Related papers: Averages of short correlations: a note
These informal notes are concerned with sums and averages in various situations in analysis.
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…
For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.
We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.
An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\ast\mu$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the…
This short note contains elementary evaluations of some Euler sums.
Properties of weighted averages are studied for the general case that the individual measurements are subject to hidden correlations and have asymmetric statistical as well as systematic errors. Explicit expressions are derived for an…
In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable…
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…
We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and…
Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If…
We prove some basic properties of quasinearly subharmonic functions and quasinearly subharmonic functions in the narrow sense.
This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the…
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…
We apply Bayesian statistics to the estimation of correlation functions. We give the probability distributions of auto- and cross-correlations as functions of the data. Our procedure uses the measured data optimally and informs about the…
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear,…
We obtain the general k-correlations for a short divisor sum related to primes.
These notes give an elementary approach to parts of the theory of standard Borel and analytic spaces.
A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply…