Related papers: Explicit Mertens' Theorems for Number Fields

We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.

For $i\in \{1,2,3\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\in \{1,2\}$ the Riemann hypothesis is equivalent to the condition…

We prove a global uniform Artin-Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proof is…

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.

In this paper, we obtained some global approximation results for general Gamma type operators.

Renyi's result on the density of integers whose prime factorizations have excess multiplicity has an analogue for polynomials over a finite field.

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…

A nonlinear scalar field theory from which an effective metric can be deduced is considered. This metric is shown to be compatible with requirements of general relativity. It is demonstrated that there is a class of solutions which fulfill…

The Duffin-Schaeffer theorem is a well-known result from metric number theory, which generalises Khinchin's theorem from monotonic functions to a wider class of approximating functions. In recent years, there has been some interest in…

We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Roger's second moment formula for the Siegel…

In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.

We prove a "discrete analogue" for Taelman's class modules of certain Conjectures formulated by R. Greenberg for cyclotomic fields.

I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic…

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient…

We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.

A conceptual summary is given of a deterministic unified field and particle theory (the metron model) developed in more mathematical detail in a four-part paper published in Physics Essays (1996/97). The model is developed from Einsteins…

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related…

A generalization of the Heisenberg algebra has been recently constructed. This generalized algebra has a characteristic function which depends on one of its generators. When this function is linear, $qJ_0+s$, it is possible to construct a…

By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.