12.1 Prime number theorem and related results

The Prime Number Theorem is a game-changer in understanding how primes are spread out. It tells us that as numbers get bigger, primes become rarer, but in a predictable way. This insight helps us estimate how many primes are in different ranges.

Beyond just counting primes, this theorem connects to deeper math mysteries. It's linked to the famous Riemann Hypothesis, which could give us even more precise info about primes. This stuff matters for real-world things like making secure codes for computers.

Asymptotic Distribution of Primes

Prime Number Theorem and Approximations

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• The prime number theorem states the prime-counting function π(x), which counts the number of primes less than or equal to x, is asymptotically equal to x / log(x) as x tends to infinity
• The logarithmic integral function Li(x) provides a more accurate approximation to π(x) than x / log(x) for large values of x
  • For example, Li(10^10) ≈ 4118054813, while π(10^10) = 455052511
• The prime number theorem implies the probability of a randomly chosen integer n being prime is approximately 1 / log(n) for large n
  • For instance, the probability of a random 10-digit number being prime is about 1 / log(10^10) ≈ 0.043

Equivalent Statements and Historical Context

• The prime number theorem is equivalent to the statement that the n-th prime number p_n is asymptotically equal to n log(n) as n tends to infinity
  • For example, the 1,000,000th prime is 15,485,863, while 1,000,000 * log(1,000,000) ≈ 15,480,790
• The prime number theorem was independently proved by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using complex analysis
  • Their proofs relied on the properties of the Riemann zeta function and its connection to the distribution of primes

Estimating Primes in Intervals

Using the Prime Number Theorem

• The prime number theorem can estimate the number of primes between two integers a and b, where b > a, by calculating the difference between the logarithmic integral function at b and a: Li(b) - Li(a)
  • For example, to estimate the number of primes between 1,000,000 and 2,000,000, calculate Li(2,000,000) - Li(1,000,000) ≈ 46,628
• For large intervals, the prime number theorem provides a good approximation for the number of primes, but for smaller intervals, the error term in the approximation becomes more significant
  • The actual number of primes between 1,000,000 and 2,000,000 is 46,262, which is close to the estimate provided by the prime number theorem

Applications and Improvements

• The Riemann hypothesis, if true, would provide a more accurate estimate for the number of primes in a given interval by bounding the error term in the prime number theorem
• Estimating the number of primes in a given interval has applications in cryptography, where large prime numbers are used in various encryption algorithms
  • RSA encryption relies on the difficulty of factoring large composite numbers into their prime factors

Error Terms in the Prime Number Theorem

Definition and Asymptotic Behavior

• The error term in the prime number theorem, π(x) - x / log(x), is denoted as E(x) and represents the difference between the actual number of primes less than or equal to x and the approximation given by the theorem
• The prime number theorem states that E(x) = o(x / log(x)), meaning that the error term grows slower than x / log(x) as x tends to infinity
  • In other words, the ratio E(x) / (x / log(x)) approaches 0 as x increases

Conjectures and Implications

• The Riemann hypothesis, if true, would imply a stronger bound on the error term: E(x) = O(sqrt(x) log(x)), where O() denotes the big-O notation
  • This would provide a more accurate estimate for the distribution of primes
• Improving the bounds on the error term in the prime number theorem has implications for understanding the distribution of primes and the gaps between consecutive primes
• The Cramér-Granville conjecture suggests that the error term E(x) is bounded by +/- sqrt(x) log(log(log(x))) for sufficiently large x, but this remains unproven

Prime Number Theorem vs Riemann Hypothesis

Riemann Hypothesis and its Consequences

• The Riemann hypothesis is a conjecture about the zeros of the Riemann zeta function, which is closely related to the distribution of prime numbers
• The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2
• If the Riemann hypothesis is true, it would imply a stronger error term in the prime number theorem and provide more accurate estimates for the distribution of primes

Related Problems and Significance

• The Riemann hypothesis has numerous consequences in various branches of mathematics, including number theory, complex analysis, and mathematical physics
  • For example, the Riemann hypothesis implies the Lindelöf hypothesis, which bounds the growth of certain L-functions
• The Goldbach conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes, is related to the distribution of primes and could be approached using the Riemann hypothesis
• The Riemann hypothesis is considered one of the most important unsolved problems in mathematics, and its proof or disproof would have significant implications for our understanding of prime numbers and their distribution
  • A proof of the Riemann hypothesis would also have implications for the security of certain cryptographic systems that rely on the difficulty of factoring large numbers