Volume 7, 2001, Number 1

φ and σ functions (see, e.g., [1]) are two of the most important arithmetic functions. They have a lot of very interesting properties. Some of them will be discussed below.

In what follows we shall study certain arithmetic functions with application to the study of some trigonometrical products.

For every n > 1, let π₂(n) denote the number of primes p such that p ≤ n and p + 2 is also a prime. In the present paper some new formulae for π₂(n) are proposed.

Let C = {C_n}_n≥1 be an arbitrary increasing sequence of natural numbers. By π_C(n) we denote the number of the terms of C being not greater than n (we agree that π_C(0) = 0). In the first part of the paper we propose six different formulae for C_n (n = 1, 2, …), which depend on the numbers π_C(k) (k = 0, 1, 2, …). Using these formulae, in the second part of the paper we obtain three different explicit formulae for the n-th prime p_n, which are the first main result of the present research. In the third part of the paper, using the formulae from the first part, we propose six explicit formulae for the n-th term of the sequence of twin primes: 3, 5, 7, 11, 13, 17, 19,… – the second main result of the paper. The last three of them are main ones for the twin primes.

Twin primes of the form h2ⁿ−1 are analysed within the modular ring ℤ₆. The values of h are odd and 3|h for the lowest valued twin prime, p₂. The other values of h fall in either the second (2₆) or fourth (4₆) class of ℤ₆, depending on the parity of n. Functional relationships are developed for the various h, n and rows within ℤ₆. All p₂ fall in class 2₆ and the larger prime of the twin pair, p₄, always falls in class 2₆. With n₄ = 1 and n₄ > 1, p₂ falls in class one (1₄) of the modular ring ℤ₄ and hence equals a unique set of squares (x² + y²). Analysis of the distribution of the (x,y) pair reveals an interesting prime sequence related to the Fibonacci sequence.