Generalization of Bertrand's Postulate for Gaussian Primes

TL;DR

This paper extends Bertrand's Postulate to Gaussian primes, providing new insights into the distribution of primes in the complex plane, building on prior generalizations for real primes.

Contribution

It formalizes the generalization of Bertrand's Postulate for Gaussian primes, a novel extension to complex primes.

Findings

Gaussian primes are distributed according to the generalized postulate

Provides bounds for the number of Gaussian primes in certain regions

Enhances understanding of prime distribution in the complex plane

Abstract

Bertrand's Postulate states about the prime distribution for the real numbers. The generalization of Bertrand's Postulate was proved by Das et al. [Arxiv 2018]. In this paper, we have formalized this idea for the Gaussian primes (or the primes on the complex plane). This result gives information about the prime distribution on the complex plane.

Equations

$$\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\},\text{ where }{i}^2=-1.$$

$$a^{\prime }=ta,b^{\prime }=tb,\text{ where }0<t<1.$$

$$s{a}^{\prime }=ar\text{ and }s{b}^{\prime }=br,$$

$$L(s,\chi )=\sum _{n=1}^{\infty }\frac{\chi (n)}{n^s}$$

$$\pi (x;4,1)\sim \pi (x;4,3)\sim \frac{1}{2}\frac{x}{\log x},$$

$$\pi (2n)-\pi (n)>0$$

$$⟹$$

$$\pi (kn)-\pi (n)\ge (k-1)$$

$$⟹$$

$$\frac{x-a}{y-b}=\frac{a-c}{b-d}$$

$$⟹$$

$$a^2+b^2<{l_1}^2+{l_2}^2<c^2+d^2.$$

$$\pi \left((k{z}_1,k{z}_2)\right)-\pi \left((z_1,z_2)\right)\ge \frac{k-1}{2},$$

$$[z,kz]=[(z_1,z_2),(k{z}_1,k{z}_2)].$$

$$\sqrt{(k{z}_1-z_1{)}^2+(k{z}_2-z_2{)}^2}=(k-1)\sqrt{{z_1}^2+{z_2}^2}.$$

$$l=$$

$$=$$

Full text

This work was done while the author was a research intern at Indian Statistical Institute, Kolkata

1. Introduction

Prime numbers are always an interesting topic for mathematicians. There is no generating formula for prime numbers for their irregular distribution. In number theory, the prime number theorem (PNT) describes the asymptotic distribution of primes among the positive integers. It gives the idea that primes become less common as they become larger. The first such distribution was found by Gauss in 1970, which states that $\pi(n)\sim\frac{n}{\log n}$, where $\pi(n)$ is the prime-counting function.

There are many results about the density of the prime numbers and it is a most interesting topic for mathematicians, especially for number theorists. In 1845, Joseph Bertrand has postulated that there exists a prime between $n$ and $2n$, where $n$ is a natural number greater than equal to 2. In the language of mathematics which states that $\pi(2n)-\pi(n)>0$ for all $n\geq 2$. There are eighteen different proofs of this result but S. Ramanujan [10] has proved it by the method of mathematical induction which is more efficient. In the paper, by Mitra et al. [11], they have generalized this idea for the interval $[n,kn]$ instead of $[n,2n]$ i.e., $\pi(kn)-\pi(n)\geq(k-1)$ for all $n\geq f(k)$ and $k\geq 2$ (where $f(k)=\lceil 1.1\ln(2.5k)\rceil,\forall k\geq 2$). In, 2018, Das et al. [3] has proved the result. Later, Steven Klee et al. [2], have proved Bertrand’s Postulate for the Gaussian Primes.

Now let us move to some other idea about the prime numbers. In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. So, we can think about the Gaussian primes and their distribution. In this paper, we have combined this idea, with the generalized Bertrand’s Postulate i.e., we state and prove the same result proved by Das et al. (for the Gaussian primes). To prove this result we use the Chebyshev’s Bias and the properties of lattice points (for the $\mathbb{R}^{2}$ plane). We study about the properties of lattice points in details in section 2

One can ask that “What is the importance of this result?” Well, an important question about the distribution of Gaussian Prime is the “Gaussian Moat” problem [12], which is also a famous unsolved problem in number theory. Bertrand’s Postulate states about the distribution of primes on the real line $\mathbb{R}$ and we have formalizes that idea (more generally) for the complex plane.

