The Riemann Hypothesis

Transcription

1 The Riemann Hypothesis by Pedro Pablo Pérez Velasco, 2011

2 Introduction Consider the series of questions: How many prime numbers (2, 3, 5, 7,...) are there less than 100, less than , less than ? How many less than any given number X? According to Zagier, astonishingly: Randomness: Where will the next prime sprout? Regularity: Precise laws govern their behaviour. Primes are, multiplicatively speaking, the building blocks from which all numbers can be made. Reason: Any number can be uniquely factored as a product of primes (except for the order). Primes relate additive and multiplicative structures.

3 Introduction Simple unknown questions about primes: Are there infinitely many pairs of primes whose difference is 2? or 4? or 2k? Is every even number greater than 2 a sum of two primes? (Goldbach's conjecture) Are there infinitely many primes which are 1 more than a perfect square? Is there some neat formula giving the next prime?

4 Sieves Techniques to estimate the size of sets of integers Erathostenes (remove multiples of primes), Atkin, Selberg, Turán, Sundaram, Legendre, Brun... Objective: Find a smooth curve that is a reasonable approximation to the staircase of primes x =number of primes less than x

5 First Approach to x Probabilistic approach due to Gauss G(x) is proportional to x divided by the number of digits of x, i.e. x x log x x x G x 1e e e e G

6 The Riemann Hypothesis First formulation If x = x 1 Li x = 2 log t dt x log x O x1 /2 log x, a second formulation would be x =Li x O x 1/ 2 log x Notice that Li x =O x log x

7 The Prime Number Theorem It is the asymptotic law of distribution of primes x = x log x o x log x or equivalently x =Li x o x/log x Previous work by Legendre, Gauss, Chebyshev,... The theorem was first proved by Hadamard and de la Vallée Poussin in 1896, extending ideas of Riemann. Many different proofs currently known. Erdös and Selberg gave an elementary proof (1949, though long) and Newman a short proof (1980, not elementary).

8 Fourier Analysis Consider the trigonometric function f t =a cos b t where a is the amplitude and is the frequency.

9 Fourier Analysis Using linear combinations of basic trigonometric functions we can approximate/define general functions f t = n=0 a n cos n t One interesting point is that by keeping track of the amplitudes and the spectrum (frequencies) we can encode most reasonable functions => compression The operation that starts with a graph and goes to its spectral picture is the Fourier transform.

10 Fourier Analysis cos t How much occurs in f(t)? This is the amplitude (Fourier transform) and is given by: f f = f t cos t dt The spectrum of f(t) is the set of frequencies where the amplitude is nonzero. The inverse operation (inverse Fourier transform) is given by f t = f cos t d

11 Distributions An integrable function is one which has values and areas under its graph. A distribution D(t) has areas under its graph (previous first condition relaxed) so it makes sense to write b a D t The space of distributions was formaly defined and first studied by Laurent Schwartz. The most prominent distribution is the Dirac function (which is not a function!)

12 Distributions The Dirac distribution enjoys many properties: It is the derivative of the Heaviside function Kinda integral evaluation: f x = f t x t dt NB: The delta function can be rigurously defined as a measure, not absolutely continuous wrt the Lebesgue measure (previous formula is an abuse of notation). Its derivative can be easily calculated using integration by parts:. 0 ' [ ]= 0 [ ' ]= ' 0 Distributions are closed under derivations.

13 Distributions Its Fourier transform is 1, and it is self-adjoint: = x e 2 i x dx=1 x = 1, =, The delta function is an example of white noise: Every frequency occurs in its Fourier series and they al occur in equal amounts The spike function is Fourier transform: d x t = x t x t /2 with d x =cos x

14 From the Primes to the Spectrum All the information of the primes staircase is not in the jumps but in where those jumps happen. x Consider the function with jumps of height log(p) for every x which is a power of p. Another formulation of the Riemann hypothesis would be: x =x O x 1 /2 log 2 x We further distort this function: x = e x

15 From the Primes to the Spectrum The process ends by symmetrizing, rescaling the function (multiplication by negative exponential) and introducing Dirac deltas in the powers of primes: t =e t/ 2 d t dt

16 From the Primes to the Spectrum n Let q = p be such that q is less than C, then C t =2 q p n /2 log p d n log p t spike function We are just considering the first C linear combinations of d x t functions. Its Fourier transform is C t =2 q p n /2 log p cos nlog p The coordinates seem to be clustered about a discrete set of positive real numbers i. They are known as the spectrum of the primes.

17 From the Primes to the Spectrum

18 From the Spectrum to the Primes To recover the primes out of the spectrum, consider the Fourier series with spectrum i represented in a logarithmic scale: H C s =1 n C cos n log s

19 Riemann's way to build x Can we construct solely from? Riemann's guess for an approximation to n x R x = n=1 n n Li x1 /n { n } n N x where is the Möbius function, with value 1 if n has an even number of distinct prime factors, -1 if the number of distinct primes is odd and 0 if not square-free. is

20 Riemann's way to build x Riemann's R(x) is a (conjecturally) much better approximation than Li(x) or G(x). Think of R(x) as the fundamental approximation to. x Riemann gave an infinite sequence of guesses, all of them are (conjecturally) square root approximations: 1 2 R j x =R j 1 x R x i j 1 x =R x R x 2 i j exact formula j=1

21 Riemann's way to build x

22 Riemann's way to build x

23 Riemann's way to build x

24 Gracias por la atención Prácticamente todo el material se ha cogido (sin permiso) del libro de B. Mazur y W. Stein What is Riemann's Hypothesis? disponible en la url