Related papers: The generalized strong recurrence for non-zero rat…

A set $G \subseteq \omega$ is $n$-generic for a positive integer $n$ if and only if every $\Sigma^0_n$ formula of $G$ is decided by a finite initial segment of $G$ in the sense of Cohen forcing. It is shown here that every $n$-generic set…

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…

This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to,…

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann…

Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…

The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this…

We consider the inverse problem of determining a general semilinear term appearing in nonlinear parabolic equations. For this purpose, we derive a new criterion that allows to prove global recovery of some general class of semilinear terms…

We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which the diffeomorphisms have arbitrarily large finite numbers of…

In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between…

A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in…

We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument.…

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence…

In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…

Cotangent sums play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of…

We prove that, for every rational $d\ne 0,\pm 1$ and every compact set $K\subset\{s\in\mathbb{C}:1/2<\Re(s)<1\}$ with connected complement, any analytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated, uniformly on $K$, by the…