The Prime Number Theorem (PNT) and the Riemann zeta function are deeply connected. This section shows how PNT's truth is equivalent to certain properties of the zeta function, like not having zeros on the line Re(s) = 1.

Understanding this link is crucial for grasping the power of analytic methods in number theory. It reveals how complex analysis can shed light on the distribution of primes, a fundamental question in mathematics.

Prime Number Theorem and Zeta Function

Fundamental Concepts and Definitions

• Prime Number Theorem (PNT) states the asymptotic distribution of prime numbers among positive integers
• PNT expresses that $\pi(x) \sim \frac{x}{\log x}$ as x approaches infinity
• Zeta function $\zeta(s)$ defined as the infinite series $\sum_{n=1}^{\infty} \frac{1}{n^s}$ for complex s with real part > 1
• Zeta function extends analytically to the entire complex plane except for a simple pole at s = 1
• Asymptotic equivalence denoted by the symbol ~ indicates that the ratio of two functions approaches 1 as the variable tends to a limit

Chebyshev Functions and Their Significance

• Chebyshev functions $\psi(x)$ and $\theta(x)$ play crucial roles in understanding prime number distribution
• $\psi(x)$ defined as the sum of $\log p$ over prime powers $p^k \leq x$
• $\theta(x)$ defined as the sum of $\log p$ over primes $p \leq x$
• PNT equivalent to statement that $\psi(x) \sim x$ or $\theta(x) \sim x$ as x approaches infinity
• Chebyshev functions provide smoother approximations to prime counting function compared to $\pi(x)$

Connections Between PNT and Zeta Function

• Riemann's explicit formula connects $\psi(x)$ to the zeros of the zeta function
• Non-vanishing of $\zeta(s)$ on the line Re(s) = 1 implies PNT
• PNT equivalent to the statement that $\zeta(s)$ has no zeros on the line Re(s) = 1
• Relationship between PNT and zeta function properties demonstrates deep connection between analytic and number-theoretic concepts

Arithmetic Functions

Möbius Function and Its Properties

• Möbius function $\mu(n)$ defined for positive integers n
• $\mu(n) = 1$ if n is a square-free positive integer with an even number of prime factors
• $\mu(n) = -1$ if n is a square-free positive integer with an odd number of prime factors
• $\mu(n) = 0$ if n has a squared prime factor
• Möbius function satisfies the identity $\sum_{d|n} \mu(d) = \begin{cases} 1 & \text{if } n = 1 \\ 0 & \text{if } n > 1 \end{cases}$
• Möbius inversion formula allows reversing certain sums involving multiplicative functions

Von Mangoldt Function and Prime Power Detection

• Von Mangoldt function $\Lambda(n)$ defined for positive integers n
• $\Lambda(n) = \log p$ if n is a power of a prime p
• $\Lambda(n) = 0$ if n is not a prime power
• Von Mangoldt function relates to the logarithmic derivative of the zeta function: $-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$
• $\psi(x) = \sum_{n \leq x} \Lambda(n)$ connects von Mangoldt function to Chebyshev function
• Von Mangoldt function plays a crucial role in explicit formulas for prime counting functions

Analytical Tools

Tauberian Theorems and Their Applications

• Tauberian theorems provide conditions under which convergence properties of a sequence can be deduced from its summability
• Wiener-Ikehara Tauberian theorem crucial in proving PNT
• Tauberian theorems allow deduction of asymptotic behavior of arithmetic functions from analytic properties of associated Dirichlet series
• Applications include proving asymptotic formulas for summatory functions of multiplicative arithmetic functions
• Tauberian theorems bridge gap between analytic and elementary methods in number theory

Mellin Transform and Complex Analysis Techniques

• Mellin transform defined as $\mathcal{M}f(s) = \int_0^{\infty} f(x) x^{s-1} dx$ for suitable functions f
• Mellin transform connects multiplicative structure of arithmetic to additive structure of complex analysis
• Inverse Mellin transform allows recovery of original function from its Mellin transform
• Mellin transform of $e^{-x}$ yields the gamma function $\Gamma(s)$
• Functional equation of zeta function derived using Mellin transform techniques
• Perron's formula, based on inverse Mellin transform, used to relate Dirichlet series to summatory functions of their coefficients