Distributions and integrals of L-functions on critical line

•Recall the Mellin transform of the distribution t(x) = ån¹ 0 an|n|-ne(nx)
is

•Let y be an
even, smooth function of compact support on R*. By
Parseval

for any s (integrand is entire, so the contour may be
shifted).

•If ye(x)
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
general irrational).

•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·T an e(nx).

•__Conclusion__: cancellation in additive sums is related to
moments.