Lecture 22

Question 1

Assume the series F(s) = ∑^∞(n=1) f(n)/n^s converges absolutely for σ > σa. Then for σ > σa and x > 0 find

• lim_(T →∞) (1/2T) ∫^(T)_(−T) F(σ + it) x^(σ+it) dt

Answer 1

• f(n) if x = n
• 0 otherwise

Question 1

Assume the series F(s) = ∑^∞(n=1) f(n)/n^s converges absolutely for σ > σa. Then for σ > σa and x > 0 find

• lim_(T →∞) (1/2T) ∫^(T)_(−T) F(σ + it) x^(σ+it) dt

Answer 1

• f(n) if x = n
• 0 otherwise

Question 2

Perron’s formula

Answer 2

Let F(s) = ∑^∞_(n=1) f(n)/n^s be absolutely convergent for σ > σ_a and let c > 0 and x > 0 be arbitrary. Then if σ > σa − c we have

• (1/2πi) ∫^(c+i∞)(c−i∞) F(s + z) (x^z/z) dz = ∑(n≤x)^(∗) f(n)/n^s

where ∑^(∗) means that the last term in the sum must be multiplied by 1/2 when x is an integer

Question 3

Perron’s formula if c > σ_a

Answer 3

(1/2πi) ∫^(c+i∞)(c−i∞) F(z)(x^z/z) dz = ∑(n≤x)^(∗) f(n).