不知名的碎片 14

L-function:

Autocorrelation function, I

Ramanujan (1915), Lewis & Zagier (2001), Bettin & Conrey (2013), Darses & Hillion (2021):

Analytic continuation to

Moments by differentiation of at (so, at ).

Autocorrelation function, II

Laplace transform of a Dirichlet character

holomorphic extension to

meromorphic extension to with simple (or removable) poles at , residue

Laurent series expansion in:

(pole); if (removable).

A probability measure

Probability measure

Moments given by Eulerian numbers:

Determination of the measure by its moments (Hamburger moments problem).

Related work

Theorem (Darses & Hillion, 2024, Acta Arith.)