Volume 6, 2000, Number 4

We examine here the class structure of odd primes within the modular ring ℤ₄ in relation to Goldbach’s Conjecture. Such analyses, together with the identification of compatible right-end digits for the Goldberg ’system’, permit a more efficient search for prime pairs. This is useful for the study of very large even numbers, the distribution of twin primes and other prime constellations, and the relative distribution of primes between the classes l₄ and 3_4.

The Cardano cubic, $y^3 – 6pqy – 3pq(p + q), p, q \in Z_{+}$, has one real zero and a complex conjugate pair. The real zero is given by $2(2pq + e)^{1 \over 2}$ or $(E + 2)(2pq)^{1 \over 2}$, in which $e, E$ are important parameters that feature in the roots of all Cardano cubics. They are functions of the coefficients of the complex conjugate pairs. We find that
$$ e = \frac{2q^2R \tan^2 \theta}{3 – \tan^2 \theta}$$
$$ E = 2 \Big \{ \Big ( \frac{3}{3 – \tan^2 \theta} \Big )^{1 \over 2} – 1 \Big \}$$
with $R = \frac{p}{q} = h(\theta)$ and 11° < $\theta$ < 60° for real zero. Furthermore, for $E$ integer, the range of $\theta$ is reduced to 52° < $\theta$ < 60°, where the functional surfaces suggest the reason the integer $E$ would only be compatible with an irrational value of $R$. This is verified algebraically. [/expand] Expressions for the Dirichlet inverse of arithmetical functions
Original research paper. Pages 118–124
P. Haukkanen
Full paper (PDF, 222 Kb) | AbstractWe express the values of the Dirichlet inverse f ^-1 in terms of the values of f without using the values of f ^-1. We use a method based on representing f ^-1 * f = δ as a system of linear equations. Jagannathan has given many of the results of this paper without proof starting from the basic recurrence relation for the values of f ^-1.