Distributions and integrals of L-functions on critical line
•Recall the Mellin transform of the distribution t(x) = ån¹0an|n|-ne(nx)
•Let y be an
even, smooth function of compact support on R*.By
•for any s(integrand is entire, so the contour may be
is an approximate identity (near x = 1), My(1/2+it) approximates the (normalized) characteristic function of the interval t 2 [-1/e,1/e].
•One can therefore learn the size of smoothed integrals of Mt(1/2+it) through
properties of the distribution t(x) near x = 1.
–When t vanishes to infinite order
near x = 1, these smoothed integrals are very small.
–This is related to cancellation in S(T,x) for particular values of x (in
this case rational, but in
•Similarly, the multiplicative convolution tFy has Mellin transform Mt(s)*My(s).Its L2-norm approximates the second moment of L(1/2+it), and is determined by the L2-norm of tFy.The latter is
controlled by the size of
smooth variants of S(T,x) = ån·Tane(nx).
•Conclusion: cancellation in additive sums is related to