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A Direct Proof of the Prime Number Theorem

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A Direct Proof of the Prime Number Theorem A Direct Proof of the Prime Number Theorem Stephen Lucas Department of Mathematics and Statistics James Madison University, Harrisonburg VA The primes . those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and one. Martin Gardner Direct PNT – p. 1/28 Outline Why me? What is the Prime Number Theorem. History of Prime Number Theorem. Transforms. A direct proof. Riemann’s and the “exact” forms. Number of distinct primes. Conclusion. Thanks to Ken Lever (Cardiff), Richard Martin (London) Direct PNT – p. 2/28 New Improving kissing sphere bounds. Applied Math Counting cycles in graphs. Deformation of half-spaces. Integer-valued logistic equation. Micro-pore diffusion in a finite volume. Fitting data to predator-prey. Froth Flotation. Representing reals using bounded Deformation of spectacle lenses. continued fractions. Pure Math Analysis of “Dreidel”. Fractals in regular graphs. Integral approximations to π. The Hamiltonian cycle problem. Research Projects Numerical Analysis Accurately finding multiple roots. Euler-Maclaurin-like summation for Simp- son’s rule. Placing a circle pack. Direct PNT – p. 3/28 New Improving kissing sphere bounds. Counting cycles in graphs. Integer-valued logistic equation. Fitting data to predator-prey. Representing reals using bounded continued fractions. Pure Math Analysis of “Dreidel”. Fractals in regular graphs. Integral approximations to π. The Hamiltonian cycle problem. Research Projects Numerical Analysis Accurately finding multiple roots. Euler-Maclaurin-like summation for Simp- son’s rule. Placing a circle pack. Applied Math Deformation of half-spaces. Micro-pore diffusion in a finite volume. Froth Flotation. Deformation of spectacle lenses. Direct PNT – p. 3/28 New Improving kissing sphere bounds. Counting cycles in graphs. Integer-valued logistic equation. Fitting data to predator-prey. Representing reals using bounded continued fractions. Analysis of “Dreidel”. Research Projects Numerical Analysis Accurately finding multiple roots. Euler-Maclaurin-like summation for Simp- son’s rule. Placing a circle pack. Applied Math Deformation of half-spaces. Micro-pore diffusion in a finite volume. Froth Flotation. Deformation of spectacle lenses. Pure Math Fractals in regular graphs. Integral approximations to π. The Hamiltonian cycle problem. Direct PNT – p. 3/28 Research Projects Numerical Analysis Accurately finding multiple roots. Euler-Maclaurin-like summation for Simp- son’s rule. New Placing a circle pack. Improving kissing sphere bounds. Applied Math Counting cycles in graphs. Deformation of half-spaces. Integer-valued logistic equation. Micro-pore diffusion in a finite volume. Fitting data to predator-prey. Froth Flotation. Representing reals using bounded Deformation of spectacle lenses. continued fractions. Pure Math Analysis of “Dreidel”. Fractals in regular graphs. Integral approximations to π. The Hamiltonian cycle problem. Direct PNT – p. 3/28 Let π(x) be the number of primes less than or equal to x. (π(x) ! 1 as x ! 1) The prime number theorem states that x π(x) π(x) ∼ or lim = 1: ln x x!1 x= ln x Another form states that x dt π(x) ∼ − (= li(x)) ln t Z0 The Prime Number Theorem A prime number is any integer ≥ 2 with no divisors except itself and one. Direct PNT – p. 4/28 The prime number theorem states that x π(x) π(x) ∼ or lim = 1: ln x x!1 x= ln x Another form states that x dt π(x) ∼ − (= li(x)) ln t Z0 The Prime Number Theorem A prime number is any integer ≥ 2 with no divisors except itself and one. Let π(x) be the number of primes less than or equal to x. (π(x) ! 1 as x ! 1) Direct PNT – p. 4/28 Another form states that x dt π(x) ∼ − (= li(x)) ln t Z0 The Prime Number Theorem A prime number is any integer ≥ 2 with no divisors except itself and one. Let π(x) be the number of primes less than or equal to x. (π(x) ! 1 as x ! 1) The prime number theorem states that x π(x) π(x) ∼ or lim = 1: ln x x!1 x= ln x Direct PNT – p. 4/28 The Prime Number Theorem A prime number is any integer ≥ 2 with no divisors except itself and one. Let π(x) be the number of primes less than or equal to x. (π(x) ! 