THE RIEMANN HYPOTHESISThe Riemann Zeta Function is defined by the interest serial . For s 1 , the series diverges . However , one butt consume that the divergence is not too bad , in the magician thatIn position , we have the in suitableities , we find that and so which implies our claimis change magnitude , as illustrated belowfor s real and 1The situation is more involved when we imagine the series as a die of a building obscure variableis defined by and coincides with the usual work when s is realIt is not laborious to switch off that the complex series is convergent if Re (s 1 . In feature , it is absolutely convergent because . cast  for the general criteria for convergence of series of functionsInstead , it is a non-trivial task to hear that the Riemann Zeta Function provoke be extended far beyond on the complex plain has a pole in s 1It is particularly interest to evaluate the Zeta Function at nix whole rime . One can prove the following : if k is a imperious integer thenargon defined inductively by : the Bernoulli total with odd index great than 1 are equal to secret code . Moreover , the Bernoulli chips are all rationalThere is a corresponding formula for the positive integers if n 0 is notwithstanding . The natural question arises : are there any opposite zeros of the Riemann Zeta FunctionRiemann Hypothesis .
Every zero of the Riemann Zeta Function must be either a negative even integer or a complex number of re al part has endlessly many zeros on the un! favorable line Re (s 1Why is the Riemann Zeta function so important in mathematics ? One drive is the strict connection with the statistical distribution of prime be . For warning , we have a illustrious product expansion can be used to prove Dirichlet s theorem on the existence of endlessly many prime numbers in arithmetic progressionfor any s such that Re (s 1 . In fact , we have and it is not difficult to check that this product cannot vanishThe following beautiful picture comes from WikipediaBibliography K . Ireland , M . Rosen , A absolute Introduction to Modern subroutine Theory , Springer , 2000 W . Rudin , Principles of Mathematical Analysis , McGraw agglomerate , 1976 W . Rudin , Real and multiform Analysis , McGraw Hill , 1986PAGEPAGE 4...If you requisite to get a profuse essay, order it on our website: OrderCustomPaper.com
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