Lindelöf conjecture and moment estimates

•__Lindel____ö____f conjecture:__ L(1/2+it) = Oe((1+|t|)e) for any

e > 0.

e > 0.

–Fundamental unsolved conjecture in analytic number theory.

–Implied by GRH.

–Equivalent to moment bounds:

s-TT |L(½+it)|2k dt = Oe(T1+e) for each fixed k ¸ 1.

s-TT |L(½+it)|2k dt = Oe(T1+e) for each fixed k ¸ 1.

•The 2k-th moment for a cusp form on GL(d) is thought to be exactly as difficult to the 2nd moment on
GL(dk).

•The cancellation conjecture – or more precisely a variant for non-cusp forms – implies the Lindelöf conjecture
(next slide), and is thus a very hard problem for d > 2.