科研之友
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摘要
Let k be any real number with k %26lt; 2. We will consider complexvalued smooth functions f, %26lt;(f)over tilde%26gt; on H of period 1, having exponential decay at infinity (i.e. they are %26lt;%26lt; e(-cy) for y = I(z) -%26gt; infinity with c%26gt; 0) and such that f |(k) W(N) = C (f) over tilde + q(g). Here |(k) is an appropriately defined Petersson slash operator in weight k, C is an element of C* is a constant and %26lt;br%26gt;q(g)(z) := integral(i infinity)(0) g(tau)(tau - (z) over bar)(-k) d tau (z is an element of H) %26lt;br%26gt;is a period integral attached to a holomorphic function g : H -%26gt; C such that both g and g|(2-k)W(N) 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 %26lt; 1, we will show that the Mellin transform M(f, s) (sigma %26gt;%26gt; 1) naturally attached to I 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 = -n where n = 0, 1, 2, ... and the residue at s = -n essentially is equal to the %26quot;n-th period%26quot; %26lt;br%26gt;integral(infinity)(0) g(it)t(n) dt %26lt;br%26gt;of g. Moreover, there again is a functional equation relating m(f, s) and M((f) over tilde, k - s).
- 出版日期2012-2
