Let r v (N) denote the number of representations of the integer N as a sum
of v square-free numbers. We obtain unconditional and conditional bounds for
the error term in the asymptotic formula for rv (N), when v > 3. The conditional
bounds are essentially best possible for v > 4. The unconditional bounds are,
for v > 3, essentially best possible with respect to the present knowledge on
the distribution of the zeros of the Riemann zeta function. Proofs are based on
the circle method. The main ingredients are a new pointwise estimate for the
exponential sum S(a) over square-free numbers and a recent bound (see [3]) for
the L2-norm of S(a) restricted to the minor arcs.
1. A. Weil [3] constructed a universal distribution t on the Weil group.The values of I at various test functions give the contributions from the zeros of L-functions which appear in the .explicitformulas.In this note, we shall construct a universal distribution zi on GL(n) and prove the explicit formula for automorphic L-functions using z/ when n-2.For n 2, to derive such a result, we must assume certain property of characters of infinite dimensional representations of GL(n) over a local field.This property, formulated as Conjecture, seems to lie slightly beyond our present knowledge of harmonic analysis.The distributions A have striking formal resemblance to Weil's one.Furthermore they are related to each other so that zl is the "direct image" of z/ for m n.This is a pleasant fact since we think that a discovery of new functorial properties related to zeros of zeta functions would be crucial for the proof of the Riemann hypothesis.