The Riemann Hypothesis, a cornerstone of number theory, has far-reaching consequences. If true, it would unlock deeper insights into prime number distribution and arithmetic functions. This section explores how the hypothesis shapes our understanding of these fundamental concepts.
From prime gaps to arithmetic functions, the Riemann Hypothesis influences numerous conjectures and theorems. We'll see how it connects to the Mertens function, L-functions, and other key ideas in analytic number theory.
Prime Number Conjectures
Distribution and Patterns of Prime Gaps
- Prime gap distribution describes spacing between consecutive prime numbers
- Cramér's model predicts average gap size increases logarithmically with prime size
- Prime k-tuples conjecture generalizes patterns in prime number sequences
- Polignac's conjecture proposes infinitely many prime pairs with any even gap
- Zhang's theorem proves bounded gaps between primes, significant progress on twin prime conjecture
- Maynard-Tao theorem improves Zhang's result, shows infinitely many pairs of primes with gap at most 246
- Twin prime conjecture states infinitely many pairs of primes differ by exactly 2 (3 and 5, 5 and 7, 11 and 13)
- Remains unproven despite substantial evidence and partial results
- Brun's theorem shows sum of reciprocals of twin primes converges to Brun's constant
- Yitang Zhang's 2013 breakthrough proved infinitely many prime pairs within 70 million
- Current best bound reduced to 246 by polymath project and James Maynard
- Cousin primes (differing by 4) and sexy primes (differing by 6) form related conjectures
Goldbach's Conjecture and Its Variants
- Goldbach's conjecture states every even integer greater than 2 is sum of two primes
- Weak Goldbach conjecture (odd Goldbach conjecture) proven by Harald Helfgott in 2013
- Vinogradov's theorem shows sufficiently large odd numbers are sum of three primes
- Ternary Goldbach conjecture states odd numbers greater than 5 are sum of three primes
- Chen's theorem proves every sufficiently large even number is sum of a prime and product of at most two primes
- Goldbach comet visualizes distribution of Goldbach partitions for even numbers
Arithmetic Functions
Mertens Function and Its Properties
- Mertens function M(n) sums Möbius function values up to n
- Defined as M(n) = ∑μ(k) for k from 1 to n
- Mertens conjecture proposed |M(n)| < √n for all n > 1, disproven in 1985
- Odlyzko and te Riele showed Mertens conjecture false for some n < exp(1.59 × 10^40)
- Mertens function related to distribution of prime numbers and Riemann hypothesis
- Asymptotic behavior of M(n) closely tied to zero-free region of Riemann zeta function
Möbius Function and Its Applications
- Möbius function μ(n) defined for positive integers based on prime factorization
- μ(n) = 1 if n is square-free with even number of prime factors
- μ(n) = -1 if n is square-free with odd number of prime factors
- μ(n) = 0 if n has a squared prime factor
- Möbius inversion formula reverses certain arithmetic sums
- Dirichlet series of Möbius function is reciprocal of Riemann zeta function
- Mertens function M(n) = ∑μ(k) for k from 1 to n
Growth and Behavior of Arithmetic Functions
- Arithmetic functions map positive integers to complex numbers
- Multiplicative functions satisfy f(ab) = f(a)f(b) for coprime a and b
- Completely multiplicative functions satisfy f(ab) = f(a)f(b) for all a and b
- Divisor function d(n) counts number of divisors of n
- Sum of divisors function σ(n) sums all positive divisors of n
- Euler's totient function φ(n) counts numbers up to n coprime to n
- Growth rates of arithmetic functions often related to prime number distribution
L-Functions and Hypotheses
Dirichlet L-Functions and Their Properties
- Dirichlet L-functions generalize Riemann zeta function to arithmetic progressions
- Defined as L(s, χ) = ∑χ(n)n^(-s) for Re(s) > 1, where χ is a Dirichlet character
- Analytic continuation extends L-functions to entire complex plane (except s = 1 for principal character)
- Functional equation relates values of L(s, χ) and L(1-s, χ̄)
- Non-trivial zeros of L-functions lie in critical strip 0 ≤ Re(s) ≤ 1
- Generalized Riemann hypothesis conjectures all non-trivial zeros have real part 1/2
- Dirichlet's theorem on primes in arithmetic progressions proved using L-functions
Lindelöf Hypothesis and Its Implications
- Lindelöf hypothesis bounds growth of Riemann zeta function on critical line
- States |ζ(1/2 + it)| = O(|t|^ε) for any ε > 0 as |t| → ∞
- Implies bounds for other L-functions and number-theoretic functions
- Weaker than Riemann hypothesis but still unproven
- Current best bound by Bourgain: |ζ(1/2 + it)| = O(|t|^13/84+ε)
- Lindelöf hypothesis would improve many results in analytic number theory
- Connections to moment problems for ζ(s) and distribution of prime numbers