The Riemann Hypothesis Essay Example for Free

The Riemann Hypothesis EssayThe Riemann Zeta mathematical influence is defined by the pursuance serial publicationHere s is a colonial number and the first obvious issue is to limit the domain of this survive, that is, the values of s where the subprogram is actually defined. First of all, it is a well known offspring in calculus that, when s is rattling, the series is merging(prenominal) for s1 (see 2). For example, a simple application of the theory of Fourier series allows to prove that . For s=1, the series diverges. However, one evict prove that the divergence is not too bad, in the sense thatIn fact, we restrain the in relateitiesSumming from 1 to , we find that and sowhich implies our claim.As a function of the real variable s, is decreasing, as illustrated below.for s real and 1The situation is more complicated when we consider the series as a function of a labyrinthian variable.Remember that a conglomerate number is a sum , where atomic number 18 real n umbers game (the real and the imaginary part of z, respectively) and , by definition. unity commonly writes There is no ordering on the complex numbers, so the above arguments do not catch sense in this setting. We remind that the complex power is defined byandTherefore, the power coincides with the usual function when s is real.It is not difficult to prove that the complex series is convergent if Re(s)1. In fact, it is absolutely convergent becausewhere z denotes as usual, the absolute value . See 2 for the general criteria for convergence of series of functions.Instead, it is a non-trivial proletariat to prove that the Riemann Zeta Function can be extended far beyond on the complex categoricalTheorem. There exists a (unique) meromorphic function on the complex plane, that coincides with , when Re(s)1. We will denote this function again byWe have to explain what meromorphic means. This means that the function is defined, and holomorphic (i.e. it is differentiable as a complex function), on the complex plane, except for a countable set of isolated points, where the function has a pole. A complex function f(z) has a pole in w if the limit exists and is finite for some integer m. For example, has a pole in s=1.It is particularly interested to evaluate the Zeta Function at negative integers. One can prove the following if k is a tyrannical integer thenwhere the Bernoulli numbers are defined inductively byNote that the Bernoulli numbers with odd index greater than 1 are equal to zero. Moreover, the Bernoulli numbers are all rational.Of course, the number is not obtained by replacing s=1-k in our accepted definition of the function, because the series would diverge in fact, it would be more appropriate to write where the superscript * denotes the meromorphic function whose values are defined, only when Re(s)1, by the series .There is a corresponding formula for the positive integers2It is a remarkable fact that the values of the Riemann Zeta Function at neg ative integers are rational. Moreover, we have seen that if n0 is even. The natural question arises are there any other zeros of the Riemann Zeta Function?Riemann Hypothesis. Every zero of the Riemann Zeta Function must be either a negative even integer or a complex number of real part = .It is hard to motivate this conjecture in an elementary setting, however the bring out point is that there exists a functional equation relating and (in fact, such a functional equation is scarce what is needed to extend to the complex plane). The point is the center of symmetry of the mapIt is also known that has continuously many zeros on the critical line Re(s)=1.Why is the Riemann Zeta function so important in mathematics? One reason is the strict connection with the distribution of prime numbers. For example, we have a celebrated crossway expansionwhere the infinite product is extended to all the prime numbers and Re(s)1. So, in some sense, the Riemann Zeta function is an analytically defi ned object, encoding virtually all the information about the prime numbers. For example, the fact that can be used to prove Dirichlets theorem on the existence of infinitely many prime numbers in arithmetic progression.The product expansion implies that for every s such that Re(s)1. In fact, we haveand it is not difficult to check that this product cannot vanish.The following beautiful picture comes from Wikipedia.Bibliography1 K. Ireland, M. Rosen, A Classical Introduction to ultramodern Number Theory, Springer, 20002 W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 19763 W. Rudin, Real and Complex Analysis, , McGraw Hill, 1986