The Prime Index Function
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DOI: 10.1007/s13226-020-0458-9
THE PRIME INDEX FUNCTION Theophilus Agama Department of Mathematics, African Institute for Mathematical science, Ghana e-mails: [email protected], [email protected] (Received 7 December 2018; after final revision 27 March 2019; accepted 24 June 2019) In this paper we introduce the prime index function ι(n) = (−1)π(n) , where π(n) is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by X
ξ(x) :=
ι(n).
n≤x
Key words : Oscilloation; period; index; prime. 2010 Mathematics Subject Classification : 42C15, 41A58.
1. I NTRODUCTION AND M OTIVATION The prime counting function is a very important and useful function in number theory and the whole of mathematics. It is connected to many open problems in mathematics, such as the Riemman hypothesis [2]. In the following sequel we introduce the prime index function ι(n), an arithmetic function which is neither additive nor multiplicative. It can be considered to be of the same class with the Liouville λ(n) function. Rather like the Liouville function defined on the prime factors of the integers, the prime index function is defined on the number of prime less than a fixed integer. It is given by ι(n) = (−1)π(n) , where π(n) is the prime counting function. Given the chaotic behaviour of the prime index-function makes it an intractable function to study. Hence we introduce as well the second prime-index function
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THEOPHILUS AGAMA
given by ξ(x) :=
X
ι(n).
n≤x
By introducing the concept of oscillation on the second prime-index function ξ, we relate results of primes in short and long intervals to oscillations. It turns out that the following, which can be considered as a sibling of Bertrand’s postulate, is true: Theorem 1.1 — Let x, y ∈ R. If ξ(x) = ξ(y), then there exist at least a prime in the interval (x, y]. In the spirit of understanding the twin prime conjecture, we obtain the following weaker result: Theorem 1.2 — There are infinitely many points y ∈ R such that ξ(y − s) = ξ(y + s) for some s ∈ N. In particular, there are infinitely many points of oscillations with period s. 2. T HE P RIME I NDEX F UNCTION Definition 2.1 — Let n ≥ 1, then we set ι(n) = (−1)π(n) , where π(n) is the prime counting function. The prime index function is an extremely usefull function. It is basically the sequence ι : N −→ {1, −1}. It is somewhat an intractable but very interesting function when we take their partial sums. Below is a table for the distribution of the prime-index function: Table 1: Values of n 1 ι(n)
2
3 4
5
6
7 8 9 10
1 −1 1 1 −1 −1 1 1 1
1
Definition 2.2 — Let x ≥ 1, then we set ξ(x) :=
X n≤x
ι(n).
11
12
−1 −1
13 14 15 16
17
1
−1 −1
1
1
1
18
THE PRIME INDEX FUNCTION
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Next we make a leap by understanding the distribution of the partial sums of the prime index function ξ(x). Below is a table that gives the distribution of the first eighteen values of ξ(x). 3. D ISTRIBUTION OF THE S ECOND P RIME I NDEX F UNCTION ξ(x)
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