Riemann Hypothesis/The hypothesis

0 100% #FFFFFF none ;Theorem 1 Proof Consider the functional equation for Zeta, Notice that for , the sine term evaluates to which evaluates to 0 for all integers , hence for all natural . Definition 1 These zeroes are referred to as trivial zeroes. As a set, Zeroes that do not lie in this set are to be referred to as non-trivial zeroes. Note 1 The above argument cannot be applied to , as is a simple pole (), as are negative arguments of (). 0 100% #FFFFFF none ;Theorem 2 All non-trivial zeroes of have a real part that lies in the interval Theorem 3 Take the inequality, Using the definition of deduced in an earlier chapter, Taking the log of both sides, using Writing as a power series, Substituting , Taking the modulus of the argument, Substituting appropriate values, If one lets , it should become apparent that, Clearly implying, Exponentiating both sides, Let's assume that has a zero at , then, As gives a pole, and gives a zero, contradicting the previously stated inequality, proving theorem 3 by contradiction . Theorem 4 Using the functional equation, By theorem 3, the RHS is non-zero, hence as is the LHS. Theorems 3 and 4 are sufficient to imply theorem 2. The conjecture Riemann, knowing that all zeroes lied in the critical strip, postulated, 0 100% #FFFFFF none ;Conjecture All non-trivial zeroes of have a real part of The above conjecture is considered to be the most notable in pure mathematics, and the most notable of Riemann's works.