Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Dynamical Zeta functions Mark Pollicott Warwick University Chalmers' Colloquium : 11 September, 2017 1 / 30 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 2 / 30 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 / 30 and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; 2 / 30 In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. 2 / 30 The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. 2 / 30 The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. 2 / 30 Riemann Zeta Function Selberg Zeta function Geodesic and Anosov flows Applications Different types of zeta functions In this talk I want to consider three types of basic zeta function. 1 The Riemann zeta function for prime numbers (for motivation only); 2 The Selberg zeta function for lengths of closed geodesics of surfaces of constant negative curvature; and 3 The Ruelle zeta function for periods of closed orbits of geodesic and Anosov flows. In particular, the Ruelle zeta function generalises the Selberg zeta function. The general philosophy The more we know about the zeta function the more we can deduce about the underlying properties of the numbers that are used in their definition. The blueprint for this comes from prime numbers and the Riemann zeta function, which we briefly recall. 2 / 30 (e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x ! +1 is given by the classical prime number theorem. Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896)) x π(x) We have that π(x) ∼ as x ! +1, (i.e., limx!+1 x = 1). log x log x The main tool to prove the Prime Number Theorem is the Riemann zeta function. Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov flows Properties of the Riemann zeta function Applications Counting Prime Numbers Consider the prime numbers 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; ··· Definition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x. 3 / 30 π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x ! +1 is given by the classical prime number theorem. Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896)) x π(x) We have that π(x) ∼ as x ! +1, (i.e., limx!+1 x = 1). log x log x The main tool to prove the Prime Number Theorem is the Riemann zeta function. Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov flows Properties of the Riemann zeta function Applications Counting Prime Numbers Consider the prime numbers 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; ··· Definition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x. (e.g., π(10) = 4, 3 / 30 π(1000) = 168, etc.) The growth of π(x) as x ! +1 is given by the classical prime number theorem. Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896)) x π(x) We have that π(x) ∼ as x ! +1, (i.e., limx!+1 x = 1). log x log x The main tool to prove the Prime Number Theorem is the Riemann zeta function. Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov flows Properties of the Riemann zeta function Applications Counting Prime Numbers Consider the prime numbers 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; ··· Definition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x. (e.g., π(10) = 4, π(100) = 25, 3 / 30 The growth of π(x) as x ! +1 is given by the classical prime number theorem. Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896)) x π(x) We have that π(x) ∼ as x ! +1, (i.e., limx!+1 x = 1). log x log x The main tool to prove the Prime Number Theorem is the Riemann zeta function. Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov flows Properties of the Riemann zeta function Applications Counting Prime Numbers Consider the prime numbers 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; ··· Definition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x. (e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) 3 / 30 The main tool to prove the Prime Number Theorem is the Riemann zeta function. Riemann Zeta Function Prime Number Theorem Selberg Zeta function The Riemann Zeta Function Geodesic and Anosov flows Properties of the Riemann zeta function Applications Counting Prime Numbers Consider the prime numbers 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; ··· Definition Given x > 0, let π(x) denote the number of primes numbers less than or equal to x. (e.g., π(10) = 4, π(100) = 25, π(1000) = 168, etc.) The growth of π(x) as x ! +1 is given by the classical prime number theorem. Theorem (Prime Number Theorem: Hadamard, de la Valle Poussin (1896)) x π(x) We have that π(x) ∼ as x ! +1, (i.e., limx!+1 x = 1).