Our result gives information about the distribution of the Gaussian primes on the complex plane. If we take a straight line on the complex plane and cut a particular segment from that line then the main result of this paper is motivated to calculate the minimum number of Gaussian prime for that particular segment under certain conditions (which is the content of section 3). For the real number line, the same concept is known as the Generalization of Bertrand’s postulate which plays an important role in the distribution of the primes on $\mathbb{R}$.

2. Preliminaries

In this section, we study about the background results we need to prove the main problem we are going to encounter in this paper. Our first target is to define the primes on the complex plane which is called ‘Gaussian prime’.

2.1. Gaussian Primes

The ring of the Gaussian integer is denoted by $\mathbb{Z}[i]$. A Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers.

Definition 1**.**

Gaussian Integer:* The Gaussian integers are the set*

[TABLE]

The ring $\mathbb{Z}[i]$ has unique factorization and it admits a well-defined notion of primality. Gaussian primes are the irreducible elements of the ring $\mathbb{Z}[i]$; they emerge from factorization of the rational primes.

Definition 2**.**

Gaussian Primes:* Gaussian primes [7] are Gaussian integers $z=a+bi$ satisfying one of the following properties.*

1. (1)

If both $a$ and $b$ are nonzero then, $a+bi$ is a Gaussian prime iff $a^{2}+b^{2}$ is an ordinary prime. 2. (2)

If $a=0$, then $bi$ is a Gaussian prime iff $|b|$ is an ordinary prime and $|b|\equiv 1\pmod{4}$. 3. (3)

If $b=0$, then a is a Gaussian prime iff $|a|$ is an ordinary prime and $|a|\equiv 3\pmod{4}$.

The Gaussian primes $a+ib$ can be viewed as two-dimensional lattice points, and this allows us to explore the distribution problems from various directions.

2.2. Lattice points and their properties

To prove our main result of this paper we need to study the lattice points. Before that, we define what is a point.

Definition 3**.**

Point:* A point [8] in an $n$-dimensional space is an $n$-tuple $(x_{1},x_{2},...,x_{n})$ consisting of $n$ coordinates.*

Definition 4**.**

Lattice Point:* A point at the intersection of two or more grid lines [1] is a point lattice.*

A point lattice is a regularly spaced array of points. In the plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, etc. Unless otherwise specified, point lattices may be taken to refer to points in a square array, i.e., points with coordinates $(m,n,\ldots)$, where $m,n,\ldots$ are integers. Such an array is often called a grid or a mesh. Point lattices are frequently simply called “lattices,” which unfortunately conflicts with the same term applied to ordered sets treated in lattice theory. Every “point lattice” is a lattice under the ordering inherited from the plane, although a point lattice may not be a sublattice of the plane since the infimum operation in the plane need not agree with the infimum operation in the point lattice. On the other hand, many lattices are not pointed lattices.

Now we have a clear picture of lattice point. An important fact about the distribution of the point lattice is mutually visible lattice points.

Remark 1**.**

In this note, we are dealing with the Gaussian prime so we need to study only about the two-dimensional integer lattices.

Definition 5**.**

Mutually visible lattice points:* Two lattice points $P$ and $Q$ are said to be mutually visible if the line segment which joins them contains no lattice points other than the end points $P$ and $Q$.*

From this definition, we can think more about the mutual visibility of the lattice points.

Though it is a standard result [5], we rewrite the proof below as we will be using some insights from this proof in our subsequent derivations.

Lemma 1**.**

Two lattice points $(a,b)$ and $(m,n)$ are mutually visible [5] if, and only if, $a-m$ and $b-n$ are relatively prime.

Proof.

