论文标题
无功能的乘法拉曼努扬系数
Multiplicative Ramanujan coefficients of null-function
论文作者
论文摘要
null功能$ 0（a）：= 0 $，$ \ in $ n中的a ramanujan扩展：$ 0（a）= \ sum_ {q = 1}^{\ infty}（\ infty}（1/q）c_q（a） $ 0（a）= \ sum_ {q = 1}^{\ infty}（1/φ（q））c_q（a）$，由Hardy给出（$φ：= $ euler的基本函数）。两者在N。A$ G：$ n $ \ rightarrow $ c中均匀地融合（不是绝对），称为Ramanujan系数Abbrev。 r.c.，iff（if and仅IF）$ \ sum_ {q = 1}^{\ infty} g（q）c_q（a）$在$ n in $ n in $ n in $ n;给定$ f：$ n $ \ rightarrow $ c，我们称$ <f> $，其r.c.s的集合，$ f $的ramanujan云。我们在Arxiv中的主要定理：1910.14640，用于Ramanujan扩展和有限的Euler产品，这意味着对乘以$ 0 $的乘法Ramanujan系数进行了完整的分类。 Ramanujan的$ g_r（q）：= 1/q $是普通算术函数$ g $，即用$ g（p）\ neq 1 $乘坐所有Primes $ p $的乘法;虽然Hardy的$ g_h（q）：= 1/φ（q）$是一个零星的$ g $，即乘法，$ g（p）= 1 $对于有限的$ p $，但没有$ p $ a $ g（p^k）= 1 $ in Integers $ k \ ge 0 $ k \ ge ge 0 $ $ k \ ge 0 $ g_ $ g_ $ g_ $ $ g_ $ $ $ $ $ $ $ $ $ $ $ p = fff $ $ p = fff $ $ $ p） $ g：$ n $ \ rightarrow $ c乘法，使得所有$ g（p^k）= 1 $，所有$ k \ ge 0 $的prime $ p $被定义为异国情调。该定义完成了乘法$ 0- $ RAMANUJAN系数的案例。异国情调的是$ 0- $ Cloud（即$ <0> $）中的一种新现象：外来的Ramanujan系数代表$ 0 $仅带有融合假设。除了融合假说外，不奇特的$ \ sum_ {q = 1}^{\ infty} g（q）g（q）= 0 $（q）= 0 $ for normal $ g \ in <0> $ in <0> $，而sporadic $ g \ in <0> $ in <0> $ in <0> $ in <0> $ nied $ sum sum \ sum _ sum _ {q，q，q，q，q，p（g）q（q）= 1} q（q）（q）（q） $ p（g）：= $所有$ p $制造$ g（p）= 1 $的产品。我们给出了许多R.C.S $ G \ in <0> $的例子;我们还证明，绝对收敛的<0> $中唯一的$ g \是异国情调的$。实际上，这些概括到弱异国情调，不一定是乘法。
The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/φ(q))c_q(a)$, given by Hardy ($φ:=$ Euler's totient function). Both converge pointwise (not absolutely) in N. A $G:$N $\rightarrow $C is called a Ramanujan coefficient, abbrev. R.c., iff (if and only if) $\sum_{q=1}^{\infty}G(q)c_q(a)$ converges in all $a\in $N; given $F:$N $\rightarrow $C, we call $<F>$, the set of its R.c.s, the Ramanujan cloud of $F$. Our Main Theorem in arxiv:1910.14640, for Ramanujan expansions and finite Euler products, implies a complete Classification for multiplicative Ramanujan coefficients of $0$. Ramanujan's $G_R(q):=1/q$ is a normal arithmetic function $G$, i.e., multiplicative with $G(p)\neq 1$ on all primes $p$; while Hardy's $G_H(q):=1/φ(q)$ is a sporadic $G$, namely multiplicative, $G(p)=1$ for a finite set of $p$, but there's no $p$ with $G(p^K)=1$ on all integers $K\ge 0$ (Hardy's has $G_H(p)=1$ iff $p=2$). The $G:$N $\rightarrow $C multiplicative, such that there's at least a prime $p$ with $G(p^K)=1$, on all $K\ge 0$, are defined to be exotic. This definition completes the cases for multiplicative $0-$Ramanujan coefficients. The exotic ones are a kind of new phenomenon in the $0-$cloud (i.e., $<0>$): exotic Ramanujan coefficients represent $0$ only with a convergence hypothesis. The not exotic, apart from the convergence hypothesis, require in addition $\sum_{q=1}^{\infty}G(q)μ(q)=0$ for normal $G\in <0>$, while sporadic $G\in <0>$ need $\sum_{(q,P(G))=1}G(q)μ(q)=0$, $P(G):=$product of all $p$ making $G(p)=1$. We give many examples of R.c.s $G\in <0>$; we also prove that the only $G\in <0>$ with absolute convergence are the exotic ones; actually, these generalize to the weakly exotic, not necessarily multiplicative.
