On the sum of the first n prime numbers

In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.Comment: 5 page

Compact Operators via the Berezin Transform

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on the boundary of the disk. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz operators, compact Hankel operators, and appropriate products of these operators.Comment: 15 pages. To appear in Indiana University Mathematics Journal. For more information, see http://math.sfsu.edu/axler/CompactBerezin.htm