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Giovanni Coppola

Publications and source records attributed to Giovanni Coppola.

At least 19 recordsLinked to original sources
arXiv2026

A new class of Correlations insisting on Ramanujan expansions

Studying Correlations with Ramanujan Expansions, we arrive to present the new class of, say, Two-Seasons Correlations, abbr. T-S, as a natural set expressing some of the features of, say, H-L-like Correlations; these are the ones that mimic the H-L ($=$Hardy-Littlewood) Correlation with shift $2k$, needed to study $2k-$twin primes following Hardy \& Littlewood Conjecture. After introducing the $3-$Hypotheses Correlations in a previous paper, we add two other, very natural, hypotheses: the fifth is a technical one, simplifying calculations; but the fourth is called 'Parity', since it deals with the parity of natural numbers we play with. In particular, we may build (devoting to this 'our mainstream', here) a single Correlation that satisfies these '5 Axioms', thus a T-S one, that 'entangles two different Correlations' (whence Two-Seasons: T-S) depending on $a$ ($=$ the shift) parity. For $a$ even, our 'Artifact' mimics the H-L Correlation, in fact $a=2k$; but, while H-L Correlation is 'negligible', say, on $a$ odd, our Artifact seems to compare at least in the order of magnitude to H-L Correlation on $a$ even, being linked to another additive problem. Namely, on $a$ even, the Artifact 'counts', say, classic solutions to: $p_1+a=p_2$, in odd primes $p_1,p_2$; while, on $a$ odd, it 'counts' solutions to: $p_1+a=2^j p_2$, again with odd primes $p_1,p_2$ and with $j\in \N$ (satisfying the natural arithmetic constraints). More in general, our T-S Correlations 'entangle' two different Diophantine equations.

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arXiv2025

Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences

In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function $F$, building upon a good decay of its coefficients $G$; this, gains $\log-$powers w.r.t. the trivial bound for $G$ and precisely $\log^{1+\eta}$, where the present parameter $\eta>0$ is real. This property alone has important consequences for all the $F$ having a G.R.E. : mainly, 1) the Eratosthenes Transform $F'$ of our $F$ is infinitesimal (see in Theorem 1); 2) when $\eta>1$ (an enhanced decay) we have uniqueness of $G$ (actually, these are the classic Wintner-Carmichael coefficients, see Th.2); 3) we get a bound for $F$ (in Th.3); 4) an important new class of arithmetic functions $F$ can't have a G.R.E. (see Th.4). These are a generalization of Correlations; which in this way, if are, say, a kind of "far from constants", may not have a G.R.E., whence, a fortiori, can't have the R.E.E.F. This is the Ramanujan Exact Explicit Formula, that we introduced with Prof. Ram Murty. On the Hardy-Ramanujan Journal, I proved that: any "fair" Correlation may be "well-approximated" by a (BH) Correlation. For a (BH) Correlation, we prove here: it has a G.R.E. IF AND ONLY IF it has the (R.E.E.F.). On the other hand, next Counterexample 1 in section 3 is a (BH) Correlation without the (R.E.E.F.); also, it provides here a (BH) Correlation behaving like in Theorem 4.

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arXiv2024

On Ramanujan smooth expansions for a general arithmetic function

We study in detail the Ramanujan smooth expansions, for arithmetic functions; we start with the most general ones, for which we supply the "$P-$local expansions", for arguments with all prime-factors $p\le P$ (namely, $P-$smooth arguments), that are also square-free; then, we supply general results for interesting subsets of arithmetic functions, regarding both their $P-$local and (global) Ramanujan smooth expansions.

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arXiv2024

General elementary methods meeting elementary properties of correlations

This is a kind of survey on properties of correlations of two very general arithmetic functions, mainly from the point of view of Ramanujan expansions. In fact, our previous papers on these links had, as a focus, the "Ramanujan coefficients" of these correlations and the resulting "R.e.e.f.", i.e., Ramanujan exact explicit formula. This holds, actually, under a variety of sufficient conditions, mainly under two conditions of convergence involving correlations' "Eratosthenes Transform", namely what we call "Delange Hypothesis" and "Wintner Assumption" (the former implying the latter). We proved Hardy-Littlewood Conjecture on $2k-$twin primes, in particular, from the first of these two (that implies convergence of classic Ramanujan expansion, whence the R.e.e.f.); more recently, we gave a more general proof, from second condition, entailing the R.e.e.f. again, but this time from another method of summation for Ramanujan expansions, we detailed in "A smooth summation of Ramanujan expansions"; in which paper (see 8th ver.) we also started to give few elementary methods for correlations. Which we deepen here, adding recent, elementary and entirely new ones.

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arXiv2023

A smooth summation of Ramanujan expansions

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.

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arXiv2020

Convergence of Ramanujan expansions, I [Multiplicativity on Ramanujan clouds]

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 $ :=${$G:N\to C|R_G \; w-d, F=R_G$}. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients $G$ : their $ $ subset is called the 'multiplicative Ramanujan cloud', $ _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 $ _M$; also giving, for all multiplicative $F$, the 'canonical Ramanujan coefficient' $G_F\in _M$, proving: Any multiplicative $F$ has a finite Ramanujan expansion with multiplicative coefficients.