It is clear that $(a,b)$ and $(m,n)$ are mutually visible if and only if $(a-m,b-n)$ is visible from the origin. Hence it suffices to prove the theorem when $(m,n)=(0,0)$. Assume $(a,b)$ is visible from the origin, and let $d=\gcd(a,b)$. We wish to prove that $d=1$. If $d>1$ then $a=da^{\prime},b=db^{\prime}$ and the lattice point $(a^{\prime},b^{\prime})$ is on the line segment joining $(0,0)$ to $(a,b)$. This contradiction proves that $d=1$.

Conversely, assume $\gcd(a,b)=1$. If a lattice point $(a^{\prime},b^{\prime})$ is on the line segment joining $(0,0)$ to $(a,b)$ we have

[TABLE]

Hence $t$ is rational, so $t=r/s$ where $r,s$ are positive integers with $\gcd(r,s)=1$. Thus

[TABLE]

so $s|ar$, $s|br$. But $\gcd(s,r)=1$ so $s|a,s|b$. Hence $s=1$ since $\gcd(a,b)=1$. This contradicts the inequality $0<t<1$. Therefore the lattice point $(a,b)$ is visible from the origin. ∎

Our target is to prove a result which is about the Gaussian prime distribution on the complex plane. It is evident that to prove this result we need to study about the prime distribution on $\mathbb{R}$.

2.3. Distribution of primes and the Riemann Hypothesis

It is a well known fact that $L$-function [16] plays an important role to the distribution of prime numbers. The most famous unsolved problem in mathematics about the distribution of primes is the Riemann Hypothesis. In 1884, a more general and stronger version was formulated by Adolf Piltz ([14], p.124.), which states that, A Dirichlet character is a completely multiplicative arithmetic function $\chi$ such that there exists a positive integer $k$ with $\chi(n+k)=\chi(n)$ for all $n$ and $\chi(n)=0$ whenever $\gcd(n,k)>1$. If such a character is given, we define the corresponding Dirichlet $L$-function by

[TABLE]

for every complex number $s$ such that $Re(s)>1$. The generalized Riemann Hypothesis asserts that, for every Dirichlet character $\chi$ and every complex number $s$ with $L(\chi,s)=0$, if s is not negative real number, then the real part of $s$ is 1/2. The case $\chi(n)=1$ for all $n$ yields the ordinary Riemann Hypothesis. If Riemann Hypothesis holds then we can replace it by an equivalent statement that is, $li[\psi(x)]>\pi(x)$ where $\psi$ is the Chebyshev function and $\pi$ is the prime counting function. A similar statement of this equation was found by Chebyshev in 1853 called ‘ Chebyshev Bias’ [17] states that

[TABLE]

i.e., half of the primes are of the form congruent to $1\pmod{4}$, and half of the form $3\pmod{4}$. Later in 1994, Rubinstein and Sarnak [13] have studied this phenomenon and proved that primes congruent to 3 modulo 4 seem predominate over those congruent modulo 1. In this literature we are studying about the minimum number of primes inside a particular segment on the complex plane. So we can use this fact without this predominating nature to prove our result.

If we see the work of Das and Paul [3], they have proved “Generalization of Bertrand’s Postulate” for natural prime numbers, under certain conditions. In this note, we study the same result for the primes in two dimensions. In particular, some facts are already known about $Gaussian$ $Primes$. So, we are going to state and prove the generalization of Bertrand’s postulate for the Gaussian primes. Roughly speaking, the main result of this paper is saying about the minimum number of Gaussian Primes for a particular segment of a straight line can contain on the complex plane. It is not possible to prove this same statement directly for the maximum number of primes on the complex plane because the existing results are not enough to provide the proof of this statement. Also, there are several interesting results on the Gaussian Primes [4, 15].

3. Generalized Bertrand Postulate for Gaussian primes

In this section, we are going to state and prove the main result of this article that is the generalization of Bertrand’s Postulate for the Gaussian primes. Before we focus on the main problem let us state the Bertand’s Postulate [9, 10] and it’s generalization (which was conjectured by Mitra et al. [11] and proved by Das et al. [3]).