1 as x ! 1) The prime number theorem states that x π(x) π(x) ∼ or lim = 1: ln x x!1 x= ln x Another form states that x dt π(x) ∼ − (= li(x)) ln t Z0 Direct PNT – p. 4/28 Plot of π(x), x= ln x, li(x) 1014 100 -1 1012 x/ln x 10 10-2 1010 s e -3 10 r m o i r r 8 10 r P E f -4 e o 10 v i r t e 6 a b 10 l e m -5 R u 10 N li(x) 104 10-6 2 10 10-7 100 10-8 101 103 105 107 109 1011 1013 1015 x Direct PNT – p. 5/28 Legendre (1808) used numerical evidence to claim x π(x) = , where lim A(x) = 1:08366 : : :. ln x + A(x) x!1 Gauss (1849, as early as 1792) used numerical evidence to conjecture that x dt π(x) ∼ − : ln t Z0 Dirichlet (1837) introduced Dirichlet series: 1 f(n) f(s) = : ns n=1 X b History of the PNT x Legendre (1798) conjectured π(x) = . A ln x + B Direct PNT – p. 6/28 Gauss (1849, as early as 1792) used numerical evidence to conjecture that x dt π(x) ∼ − : ln t Z0 Dirichlet (1837) introduced Dirichlet series: 1 f(n) f(s) = : ns n=1 X b History of the PNT x Legendre (1798) conjectured π(x) = . A ln x + B Legendre (1808) used numerical evidence to claim x π(x) = , where lim A(x) = 1:08366 : : :. ln x + A(x) x!1 Direct PNT – p. 6/28 Dirichlet (1837) introduced Dirichlet series: 1 f(n) f(s) = : ns n=1 X b History of the PNT x Legendre (1798) conjectured π(x) = . A ln x + B Legendre (1808) used numerical evidence to claim x π(x) = , where lim A(x) = 1:08366 : : :. ln x + A(x) x!1 Gauss (1849, as early as 1792) used numerical evidence to conjecture that x dt π(x) ∼ − : ln t Z0 Direct PNT – p. 6/28 History of the PNT x Legendre (1798) conjectured π(x) = . A ln x + B Legendre (1808) used numerical evidence to claim x π(x) = , where lim A(x) = 1:08366 : : :. ln x + A(x) x!1 Gauss (1849, as early as 1792) used numerical evidence to conjecture that x dt π(x) ∼ − : ln t Z0 Dirichlet (1837) introduced Dirichlet series: 1 f(n) f(s) = : ns n=1 X b Direct PNT – p. 6/28 Riemann (1860) introduced the Riemann zeta function: 1 1 1 −1 ζ(s) = = 1 − ; <(s) > 1: ns ps n=1 p X Y History of the PNT (cont’d) Chebyshev (1851) introduced θ(x) = ln p; (x) = ln p; ≤ m≤ pXx pXx and showed that the PNT is equivalent to θ(x) (x) lim = 1; lim = 1: x!1 x x!1 x Direct PNT – p. 7/28 History of the PNT (cont’d) Chebyshev (1851) introduced θ(x) = ln p; (x) = ln p; ≤ m≤ pXx pXx and showed that the PNT is equivalent to θ(x) (x) lim = 1; lim = 1: x!1 x x!1 x Riemann (1860) introduced the Riemann zeta function: 1 1 1 −1 ζ(s) = = 1 − ; <(s) > 1: ns ps n=1 p X Y Direct PNT – p. 7/28 1 Only singularity at s = 1, ζ(s) = + γ + γ (s − 1) + · · ·. s − 1 1 Showed using residue calculus that xρ ζ0(0) 1 (x) = x − − − ln 1 − x−2 ; ρ ζ(0) 2 ρ X where ρ are the non-trivial zeros of ζ(s), so the PNT is equivalent to 1 xρ lim = 0: x!1 x ρ ρ X 1 Riemann hypothesis: <(ρ) = . 2 Riemann Zeta Function Integrals extend to whole plane by analytic continuation. Direct PNT – p. 8/28 Showed using residue calculus that xρ ζ0(0) 1 (x) = x − − − ln 1 − x−2 ; ρ ζ(0) 2 ρ X where ρ are the non-trivial zeros of ζ(s), so the PNT is equivalent to 1 xρ lim = 0: x!1 x ρ ρ X 1 Riemann hypothesis: <(ρ) = . 2 Riemann Zeta Function Integrals extend to whole plane by analytic continuation. 1 Only singularity at s = 1, ζ(s) = + γ + γ (s − 1) + · · ·. s − 1 1 Direct PNT – p. 8/28 1 Riemann hypothesis: <(ρ) = . 2 Riemann Zeta Function Integrals extend to whole plane by analytic continuation. 1 Only singularity at s = 1, ζ(s) = + γ + γ (s − 1) + · · ·. s − 1 1 Showed using residue calculus that xρ ζ0(0) 1 (x) = x − − − ln 1 − x−2 ; ρ ζ(0) 2 ρ X where ρ are the non-trivial zeros of ζ(s), so the PNT is equivalent to 1 xρ lim = 0: x!1 x ρ ρ X Direct PNT – p. 8/28 Riemann Zeta Function Integrals extend to whole plane by analytic continuation. 1 Only singularity at s = 1, ζ(s) = + γ + γ (s − 1) + · · ·. s − 1 1 Showed using residue calculus that xρ ζ0(0) 1 (x) = x − − − ln 1 − x−2 ; ρ ζ(0) 2 ρ X where ρ are the non-trivial zeros of ζ(s), so the PNT is equivalent to 1 xρ lim = 0: x!1 x ρ ρ X 1 Riemann hypothesis: <(ρ) = . 2 Direct PNT – p. 8/28 First Proof of the PNT Hadamard & de la Valèe Poussin (1896, independently) showed 9a; t0 1 such that ζ(σ + it) 6= 0 if σ ≥ 1 − , jtj ≥ t , so a log jtj 0 1=14 (x) = x + O xe−c(log x) : Direct PNT – p.
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