Related papers: On Riemann's Theorem About Conditionally Convergen…
The aim of this note is to remove an implausible assumption in Moser's theorem \cite{JM} to establish our new theorem 1 which gives a lower estimate for the sum $p+c^2\rho$ on Riemann hypothesis. Corollary 1 gives a rather plausible…
In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient…
Let $x \in \mathbb{R}$ be arbitrary and consider the `greedy' approximation of $x$ by signed harmonic sums: given $a_n = \sum_{k \leq n} \varepsilon_k/k$ with $\varepsilon_k \in \left\{-1,1\right\}$, we set $\varepsilon_{n+1} = 1$ if $a_n…
The addition relation for the Riemann theta functions and for its limits, which lead to the appearance of exponential functions in soliton type equations is discussed. The presented form of addition property resolves itself to the…
We generalize Warnaar's elliptic extension of a Macdonald multiparameter summation formula to Riemann surfaces of arbitrary genus.
We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…
This paper shows the equivalence of the Riemann hypothesis to an sequence of elementary inequalities involving the harmonic numbers H_n, the sum of the reciprocals of the integers from 1 to n. It is a modification of a criterion due to Guy…
This study involves definitions for multiple-counting regular and summation sequences of rho. My paper introduces and proves recurrent relationships for multiple-counting sequences and shows their association with Fermat's little theorem. I…
In this work, we generalize the probability simplex constraint to matrices, i.e., $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_K = \mathbf{I}$, where $\mathbf{X}_i \succeq 0$ is a symmetric positive semidefinite matrix of size…
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…
Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper…
Assuming the validity of Riemann Hypothesis (RH), we derive the explicit bilateral estimates ("narrow passage") of the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals. These results are reversable,…
We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent…
We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.
We extend a law of the single logarithm for delayed sums by Lai to delayed sums of random fields. A law for subsequences, which also includes the one-dimensional case, is obtained in passing.
We obtain a necessary and sufficient condition for the convergence of independent products on Lie groups, as a natural extension of Kolmogorov's three-series theorem. Application to independent random matrices is discussed.
In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorr's theorem. We also study randomness from the view point of…
In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…
We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.