Abstract
We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function Mf(s) so that we have Mell(φ ⋆ f,s) = Mell(φ,s)Mf(s) for all test functions φ in C 1 c (R � ) or C 1 c (Q � p). We show that if f is a non degenerate second degree character on R or Qp, as defined by Weil, then the weak Mellin transform of f satisfies a functional equation and cancels only for ℜ(s) = 1 . We then show that if f is a non degenerate second degree character defined on the adele ring AQ, the same statement is equivalent to the Riemann hypothesis. Various generalizations are provided.
Highlights
It has been a standard practice, since Riemann’s 1859 paper [5], when considering the zeta function ζ(s) n≥1 1 ns p −1 ps to add a “gamma factor” Γ( s 2 s )in order to build a“completed zeta function” π2 Ξ(s)
We show that if f is a non degenerate second degree character defined on the adele ring AQ, the same statement is equivalent to the Riemann hypothesis
The Mellin transforms of these functions are not well defined in the usual sense, but it is possible to extend the definition of the Mellin transform using the fact that if f ⋆ g is the multiplicative convolution product, we have Mell(f ⋆ g) = Mell(f ) Mell(g) : We say that a function f on R or Qp has a well defined “weak Mellin transform” at the character |x|s if there exists a function Mf (s) so that for any smooth test function φ with compact support in the multiplicative group R∗ or Q∗p, we have the equality Mell(φ ⋆ f, s) = Mell(φ, s)Mf (s)
Summary
It has been a standard practice, since Riemann’s 1859 paper [5], when considering the zeta function ζ(s) n≥1. It is immediate that if φ is a function in Cc∞(GLn(L)) satisfying the same condition, we have Using this formula, it is possible to give a definition for the Mellin transform of a non degenerate second degree character on Ln similar to the definition given for n = 1. If we note ζf (s) the weak Mellin transform of a non degenerate second degree character f defined on Ln and if the endomorphism associated to f is a dilation, ζf (s) and ζf (n − s) are related by a and all the zeroes of ζfv the axis. If we note ζf (s, π) the weak Mellin transform of a non degenerate second degree character f defined on Rn in this way and if the endomorphism associated to f is a dilation, we prove again that of ζfv (s, π). If μ is an element of S′(L), the weak Fourier transform of μ is defined, following Schwartz, by the usual formula
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