PDF] Approximate formulas for some functions of prime numbers ... - Free Printable

Problem Analysis

The provided table, Table III, lists values of \( n \) along with the corresponding values of \( \psi(n) - \vartheta(n) \). Here:
- \( \psi(n) \) is the second Chebyshev function, which sums the logarithms of all prime powers less than or equal to \( n \).
- \( \vartheta(n) \) is the first Chebyshev function, which sums the logarithms of all primes less than or equal to \( n \).

The task appears to involve analyzing the behavior of \( \psi(n) - \vartheta(n) \) as \( n \) increases. This difference is significant in number theory, particularly in studying the distribution of prime numbers.

Key Observations from the Table

1. Structure of the Table:
- The table is divided into two columns for \( n \) and \( \psi(n) - \vartheta(n) \).
- Values of \( n \) are listed in ascending order.
- The differences \( \psi(n) - \vartheta(n) \) are given in a specific format (e.g., "0.69314 71805 59945" for \( n = 4 \)).

2. Behavior of \( \psi(n) - \vartheta(n) \):
- As \( n \) increases, \( \psi(n) - \vartheta(n) \) generally grows.
- The growth appears to be irregular but shows a trend of increasing magnitude.

3. Mathematical Context:
- The difference \( \psi(n) - \vartheta(n) \) captures the contribution of prime powers (other than primes themselves) to the sum of logarithms.
- For large \( n \), this difference is influenced by the density of prime powers and their logarithmic contributions.

Solution Approach

The problem likely asks us to:
1. Analyze the trend of \( \psi(n) - \vartheta(n) \) as \( n \) increases.
2. Interpret the significance of the observed values in the context of number theory.
3. Identify any patterns or notable features in the data.

#### Step 1: Trend Analysis
From the table:
- For small \( n \) (e.g., \( n = 4 \)), \( \psi(n) - \vartheta(n) \) is relatively small (e.g., 0.69314...).
- As \( n \) increases, \( \psi(n) - \vartheta(n) \) grows. For example:
- At \( n = 100 \), \( \psi(n) - \vartheta(n) \approx 100.28553... \).
- At \( n = 1000 \), \( \psi(n) - \vartheta(n) \approx 122.02043... \).
- At \( n = 10000 \), \( \psi(n) - \vartheta(n) \approx 253.93933... \).

This indicates that the difference \( \psi(n) - \vartheta(n) \) increases with \( n \), though not linearly.

#### Step 2: Significance in Number Theory
The difference \( \psi(n) - \vartheta(n) \) reflects the additional contribution of prime powers (beyond just primes) to the sum of logarithms. This is important because:
- Primes are the building blocks of integers, but prime powers also play a role in the distribution of integers.
- The growth of \( \psi(n) - \vartheta(n) \) suggests that prime powers become more frequent or significant as \( n \) increases.

#### Step 3: Patterns and Notable Features
1. Irregular Growth:
- The increase in \( \psi(n) - \vartheta(n) \) is not smooth. There are fluctuations, likely due to the irregular distribution of prime powers.

2. Asymptotic Behavior:
- It is known from number theory that \( \psi(n) \sim n \) and \( \vartheta(n) \sim n \) as \( n \to \infty \). Thus, \( \psi(n) - \vartheta(n) \) is expected to grow, but its exact rate depends on the density of prime powers.

3. Notable Values:
- Certain values of \( n \) (e.g., powers of 10 or highly composite numbers) might show more pronounced changes in \( \psi(n) - \vartheta(n) \).

Final Answer

The analysis of Table III reveals that \( \psi(n) - \vartheta(n) \) generally increases as \( n \) increases, reflecting the growing contribution of prime powers to the sum of logarithms. The irregular growth highlights the complex distribution of prime powers.

\[
\boxed{\text{The difference } \psi(n) - \vartheta(n) \text{ increases with } n, \text{ reflecting the contribution of prime powers.}}
\]