On Absolutely Convergent Exponential Sums

On Absolutely Convergent Dirichlet Series

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

Note on Absolutely Convergent Dirichlet Series

is it true that \f(s)\ s^k>0 for cr^O implies that (/(s))-1 is also of the form (1)? In this note, an affirmative answer is supplied.3 Let P be the semigroup of positive integers under multiplication, and let h(P) be the class of all complex functions a on P, a= {an}»~i, for which ||a|| = y^°-i \an\ is finite. We obtain a commutative Banach algebra by defining (aa)n = aan for complex a, (a+b)n ...

Exponential Sums on A

Let d = degree of f and write f = f (d) + f (d−1) + · · ·+ f , where f (j) is homogeneous of degree j. A by now classical theorem of Deligne[2, Théorème 8.4] says that if (p, d) = 1 and f (d) = 0 defines a smooth hypersurface in P, then L(A, f ; t) n+1 is a polynomial of degree (d − 1), all of whose reciprocal roots have absolute value equal to q. This implies the estimate |S(A(Fqi), f)| ≤ (d− ...

On Exponential Sums

Exponential Sums On

where f (j) is homogeneous of degree j. Theorem 1.4. Suppose (p, δ) = 1 and f (δ) = 0 defines a smooth hypersurface in P. Then L(A, f ; t) n+1 is a polynomial of degree (δ − 1), all of whose reciprocal roots have absolute value q. For exponential sums on A, several generalizations of Deligne’s result have been proved ([1, 2, 5, 7]). In all these theorems, the hypothesis implies that f , regarde...