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We give non‐trivial bounds for the bilinear sums∑   u  =  1   U∑   v  =  1   Vα  u    β  v    e  p    (   u  /  f   (  v  )    )   ,wheree  p    (  z  )is a non‐trivial additive character of the prime finite field    F  p    of   p   elements, with integers   U  ,   V  , a polynomial    f  ∈   F  p    [  X  ]and some complex weights{    α  u    }    ,{    β  v    }    . In particular, for    f  (  X  )  =  a  X  +  b   , we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of    F  p   .

ON BILINEAR EXPONENTIAL AND CHARACTER SUMS WITH RECIPROCALS OF POLYNOMIALS