论文标题
Ramanujan扩展的收敛性，我[Ramanujan云上的多重性]
Convergence of Ramanujan expansions, I [Multiplicativity on Ramanujan clouds]
论文作者
论文摘要
我们称$ r_g（a）：= \ sum_ {q = 1}^{\ infty} g（q）c_q（a）$ of系数$ g：$ n $ \ to $ c的'ramanujan系列'，其中$ c_q（a）$是众所周知的ramanujan sum。 We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define $G$ a 'Ramanujan coefficient' when $R_G(a)$ converges pointwise, in all natural $a$. Then, $R_G:$N$\to$C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed $F:$N$\to$C is美元 我们的第一个主要结果，即$ g $乘法的“有限收敛定理”，以及其他属性等于“ $ r_g $定义良好”，将收敛测试降低到有限的集合，即$ r_g $ w-w-d等于：$ r_g（a）$ r_g（a）$ revenges $ r_g（a）$ a $ a $ a $ a $ n $ n（g）callue n cam nevient y ram（g） 我们的第二个主要结果是“有限的Euler产品显式公式”，用于乘法Ramanujan系数$ G $，将$ f = r_g $作为有限的Euler产品写；因此，$ f $是半杂质功能（后面定义），该产品是$ f $的Selberg分解。特别是，我们有：$ f（a）= r_g（a）$绝对收敛，是有限的（长度取决于$ a $的非零$ p-$ adic估计）。 我们的第三个主要结果称为“乘法Ramanujan云”，研究了$ <f> _m $的重要子集；同样，对于所有乘法$ f $，<f> _m $中的$ g_f \ $ g_f \ in <f> _m $，证明：任何乘法$ f $都具有有限的Ramanujan扩展，并具有多重系数。
We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define $G$ a 'Ramanujan coefficient' when $R_G(a)$ converges pointwise, in all natural $a$. Then, $R_G:$N$\to$C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed $F:$N$\to$C is $<F>:=${$G:N\to C|R_G \; w-d, F=R_G$}. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients $G$ : their $<F>$ subset is called the 'multiplicative Ramanujan cloud', $<F>_M$. Our first main result, the "Finiteness convergence Theorem", for $G$ multiplicative, among other properties equivalent to "$R_G$ well defined", reduces the convergence test to a finite set, i.e., $R_G$ w-d is equivalent to: $R_G(a)$ converges for all $a$ dividing $N(G)\in$N, that we call the "Ramanujan conductor". Our second main result, the "Finite Euler product explicit formula", for multiplicative Ramanujan coefficients $G$, writes $F=R_G$ as a finite Euler product; thus, $F$ is a semi-multiplicative function (following Rearick definition) and this product is the Selberg factorization for $F$. In particular, we have: $F(a)=R_G(a)$ converges absolutely, being finite (of length depending on non-zero $p-$adic valuations of $a$). Our third main result, called the "Multiplicative Ramanujan clouds", studies the important subsets of $<F>_M$; also giving, for all multiplicative $F$, the 'canonical Ramanujan coefficient' $G_F\in <F>_M$, proving: Any multiplicative $F$ has a finite Ramanujan expansion with multiplicative coefficients.
