A smooth summation of Ramanujan expansions

TL;DR

This paper introduces the concept of smooth summation for Ramanujan series, demonstrating convergence under certain assumptions and exploring implications for prime number conjectures.

Contribution

It defines smooth Ramanujan series via prime factor restrictions and proves their convergence under Wintner assumptions, distinguishing them from traditional Ramanujan series.

Findings

Smooth Ramanujan series converge under Wintner assumptions.

Smooth series differ from classical Ramanujan series.

Application to correlations and twin primes conjecture.

Abstract

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.