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arXiv2020

Multiplicative Ramanujan coefficients of null-function

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 $ $, 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.

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arXiv2019

Finite and infinite Euler products of Ramanujan expansions

All the $F:$N$\rightarrow $C having Ramanujan expansion $F(a)=\sum_{q=1}^{\infty}G(q)c_q(a)$ (here $c_q(a)$ is the Ramanujan sum) pointwise converging in $a\in $N, with $G:$N$\rightarrow $C a multiplicative function, may be factored into two Ramanujan expansions, one of which is a finite Euler product : details in our Main Theorem. This is a general result, with unexpected and useful consequences, esp., for the Ramanujan expansion of null-function, say 0. The Main Theorem doesn't require other analytic assumptions, as pointwise convergence suffices; this depends on a general property of Euler $p-$factors (the factors in Euler products) for the general term $G(q)c_q(a)$; namely, once fixed $a\in $N (and prime $p$), the $p-$Euler factor of $G(q)c_q(a)$ (involving all $p-$powers) has a finite number of non-vanishing terms (depending on $a$) : see our Main Lemma. In case we also add some other hypotheses, like the absolute convergence, we get more classical Euler products: the infinite ones. For the Ramanujan expansion of 0 this strong hypothesis makes the class of 0 Ramanujan coefficients much smaller; also excluding Ramanujan's $G(q)=1/q$ and Hardy's $G(q)=1/φ(q)$ ($φ$ is Euler's totient function). Our Main Theorem, instead, suffices to classify all the multiplicative Ramanujan coefficients for 0, so we also announce and (partially) prove this Classification.

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arXiv2019

A smooth shift approach for a Ramanujan expansion

All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast μ$ supported in $Q-$smooth numbers, have a kind of unique Ramanujan expansion; also, these Ramanujan coefficients decay very well to $0$ and have two explicit expressions (in the style of Carmichael and Wintner). This general result, then, is applied to the shift-Ramanujan expansions, i.e., the expansions for correlations with respect to the shift, whence the title.

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arXiv2018

A map of Ramanujan expansions

A map is a panorama in small scale. In this half-survey, half-research paper we give general results on Ramanujan expansions. We don't include the ocean of results from the literature on the two classes (see Schwarz-Spilker Book, also Lucht's survey for these) of additive and multiplicative functions while we include, say, the two new (not simply connected) lands of finite Ramanujan expansions (see my paper, with Murty & Saha) and of shift-Ramanujan expansions (see my subsequent paper, with Murty) .

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arXiv2017

An elementary property of correlations

We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan expansion is with respect to a>0). We prove that, assuming Delange Hypothesis (DH) for the expansion, we get say Ramnujan exact explicit formula (R.e.e.f.). A noteworthy case, of course, is $f=g=Λ$, the von Mangoldt function, so $C_{Λ,Λ}(N,2k)$, for natural $k$, regards $2k-$twin primes; assuming $(DH)$ for them, we get (from corresponding R.e.e.f.) the proof, easily, of Hardy-Littlewood Conjecture for them.

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arXiv2017

Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, II

We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formulæ. Here, the integer $h\ge 0$ is called, as usual, the {\it shift} of the convolution sum. We deepen the study of finite Ramanujan expansions of general $f,g$ for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of $f$ and $g$, namely in terms of its shift $h$ and we compare this \lq \lq shifted Ramanujan expansion\rq \rq, with our previous finite expansions in terms of the $f$ and $g$ arguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formulæ.

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arXiv2016

Sieve functions in arithmetic bands, II

An arithmetic function $f$ is called a $sieve$ $function$ of $range$ $Q$ if its Eratosthenes transform $g=f\astμ$ has support in $[1,Q]$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). We continue our study of the distribution of such functions over short $arithmetic$ $bands$, $n\equiv ar+b\, (\bmod\,q)$, with $1\le a\le H=o(N)$ and $r,b$ integers such that g.c.d.$(r,q)=1$. In particular, we discuss the optimality of some results.

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arXiv2016

Averages of short correlations: a note

We give a completely elementary study for averages of short correlations of so-called sieve functions (a pretty general class of arithmetic functions).

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arXiv2016

Sieve functions in arithmetic bands

An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\astμ$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the distribution of $f$ over short {\it arithmetic bands} $\cup_{1\le a\le H}\{n\in(N,2N]: n\equiv a\, (\bmod\,q)\}$, with $H=o(N)$, and give applications to both the correlations and to the so-called weighted Selberg integrals of $f$, on which we have concentrated our recent research.

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arXiv2016

Symmetry and short interval mean-squares

The weighted Selberg integral is a discrete mean-square, that is a generalization of the classical Selberg integral of primes to an arithmetic function $f$, whose values in a short interval are suitably attached to a weight function. We give conditions on $f$ and select a particular class of weights, in order to investigate non-trivial bounds of weighted Selberg integrals of both $f$ and $f\astμ$. In particular, we discuss the cases of the symmetry integral and the modified Selberg integral, the latter involving the Cesaro weight. We also prove some side results when $f$ is a divisor function.

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