Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4594365 | Journal of Number Theory | 2012 | 13 Pages |
Abstract
Let k be any real number with k<2. We will consider complex-valued smooth functions f,fË on H of period 1, having exponential decay at infinity (i.e. they are âªeâcy for y=â(z)ââ with c>0) and such that f|kWN=CfË+qg. Here |k is an appropriately defined Petersson slash operator in weight k, CâCâ is a constant andqg(z):=â«0iâg(Ï)(Ïâz¯)âkdϯ(zâH) is a period integral attached to a holomorphic function g:HâC such that both g and g|2âkWN have period 1, have only positive terms in their Fourier expansions and the Fourier coefficients are of polynomial growth. An arbitrary power of a non-zero complex number is defined by means of the principal branch of the complex logarithm. Under the assumption that k<1, we will show that the Mellin transform M(f,s) (Ïâ«1) naturally attached to f has meromorphic continuation to C and we will establish an explicit formula for it (Section 2, Theorem 1). There are possible simple poles at the points s=ân where n=0,1,2,⦠and the residue at s=ân essentially is equal to the “n-th period”â«0âg(it)tndt of g. Moreover, there again is a functional equation relating M(f,s) and M(fË,kâs).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
YoungJu Choie, Winfried Kohnen,