By M. Ram Murty, V. Kumar Murty
ISBN-10: 8132207696
ISBN-13: 9788132207696
Preface.- bankruptcy 1. The Legacy of Srinivasa Ramanujan.- bankruptcy 2. The Ramanujan tau function.- bankruptcy three. Ramanujan's conjecture and l-adic representations.- bankruptcy four. The Ramanujan conjecture from GL(2) to GL(n).- bankruptcy five. The circle method.- bankruptcy 6. Ramanujan and transcendence.- bankruptcy 7. mathematics of the partition function.- bankruptcy eight. a few nonlinear identities for divisor functions.- bankruptcy nine. Mock theta capabilities and ridicule modular forms.- bankruptcy 10. best numbers and hugely composite numbers.- bankruptcy eleven. Probabilistic quantity theory.- bankruptcy 12. The Sato-Tate conjecture for the Ramanujan tau-function.- Bibliography.- Index
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Preface. - bankruptcy 1. The Legacy of Srinivasa Ramanujan. - bankruptcy 2. The Ramanujan tau functionality. - bankruptcy three. Ramanujan's conjecture and l-adic representations. - bankruptcy four. The Ramanujan conjecture from GL(2) to GL(n). - bankruptcy five. The circle strategy. - bankruptcy 6. Ramanujan and transcendence. - bankruptcy 7.
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The survey article [141] provides an excellent introduction to this chain of ideas, and we recommend this to the reader. But the origins of the chain go back to the 1916 paper of Ramanujan which acted as a catalyst for this development. In his epic paper of 1916, Ramanujan [162] considered the function ∞ 1 − qn (z) = q 24 , q = e2πiz . n=1 As we have seen, expanding the right-hand side as a power series in q defines the celebrated Ramanujan τ -function: ∞ 1 − qn q n=1 24 ∞ = τ (n)q n . n=1 In his paper, Ramanujan made three conjectures concerning τ (n): (1) τ (mn) = τ (m)τ (n), for (m, n) = 1, (2) for p prime and a ≥ 1, τ (p a+1 ) = τ (p)τ (p a ) − p 11 τ (p a−1 ), (3) |τ (p)| ≤ 2p 11/2 .
One can renormalize and show that √ an (f ) yKir 2π |n|y e2πinx f (x, y) = a0 (f )y s + a0 (f )y 1−s + n=0 where Kir (y) = 1 2 ∞ −∞ e−y cosh t−irt dt 3 Upper Bound for Fourier Coefficients and Eigenvalue Estimates 47 with λ = 1/4 + r 2 . Maass [117] proved that the series n=0 an (f ) |n|s extends to a meromorphic function for all s ∈ C analytic everywhere except possibly at s = 0 and s = 1, and satisfies a functional equation. The analog of the Ramanujan conjecture in this context is that for any > 0, an (f ) = O(n ).
We now show that this is a contradiction. We do this by showing that some Fourier coefficient of E(z, (1 + it0 )/2) is non-zero. That is, we need to check ∞ e−πry(u+u −1 ) 0 du = 0. u1+it0 If we set u = eθ , we have to show that ∞ −∞ e−πry(e θ +e−θ )−it 0θ dθ = 0. In other words, it suffices to show that ∞ e−πry(e θ +e−θ ) cos t0 θ dθ = 0. 0 This integral is of the form ∞ 0 e−y(a θ +a −θ ) cos θ dθ, a > 1. 54 4 The Ramanujan Conjecture from GL(2) to GL(n) We would like to determine its behaviour as y tends to infinity.
Mathematical legacy of srinivasa ramanujan by M. Ram Murty, V. Kumar Murty
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