arXiv subjects
Publications and source records attributed to Giovanni Coppola.
We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.
For two arithmetical functions $f$ and $g$ with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum $\sum_{n \le N} f(n) g(n+h)$ under some suitable conditions (see http://arxiv.org/abs/1506.01945). In this follow up article we improve these results with a weakened hypothesis which is in some sense minimal.
An arithmetic function $f$ is called a sieve function of range $Q$, if it is the convolution product of the constantly $1$ function and $g$ such that $g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon>0$, for $q\leq Q$, and $g(q)=0$ for $q>Q$. Here we establish a new result on the autocorrelation of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain non-trivial asymptotic formulæ that are actually unreachable by the standard approach of the distribution of $f$ in the arithmetic progressions. Moreover, we apply our asymptotic formulæ to obtain new bounds for the so-called Selberg integral and symmetry integral of $f$, which are basic tools for the study of the distribution of $f$ in short intervals.
For a real-valued and essentially bounded arithmetic function $f$, i.e., $f(n)\ll_{\varepsilon}\!n^{\varepsilon},\,\forall\varepsilon\!>\!0$, we \enspace give some optimal links between non-trivial bounds for the sums $\sum_{h\le H}\sum_{N<n\le 2N}f(n)f(n-h)$, $\sum_{N<x\le 2N} \big| \sum_{x<n\le x+H}f(n)\big|^2$ and $\sum_{N<n\le 2N} \big| \sum_{0\le |n-x|\le H}\big(1-{|n-x|\over H}\big)f(n)\big|^2$, with $H=o(N)$ as $N\to\infty$.
First we generalize a famous lemma of Gallagher on the mean square estimate for exponential sums by plugging a weight in the right hand side of Gallagher's original inequality. Then we apply it in the special case of the Cesaro weight, in order to establish some results mainly concerning the classical Dirichlet polynomials and the Selberg integrals of an arithmetic function $f$, that are tools for studying the distribution of $f$ in short intervals. Furthermore, we describe the smoothing process via self-convolutions of a weight, that is involved into our Gallagher type inequalities, and compare it with the analogous process via the so-called correlations. Finally, we discuss a comparison argument in view of refinements on the Gallagher weighted inequalities according to different instances of the weight.
First we prove a modified version of the famous Lemma on the mean square estimate for exponential sums, by plugging the Cesaro weights in the right hand side of Gallagher's inequality. Then we apply it, in order to establish a mean value estimate for the Dirichlet polynomials.
The present paper is a dissertation on the possible consequences of a conjectural bound for the so-called \thinspace modified Selberg integral of the divisor function $d_3$, i.e. a discrete version of the classical Selberg integral, where $d_3(n)=\sum_{abc=n}1$ is attached to the Cesaro weight $1-|n-x|/H$ in the short interval $|n-x|\le H$. Mainly, an immediate consequence is a non-trivial bound for the Selberg integral of $d_3$, improving recent results of Ivić based on the standard approach through the moments of the Riemann zeta function on the critical line. We proceed instead with elementary arguments, by first applying the "elementary Dispersion Method" in order to establish a link between "weighted Selberg integrals" \thinspace of any arithmetic function $f$ and averages of correlations of $f$ in short intervals. Moreover, we provide a conditional generalization of our results to the analogous problem on the divisor function $d_k$ for any $k\ge 3$. Further, some remarkable consequences on the $2k-$th moments of the Riemann zeta function are discussed. Finally, we also discuss the essential properties that a general function $f$ should satisfy so that the estimation of its Selberg integrals could be approachable by our method.
We give a new non-trivial upper bound for the Selberg integral of the three-divisor function $d_3(n)$. Our method applies our recent conjecture together with Laporta, for the modified Selberg integral of $d_3(n)$, and a kind of modified Gallagher Lemma for the exponential sums.
