2020/12/21 by Giovanni Coppola, Coppola, Giovanni
Mathematics · #11N05 #11N37 #11P32 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2012.11231
openalex publication_date 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We studied Ramanujan series ∑q=1∞G(q)cq(a), where cq(a) is the well-known Ramanujan sum and the complex numbers G(q), as q∈N, are the Ramanujan coefficients; of course, we mean, implicitly, that the series converges pointwise, in all natural a, as its partial sums ∑q≤ QG(q)cq(a) converge in C, when Q→ ∞. 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 ∑q∈ (P)G(q)cq(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→ ∞. Notice that this kind of partial sums over P-smooth numbers (i.e., in (P), see the above) make up an infinite sum, themselves, ∀ P∈P fixed, in general; however, our summands contain cq(a), that has a vertical limit, i.e. it's supported over indices q∈N for which the p-adic valuations of, resp., q and a, namely vp(q), resp., vp(a) satisfy vp(q)≤ vp(a)+1 and this is true ∀ p≤ P (P's fixed). In other words, ∀ G:N → C, here, ∑q∈ (P)G(q)cq(a) is a finite sum, ∀ a∈ N, ∀ P∈ P fixed: we will call ∑q=1∞G(q)cq(a) a 'smooth Ramanujan series' if and only if ∃ limP ∑q∈ (P)G(q)cq(a)∈ C, ∀ a∈ 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.