论文标题
Ramanujan扩展的平稳总结
A smooth summation of Ramanujan expansions
论文作者
论文摘要
我们研究了ramanujan系列$ \ sum_ {q = 1}^{\ infty} g（q）c_q（a）$，其中$ c_q（a）$是众所周知的ramanujan sum和complexpe drumend $ g（q）$，as $ q \ as $ q \ in $ n，是$ n $ n，是Ramanujan系数；当然，我们的意思是暗示，该系列在所有天然$ a $中均匀地收敛，因为它的部分总和$ \ sum_ {q \ le q} g（q）c_q（a）$ c_q（a）$在$ q \ to \ infty $时在c中收敛。由我们最近对Ramanujan系列的无限和有限欧拉产品的研究进行的激励，在该系列中我们假设$ g $乘法，我们研究了一种（部分）平滑的总和。这些是$ \ sum_ {q \ in（p）} g（q）c_q（a）$，其中indices $ q $ in $（p）$表示所有质量因子$ p $的$ p $ $ q $均为$ p $（固定）;然后，我们将限制超过$ p \ to \ infty $。请注意，这种部分总和超过$ p- $平滑的数字（即，在$（p）$中，请参见上面）构成了无限的和$ \ forall p \ in $ p固定的$ p \ in $ p固定；但是，我们的汇总包含$ c_q（a）$，具有垂直限制，即，在$ n的$ n中，它得到了支持的$ q \ q \ n $ q \ n $ p- $ adic的估值，resp。，$ q $和$ a $，即$ v_p（q）$，resp。，resp。 p \ le p $（$ p $固定）。换句话说，$ \ forall g：$ n $ \ rightarrow $ c，在此处，$ \ sum_ {q \ in（p）} g（q）c_q（a）$是有限的总和，$ \ forall a \ in $ n，$ n，$ \ in $ p \ in $ p in $ p in $ p \ in $ p fordect：我们将prect $ \ sum_ sum_________________________且仅当$ \存在\ lim_p \ sum_ {q \ in（p）} g（q）c_q（a）\ in $ c，$ \ in $ n。请注意一个非常重要的属性：流畅的Ramanujan系列和Ramanujan系列不必相同。我们证明：平滑的Ramanujan系列在Wintner假设下收敛。 （对于Ramanujan系列而言，这不一定是正确的。）我们将其应用于相关性和Hardy-Littlewood“ 2k- $ $ twin Primes”的猜想。
We studied Ramanujan series $\sum_{q=1}^{\infty}G(q)c_q(a)$, where $c_q(a)$ is the well-known Ramanujan sum and the complex numbers $G(q)$, as $q\in$N, are the Ramanujan coefficients; of course, we mean, implicitly, that the series converges pointwise, in all natural $a$, as its partial sums $\sum_{q\le Q}G(q)c_q(a)$ converge in C, when $Q\to \infty$. Motivated by our recent study of infinite and finite Euler products for the Ramanujan series, in which we assumed $G$ multiplicative, we look at a kind of (partial) smooth summations. These are $\sum_{q\in (P)}G(q)c_q(a)$, where the indices $q$ in $(P)$ means that all prime factors $p$ of $q$ are up to $P$ (fixed); then, we pass to the limit over $P\to \infty$. Notice that this kind of partial sums over $P-$smooth numbers (i.e., in $(P)$, see the above) make up an infinite sum, themselves, $\forall P\in$P fixed, in general; however, our summands contain $c_q(a)$, that has a vertical limit, i.e. it's supported over indices $q\in$N for which the $p-$adic valuations of, resp., $q$ and $a$, namely $v_p(q)$, resp., $v_p(a)$ satisfy $v_p(q)\le v_p(a)+1$ and this is true $\forall p\le P$ ($P$'s fixed). In other words, $\forall G:$N $\rightarrow$ C, here, $\sum_{q\in (P)}G(q)c_q(a)$ is a finite sum, $\forall a\in $N, $\forall P\in $P fixed: we will call $\sum_{q=1}^{\infty}G(q)c_q(a)$ a 'smooth Ramanujan series' if and only if $\exists \lim_P \sum_{q\in (P)}G(q)c_q(a)\in $C, $\forall a\in $N. Notice a very important property : smooth Ramanujan series and Ramanujan series need not to be the same. We prove : smooth Ramanujan series converge under Wintner Assumption. (This is not necessarily true for Ramanujan series.) We apply this to correlations and to the Hardy--Littlewood "$2k-$Twin Primes" Conjecture.
