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Nine Chapters of Analytic Number Theory in Isabelle/HOL Nine Chapters of Analytic Number Theory in Isabelle/HOL Manuel Eberl Technische Universität München 12 September 2019 + + 58 Manuel Eberl Rodrigo Raya 15 library unformalised 173 18 using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters Material • Various number-theoretic functions (executable and with many properties proven): − Euler’s totient j and Carmichael’s l − Divisor function sx − Möbius’s m − Liouville’s l − Prime-counting functions: p(n), J(n), y(n) • Dirichlet series (both formal and analytic) • Multiplicative characters • Riemann’s z, Hurwitz’s z, Dirichlet’s L functions Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. Interesting results Prime Number Theorem jfp j p prime ^ p ≤ xgj ∼ x log x Dirichlet’s Theorem gcd(k,m) = 1 =) fp j p prime ^ p ≡ k (mod m)g is infinite Asymptotics y(n) • lcm(1,...,n) = e where y(x) ∼ x • On average, an integer n has log n + 2g − 1 divisors. 2 • The set of square-free integers has density 6/p ≈ 60.8%. • The set of fractions p/q for p,q prime is dense in R. • 1 Prime harmonic series: ∑p≤x p = log log x + M + O(1/log x) • They form a commutative ring (plus many other operations: division, derivative, exponential, logarithm, etc.) • They have a deep connection to number-theoretic functions. • When they converge, the corresponding complex functions often contain useful information. • Transfer between the analytic world and the algebraic world is often possible (in both directions). • Even when they do not converge, they can be very useful. Dirichlet series A Dirichlet series is a formal series of the form: ¥ a F (s) = n ∑ s n=1 n They are the ‘number-theoretic analogue’ of formal power series: • They have a deep connection to number-theoretic functions. • When they converge, the corresponding complex functions often contain useful information. • Transfer between the analytic world and the algebraic world is often possible (in both directions).
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