We prove a non-trivial result for the,say,modified Selberg integral $\modSel_3(N,h)$, of the divisor function $d_3(n):= \sum_{a}\sum_{b}\sum_{c, abc=n}1$; this integral is a slight modification of the corresponding Selberg integral, that gives the expected value of the function in short intervals. We get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a byproduct, we obtain indications on the way in which it may be proved the weak sixth moment of the Riemann zeta function.(This was OLD abstract)
We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_Λ(N,h)$, of the von Mangoldt function $Λ(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0 otherwise). We get $I_Λ(N,h)\ll NhL^5+Nh^{21/20}L^2$, with $L:=\log N$; as a Corollary, we bound non-trivially the Selberg integral of the primes, i.e. the mean-square of $\sum_{x<n\le x+h}Λ(n)-h$, over $x\in [N,2N]$, to get the "Prime Number Theorem in almost all short intervals" of (log-powers!) length $h\ge L^{11/2+ε}$. We trust here in the improvement of the exponent, say $c<11/2$.
We prove a kind of "almost all symmetry" result for the Liouville function $λ(n):=(-1)^{Ω(n)}$, giving non-trivial bounds for its "symmetry integral", say $I_λ(N,h)$ : we get $I_λ(N,h)\ll NhL^3+Nh^{21/20}$, with $L:=\log N$. We also give similar results for other related arithmetic functions, like the Möbius function $μ(n)$ ($=λ(n)$ on square-free $n$).
We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N 0$ we have $g(n)\ll_ε n^ε$, and supported in $[1,Q]$, with $Q\ll N$ (so, the exponent of $Q$ relative to $N$, say the level $λ:=(\log Q)/(\log N)$ is $λ< 1$), where the symmetry sum weights the $f-$values in (almost all, i.e. all but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$ (with positive/negative sign at the right/left of $x$), with mild restrictions on $h$ (say, $h\to \infty$ and $h=o(\sqrt N)$, as $N\to \infty$).
We give a kind of \lq \lq approximate majorant principle\rq \rq \thinspace result for the \lq \lq modified Selberg integral\rq \rq, say $\modSel_f(N,h)$, of essentially bounded $f:\N \rightarrow \R$ (i.e., bounded by arbitrary small powers); i.e., we get an upper bound, in terms of the modified Selberg integral of a related function $F$ (with $|f\ast μ|\ll F\ast μ$, in the supports intersection), getting a \lq \lq square-root cancellation\rq \rq \thinspace for the error-terms. Here $\modSel_f(N,h)$ is the mean-square (in $N<x\le 2N$) of the \lq \lq averaged short sum\rq \rq \thinspace of, say, $f:=g\ast \1$, minus its expected value; i.e., ${1\over h}\sum_{m\le h}\sum_{0\le |n-x|<m}f(n)-M_f(x,h)$, with expected value $M_f(x,h)$ (say, $\approx h\sum_{d\le x}g(d)/d$); so, this mean-square weights, on average, the $f-$values in (almost all, i.e. all, but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$, with mild restrictions on \thinspace $h$ \thinspace (say, \thinspace $h\to \infty$ \thinspace and \thinspace $h=o(N)$, when \thinspace $N\to \infty$).
We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some arithmetic functions $f:\N \rightarrow \R$; we obtain from lower bounds of $I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \lq \lq Selberg integral\rq \rq \thinspace of $f$, say $J_f$ (both these integrals give informations about $f$ in almost all the short intervals $[x-h,x+h]$, when $N\le x\le 2N$). In particular, when \thinspace $f=d_k$, the divisor function (having Dirichlet series \thinspace $ζ^k$, with \thinspace $ζ$ \thinspace the Riemann zeta function), where $k\ge 3$ is integer, we give lower bounds for the Selberg integrals, say \thinspace $J_k=J_{d_k}$, of the \thinspace $d_k$. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give $I_f$ lower bounds.
We deeply appreciate the papers of Ivić on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the former paper, we apply an elementary approach (based on arithmetic averages) in order to get information on the reverse link to the second work.
We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.
We study the arithmetic (real) function, with f 'essentially bounded'. In particular, we obtain non-trivial bounds, through f 'correlations', for the 'Selberg integral' and the 'symmetry integral' of f in almost all short intervals $[x-h,x+h]$, $N\le x\le 2N$, beyond the 'classical' level, up to a very high level of distribution (for $h$ not too small). This time we go beyond Large Sieve inequality. Precisely, our method applies Weil bound for Kloosterman sums.
We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the "classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary.