1	Welcome
2	Prologue In August 1859, a 32-year-old, timid, bashful, sensitive, diffident soul, with a horror for speaking in public, was presenting a paper to the Berlin Academy about the density of prime numbers on the real line. The brilliant mathematician giving the historic lecture was Bernhard Riemann, and in the course of the talk he made an incidental remark, which to this day has remained an enigma, known as the Riemann hypothesis. The hypothesis simply states that Riemann zeta function, an elegantly expressed analytic function, has all the complex zeros on a vertical line in the complex plane.
3	Where are the zeros of zeta of s? a poem gifted by Tom M. Apostol presentation by: Kannan Nambiar (click anywhere on the slide to start)
4	verse 1 Where are the zeros of zeta of s? G.F.B Riemann has made a good guess: "They're all on the critical line," stated he, "And their density's one over two pi log T."
5	verse 2
6	verse 3 The efforts of Landau and Bohr and Cramer, Hardy and Littlewood and Titchmarsh are there. In spite of their effort and skill and finesse, In locating the zeros there's been no success.
7	verse 4 In 1914 G.H.Hardy did find, An infinite number that lie on the line. His theorem, however, won't rule out the case, That there might be a zero at some other place.
8	verse 5 Let P be the function pi minus Li, The order of P is not known for x high. If square root of x times log x we could show, Then Riemann's conjecture would surely be so.
9	verse 6 Related to this is another enigma, Concerning the Lindelof function mu sigma, Which measures the growth in the critical strip; On the number of zeros it gives us a grip.
10	verse 7 But nobody knows how this function behaves. Convexity tells us it can have no waves. Lindelof said that the shape of its graph Is constant when sigma is more than one-half.
11	verse 8 Oh, where are the zeros of zeta of s? We must know exactly. It won't do to guess. In order to strengthen the prime number theorem, The integral's contour must never go near 'em.
12	verse 9 Andre Weil has improved on old Riemann's fine guess By using a fancier zeta of s. He proves that the zeros are where they should be, Provided the characteristic is p.
13	verse 10 There is a moral to draw from this long tale of woe That every young genius among you must know: If you tackle a problem and seem to get stuck, Just take it mod p and you'll have better luck.
14	The End