Proposition 1**.**

Bertrand’s Postulate:* For every $n>1$ there is always at least one prime $p$ such that*

[TABLE]

Proposition 2**.**

Generalization of Bertrand’s Postulate:* There exist at least $(k-1)$ primes between the integers $n$ and $kn$ for all $k\geq 2$ and $n\geq f(k)$ where $f(k)=\lceil 1.1\ln(2.5k)\rceil$. In particular,*

[TABLE]

where $p_{j}$’s are primes for $j=i_{1},i_{2},\ldots,i_{k-1}$ and $i_{1},i_{2},\ldots,i_{k-1}$ are positive integers.

Observe the statement of Bertand’s Postulate it is clear that we need to define the concept of $gap$ between two complex number to state and prove our main result. Bertrand’s postulate needs the concept of gap in $\mathbb{R}$ (that is simple distance between two reals for $\mathbb{R}$) and we are extending this result for $\mathbb{R}^{2}$ so let us define it.

Definition 6**.**

Gap** on the complex plane:*** Let $(a,b)$ and $(c,d)$ are two integer lattices on the complex plane. Join this two lattice point by a straight line. This line segment (as shown in figure 1) is called the gap between the lattice points $(a,b)$ and $(c,d)$ on the complex plane, which is denoted by $[(a,b),(c,d)]$111We use this third bracket and write two pair of coordinates to denote the gap between them as we have defined it in definition 6..*

Some facts about the Gap:

• •

The length of the segment between $(a,b)$ and $(c,d)$ is the Euclidean distance i.e., $\sqrt{(a-c)^{2}+(b-d)^{2}}$.

• •

There are $\gcd(a-c,b-d)$ many integer lattices lie on this gap. This property is coming from Lemma 1.

• •

The integer lattices which lie on this segment satisfy the line equation,

[TABLE]

If we plot them on $\mathbb{R}$ (i.e., taking the modulus of these integer lattices) then for every integer lattices $(l_{1},l_{2})$ ($\forall l_{1},l_{2}\geq 1$) lie between $(a,b)$ and $(c,d)$ will satisfy the inequality

[TABLE]

Now we are ready to state and prove our main result.

Main Theorem 1**.**

Generalization of Bertrand’s Postulate for the Gaussian Primes:* Let $z$ be an Gaussian integer with $z=z_{1}+iz_{2}$ (where $z_{1},z_{2}\geq 0$). Define the function $f(k)=\lceil 1.1\ln(2.5k)\rceil$ for all $k\geq 2$ and $\gcd(z_{1},z_{2})=f(k)$. Then there exist at least $\frac{k-1}{2}$ many Gaussian primes between the gap $[(z_{1},z_{2}),(kz_{1},kz_{2})]$ for the complex number $z=z_{1}+iz_{2}$ and $\forall k\geq 2$, such that*

[TABLE]

$\forall k\geq 2\text{ and }\gcd(z_{1},z_{2})=f(k)$.

Proof.

To prove the theorem we use Chebyshev Bias [17] and the properties of integer lattices (on $\mathbb{R}^{2}$). It is clear from the definition of the Gaussian Prime that there are two possible cases.

Case(I): Let $z$ be an Gaussian integer with $z=z_{1}+iz_{2}$ and $z_{1},z_{2}\neq 0$. It is given that $\gcd(z_{1},z_{2})=f(k)$, where $f(k)=\lceil 1.1\ln(2.5k)\rceil$. Then from the definition 6 we can write the gap between the Gaussian integer $[z,kz]$ is given by222$k$ is a positive integer greater than equal to 2, after multiplying $z$ with $k$, $z$ will remain a complex number.

[TABLE]

So the Euclidean distance between this two coordinate is,

[TABLE]

By the given condition we also have $\gcd(z_{1},z_{2})=f(k)$. We can write $z_{1}=af(k)$ and $z_{2}=bf(k)$ for some positive integer $a,b$ with the condition $\gcd(a,b)=1$.

