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The Riemann Hypothesis by Pedro Pablo Pérez Velasco, 2011 Introduction Consider the series of questions: ● How many prime numbers (2, 3, 5, 7, ...) are there less than 100, less than 10.000, less than 1.000.000? ● 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. 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? 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 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 −G 1e02 25.75 25 0.75 1e03 166.5 167 -0.5 1e06 78626 78498 126 3e06 216745 216970 -135 The Riemann Hypothesis ● ● First formulation If x Li x =∫2 x 1 /2 x= O x log x log x 1 dt , a second formulation would be log t x=Li xO x Notice that x Li x =O log x 1/ 2 log x The Prime Number Theorem ● It is the asymptotic law of distribution of primes x x x= o log x log x or equivalently x=Li xo 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). ● Fourier Analysis ● Consider the trigonometric function f t=a cos bt where a is the amplitude and is the frequency. Fourier Analysis Using linear combinations of basic trigonometric functions we can approximate/define general functions ● ∞ f t=∑ a n cos n t n=0 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. ● Fourier Analysis How much „ cos t “ occurs in f(t)? This is the amplitude (Fourier transform) f and is given by: ● ∞ 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 −∞ 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 ● ∫ Dt a 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!) ● Distributions The Dirac distribution enjoys many properties: It is the derivative of the Heaviside function. ● ● 02 1 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. 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 d x t= x t x −t / 2 with Fourier transform: ● d x =cos x From the Primes to the Spectrum All the information of the primes staircase is not in the jumps but in where those jumps happen. ● Consider the function x with jumps of height log(p) for every x which is a power of p. ● Another formulation of the Riemann hypothesis would be: ● 1 /2 x=xO x ● 2 log x We further distort this function: x x= e 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/ 2 t=e d t dt From the Primes to the Spectrum ● Let q = p n be such that q is less than C, then C t =2 ∑ p−n /2 log p d n log p t spike function q We are just considering the first C linear combinations of d x t functions. Its Fourier transform is ● C t =2 ∑ p−n /2 log p cos n log p q The coordinates seem to be clustered about a discrete set of positive real numbers i . They are known as the spectrum of the primes. ● From the Primes to the Spectrum 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 ∑ cosn · log s nC Riemann's way to build x x solely from {n}n ∈ℕ? ● Can we construct ● Riemann's guess for an approximation to x is ∞ n 1 /n R x=∑ Li x n n=1 where n 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. 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: ● R j x=R j−1 x −R x ∞ x=R x −∑ R x j=1 1 i j 2 1 i j 2 exact formula Riemann's way to build x Riemann's way to build x Riemann's way to build x 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 http://wstein.org/rh/rh/rh.pdf