的dirichlet字符的数字$ l（1，χ）$，$ q $与$ \ mathbb {q} $相比，$ q $是线性独立的。最著名的结果是由于面包师，桦木和围裙，当$ q $共同出现至$φ（q）$时，它肯定了这一点。在本文中，我们将其结果扩展到任何任意的模量家庭。更确切地说，对于一个正整数$ Q $，让$ x_q $表示所有$ l（1，χ）$值的集合为$χ$，而$ q $的非客气dirichlet字符有所不同。然后，对于任何有限的成对共阵发自然数$ q_i，1 \ le i \ le \ el \ el $ a $（q_1 \ cdots q _ {\ ell}，〜φ（q_1）\cdotsφ（q _ {\ ell}）= 1 $在$ \ mathbb {q} $上线性独立。在此过程中，我们还扩展了Okada的结果，内容涉及$ \ Mathbb {q} $的线性独立性，以及由于Mirty-Murty大约$ \ overline {\ mathbb {q}} $ linear offline {\ mathbb {q}} $线性独立性的$ l（1，χ）$ values。最后，我们证明了这种$ \ mathbb {q} $线性独立性的Erdösian函数的$ l $值，具有不同的质量时期$ p_i $，$ 1 \ le i \ le i \ le \ le \ ell $ with $（p_1 \ cdots p _ {\ ell}
It is an open question of Baker whether the numbers $L(1, χ)$ for non-trivial Dirichlet characters $χ$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms this when $q$ is co-prime to $φ(q)$. In this article, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer $q$, let $X_q$ denote the set of all $L(1,χ)$ values as $χ$ varies over non-trivial Dirichlet characters with period $q$. Then for any finite set of pairwise co-prime natural numbers $q_i, 1\le i \le \ell$ with $(q_1 \cdots q_{\ell}, ~φ(q_1)\cdots φ(q_{\ell}))=1$, we show that the set $X_{q_1} \cup \cdots \cup X_{q_l}$ is linearly independent over $\mathbb{Q}$. In the process, we also extend a result of Okada about linear independence of the cotangent values over $\mathbb{Q}$ as well as a result of Murty-Murty about $\overline{\mathbb{Q}}$ linear independence of such $L(1, χ)$ values. Finally, we prove $\mathbb{Q}$ linear independence of such $L$ values of Erdösian functions with distinct prime periods $p_i$ for $1\le i \le \ell$ with $(p_1 \cdots p_{\ell}, ~ φ( p_1\cdots p_{\ell}) )= 1$.