If we join the integer lattices $(z_{1},z_{2})$ and $(kz_{1},kz_{2})$ by a straight line then from lemma 1, we can say that there are $l$ many integer lattices on this straight line where we can compute $l$,

[TABLE]

So, there exist $(k-1)f(k)$ many integer lattices between the gap $[(z_{1},z_{2}),(kz_{1},kz_{2})]$.

Proposition 2 states that there exist at least $(k-1)$ primes between $f(k)$ and $kf(k)$. The length of the segment between the integer $f(k)$ and $kf(k)$ is $(k-1)f(k)$. In particular, it says that from the point $f(k)$ (for all $k\geq 2$), if we take a segment of value $(k-1)f(k)$ then at least $(k-1)$ many primes are there in that particular segment. Observe this picture on the complex plane then the integer lattices lie between the gap $[(z_{1},z_{2}),(kz_{1},kz_{2})]$ can be plotted on the real number line as an integer by taking their modulus. So, from proposition 2 we can say that there exist at least $(k-1)$ many primes between ${z_{1}}^{2}+{z_{2}}^{2}$ and ${(kz_{1})}^{2}+{(kz_{2})}^{2}$. We know that only the primes of the form $1\pmod{4}$ can be plotted on the complex plane (by definiton 2). Chebyshev Bias states that almost half of these primes are of the form congruent to $1\pmod{4}$ i.e., $\frac{k-1}{2}$ many complex Gaussian primes are there. So, there are at least $\frac{k-1}{2}$ Gaussian primes between the gap$[(z_{1},z_{2}),(kz_{1},kz_{2})]$ for all $k\geq 2$. Hence, we have proved the statement for the Gaussian prime on the complex plane.

Now we prove it for the Gaussian primes of the form congruent to $3\pmod{4}$, that is our second case.

Case(II): In this case, we prove the statement for the given Gaussian integer $z$ with $Re(z)=0$ or $Im(z)=0$, but both cases are symmetric. It is enough to prove this statement for $Re(z)=0$ and the other case follows similarly. If we join the statements of proposition 2 and Chebyshev Bias then there exist at least $\frac{k-1}{2}$ many primes of the form $3\pmod{4}$ between the gap $[z_{1},kz_{1}]$, by the given condition that $z_{1}$ is a multiple of $f(k)$.

Hence, we have proved the theorem. ∎

Remark 2**.**

We can observe the geometrical view : cut a segment on $\mathbb{R}$ and rotate it anti-clockwise with positive angle $\theta<90^{\circ}$ and then shift it on the complex plane with the starting point at $(a,c)$ and ending point at $(b,d)$. There are $(a^{2}-c^{2})+(b^{2}-d^{2})$ many positive integers on $\mathbb{R}$ and trivially $\gcd(a-c,b-d)<(a^{2}-c^{2})+(b^{2}-d^{2})$. So all the integers will not be there after shifting which is same as saying that only those integers satisfy the line equation 3.1 will lie on the segment after shifting333We have described this for the first quadrant. For the other quadrants due to the change of sign and the direction of the rotation, the sign of $\theta$ can be changed.. If we do reverse of this process then we can see that all the integer lattices lie on the gap $[(a,b),(c,d)]$ will lie in the segment on $\mathbb{R}$ bounded by $a^{2}+b^{2}$ and $c^{2}+d^{2}$, i.e., all the integer lattices $(l_{1},l_{2})$ ($l_{1},l_{2}$ are positive integers) lie on the gap $[(a,b),(c,d)]$ will satisfy the inequality 3.2.

4. Conclusion

In this article, we have formalized the idea of generalized Bertrand’s Postulate for the Gaussian primes, which says about the distribution of the Gaussian primes on the complex plane and prove it with the help of Chebyshev Bias.

Acknowledgments

I would like to express my very great appreciation to Prof. Goutam Paul (from CSRU unit, Indian Statistical Institute, Kolkata) for his valuable and constructive suggestions during the planning and development of this research work. His willingness to give his time so generously has been very much appreciated.