Problem 21

Question

Using the prime number theorem, show that $p_{n} \sim n \log n,$ where $p_{n}$ denotes the $n$ th prime.

Step-by-Step Solution

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Answer

Answer: The relationship between the nth prime number, $p_n$, and $n \log n$ is that $p_n$ is asymptotic to $n \log n$. This is derived from the prime number theorem, which states that the number of primes less than or equal to a positive integer x, is approximately equal to $\frac{x}{\log x}$.

1Step 1: Consider the Prime Number Theorem

The prime number theorem states that the number of primes less than or equal to a positive integer x, denoted as $\pi(x)$, is approximately equal to $\frac{x}{\log x}$. In other words, $\pi(x) \sim \frac{x}{\log x}$.

2Step 2: Define the nth Prime Number

Let $p_n$ be the nth prime number. By definition, it is the smallest positive integer such that there are exactly n primes less than or equal to $p_n$. In other words, we have $\pi(p_n) = n$.

3Step 3: Approximate the nth Prime Number Using the Prime Number Theorem

Using the prime number theorem, we can approximate the value of $p_n$. Since $\pi(p_n) = n$, we can say that $n \sim \frac{p_n}{\log p_n}$, which is derived from the approximation $\pi(x) \sim \frac{x}{\log x}$.

4Step 4: Rearrange the Equation to Show the Relationship between $p_n$ and $n \log n$

Now, we will rearrange the equation from step 3, so that we can show the relationship between $p_n$ and $n \log n$. Multiply both sides of the equation by $\log p_n$: $n \log p_n \sim p_n$. Divide both sides by $n$: $p_n \sim n \log n.$ Thus, we have shown that the nth prime number, $p_n$, is asymptotic to $n \log n$.

Key Concepts

nth prime numberasymptotic behaviorlogarithmic approximation

nth prime number

The concept of the "nth prime number" revolves around identifying the sequence of prime numbers as integers greater than 1 that have no divisors other than 1 and themselves. In simple terms, these are numbers like 2, 3, 5, 7, 11, and so on, listed in increasing order. The nth prime number, denoted by $p_n$, is the prime located at the position $n$ in this ordered list. For instance, $p_1$ is 2, $p_2$ is 3, $p_3$ is 5, and so forth.

To find the nth prime, you essentially list primes until you reach the $n$th position. Though simple in concept, calculating large nth primes manually is daunting due to their unpredictable spacing as numbers grow larger. This leads us to employ the prime number theorem to estimate the nth prime without computing all previous primes directly.

asymptotic behavior

Asymptotic behavior is a way of describing how a function behaves as its variable gets exceedingly large. When we say two variables are asymptotically equal, it means that their ratio approaches 1 as they both tend to infinity.

In the context of prime numbers, the Prime Number Theorem provides us a quintessential example of asymptotic behavior. It tells us that if $\pi(x)$ represents the number of primes less than or equal to $x$, then $\pi(x) \sim \frac{x}{\log x}$. This means that as $x$ becomes very large, $\pi(x)$ and $\frac{x}{\log x}$ grow at nearly the same rate and their ratio nears 1.

For large numbers, this approximation becomes particularly helpful and accurate.
Such behavior highlights that while the exact position of $n$th primes might not be computable without counting, their density can be estimated reasonably well using asymptotic tools.

Understanding asymptotic behavior helps in simplifying problems involving large numbers, by focusing on their growth patterns rather than exact values.

logarithmic approximation

Logarithmic approximation is an essential tool in number theory and is frequently used to simplify calculations involving large numbers. In the case of prime numbers, the logarithmic function helps relate numbers and their properties in terms of growth rates and density.

A logarithm is a mathematical operation that determines how many times one number (known as the base) must be multiplied by itself to reach another number. When we say that $p_n \sim n \log n$, we are using a logarithmic approximation to describe the growth of the nth prime number in terms of $n$.

This approximation stems from rearranging the Prime Number Theorem: $\pi(p_n) = n$, which implies $n \sim \frac{p_n}{\log p_n}$.
Multiplying through by $\log p_n$, we arrive at $n \log p_n \sim p_n$.
The approximation $p_n \sim n \log n$ becomes especially more effective as $n$ increases.

Thus, logarithmic approximations streamline complex computations by providing a simpler model of behavior, especially valuable in large-scale estimations like that of distributing prime numbers.

Problem 20

Problem 22

Other exercises in this chapter

Problem 19

Design and analyze an algorithm that on input $n$ outputs the table of values $\tau(k)$ for $k=1, \ldots, n,$ where $\tau(k)$ is the number of positive

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Problem 20

Using the prime number theorem, show that $\vartheta(x) \sim x$.

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Problem 22

Using the prime number theorem, show that Bertrand's postulate can be strengthened (asymptotically) as follows: for every $\varepsilon>0,$ there exist positiv

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Problem 16

Use Stirling's approximation (previous exercise) to show that $$ \left(\begin{array}{c} 2 m \\ m \end{array}\right)=\Theta\left(2^{2 m} / \sqrt{m}\right). $$

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