arXiv subjects
Publications and source records attributed to Marek Wolf.
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be independent results can be unified under a single equation. We apply our method to derive several new results in the above areas of number theory.
We derive heuristically the approximate formula for the difference $\sqrt{p_{n+1}} - \sqrt{p_n}$, where $p_n$ is the n-th prime. We find perfect agreement between this formula and the available data from the list of maximal gaps between consecutive primes.
We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones.
We collected rich series of RV measurements covering last 110 years and photometric observations from the past 6 primary eclipses, complemented them by our new observations and derived a new precise ephemeris and an orbital solution of epsilon Aur.
We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin's constant, what suggests that gamma_l are irrational. Next we have calculated the n-th square roots of the denominators q_n of the convergents of the continued fractions obtaining values close to the Khinchin-Levy constant, again supporting the common believe that gamma_l are irrational.
We have calculated numerically geometrical means of the denominators of the continued fraction approximations to the Brun constant B2. We get values close to the Khinchin constant. Next we calculated the n-th square roots of the denominators of the n-th convergents of these continued fractions obtaining values close to the Khinchin-Levy constant. These two results suggests that B2 is irrational, supporting the common believe that there is an infinity of twins.
The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $π_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this difference is investigated. The analogs of the Brun's constant and the Skewes number are calculated. An analog of the B conjecture of Hardy--Littlewood is formulated. It is argued that there is no Chebyshev bias for primes of the form $q=m^2+1$. All encountered integrals we were able to express by the logarithmic integral.
In this paper we are going to describe the results of the computer experiment, which in principle can rule out the Riemann Hypothesis. We use the sequence $c_k$ appearing in the \BD criterion for the RH. Namely we calculate $c_{100000}$ with thousand digits of accuracy using two different formulas for $c_k$ with the aim to disproof the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996 decimal place (accuracy of $10^{-996}$). The reported here experiment can be of interest for developers of Mathematica and PARI/GP.
We present formulae allowing calculation of numerical values of the numbers $c_k$ appearing in the Baez-Duarte criterion for the Riemann Hypothesis for arbitrary large $k$. We present plots of $c_k$ for $k \in (1, 10^9)$.
We investigate the relation between the Riesz and the B{á}ez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing in the B{á}ez-Duarte formulation. It is shown that $R(x)$ can be expressed by $c_k$, and, vice versa, the sequence $c_k$ can be obtained from the values of $R(x)$ at integer arguments. Also, we give some relations involving $c_k$ and $R(x)$, and value of the alternating sum of $c_k$.
The results of the computer investigation of the sign changes of the difference between the number of twin primes $π_2(x)$ and the Hardy--Littlewood conjecture $c_2\Li_2(x)$ are reported. It turns out that $π_2(x) - c_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{42}$ there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of $π_2(x) - c_2\Li_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$.
The formula $\lim_{N\to\infty}\sum_{p<N,p,p+2 both prime} \log(p)\log(p+2) = C_2$ is tested on the computer up to $N=2^{40}\approx 1.1\times 10^{12}$ and very good agreement is found.
The computer data up to $2^{44}\approx 1.76\times 10^{13}$ on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling \cite{Kelly1}, \cite{Kelly2} is provided and compared with the ``experimental'' values.