T2. 4. Perron's formula

Question 1

Define a Dirichlet series

Answer 1

A series of the form SUM_n=1,∞ a_n n^-s

where a_n is a sequence of complex numbers

Question 2

Define the abscissa of (absolute) convergence

Answer 2

The minimum (maximum) σ ∈[-∞,∞] such that the Dirichlet series converges (diverges) for Re(s) > σ_a.

Question 3

Relate the abscissa of absolute and convergence

Answer 3

σ_a + 1 ≤ σ_c ≤ σ_a

Question 4

State the truncated Perron’s formula

Answer 4

SUM_n≤x a_n = 1/2πi INT F(s) x^s ds/s

between c±iT for a choice of T<∞