Related papers: Cumulative Riemann sums, distribution functions, a…

We generalize the property that Riemann sums of a continuous function corresponding to equidistant subdivision of an interval converge to the integral of that function, and we give some applications of this generalization.

Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1/\log T) - N(t)$, a count of zeros of the Riemann zeta function in a box of height $1/\log T$. Conditioned on the Riemann hypothesis, we…

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$ related to the mean value of a generalized moment of Riemann's zeta function. Analytically, we predict finite, but discontinuous values and verify the prediction…

We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the…

For non-decreasing real functions $f$ and $g$, we consider the functional $ T(f,g ; I,J)=\int_{I} f(x)\di g(x) + \int_J g(x)\di f(x)$, where $I$ and $J$ are intervals with $J\subseteq I$. In particular case with $I=[a,t]$, $J=[a,s]$, $s\leq…

This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for…

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…

A sharp, distribution free, non-asymptotic result is proved for the concentration of a random function around the mean function, when the randomization is generated by a finite sequence of independent data and the random functions satisfy…

For stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of…

Fujii investigated the uniform distribution of various sequences associated with the non-trivial zeros of the Riemann zeta function by evaluating certain exponential sums over these zeros. In this paper, we present analogous results for a…

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial…

The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient…

An inequality, which combines the concept of completely monotone functions with the theory of divided differences, is proposed. It is a straightforward generalization of a result, recently introduced by two of the present authors.

We give a simple proof of a well-known theorem of G\'al and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in G\'al's theorem,…

The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the…

In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for…

Let $\sigma,t\in{\mathbb{R}}$, $s=\sigma+\mathrm{{i}}t$, $\Gamma (s)$ be the Gamma function, $\zeta(s)$ be the Riemann zeta function and $\xi(s):=s(s-1)\pi ^{-s/2}\Gamma(s/2)\zeta(s)$ be the complete Riemann zeta function. We show that…

The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with…

Riemann sums, a classical method for approximating the definite integral of a function, have been extensively studied in the past. However, their monotonic properties, while still of great importance, particularly in approximation theory…

A formal sum $\sum_n f(S_n)$ may be seen as the integral $\int f dN$ with respect to random point process $N(A)=|\{n:S_n\in A\}|$. We study its convergence beyond the well known context of Lebesgue integrable functions, admitting…