Lecture 19

Question 1

What is an Euler Product

Answer 1

Let f be a multiplicative arithemetical function such that the series ∑f(n) is absolutely convergent. Then the sum of the series can be expressed as an absolute convergent
infinite product

• ∑_(n=1)^∞ f(n) = ∏_p (1 + f(p) + f(p^2) + . . . )

extended over all primes

Question 2

Euler product for completely multiplcative function

Answer 2

∑_(n=1)^∞ f(n) = ∏_p 1/(1 − f(p))

Question 3

∑_(n=1) f(n)/n^s

If f is multiplicative express as an Euler product

Answer 3

∏_p (1 + f(p)/p^s + f(p^2)/p^(2s) + . . . ) if σ > σa

Question 4

∑_(n=1) f(n)/n^s

If f is completely multiplicative express as an Euler product

Answer 4

∏_p (1 − f(p)/p^s)^(−1) if σ > σa.

Question 5

Let s0 = σ0 + it0 and assume that the Dirichlet series ∑f(n)/n^(s_0) has bounded partial sums, say

• ∣∑_(n≤x) f(n)/n^(s_0)∣ ≤ M, for all x ≥ 1. Then for each σ > σ0, give the bound of
• ∣ ∑_(a<n≤b) f(n)/n^s∣

Answer 5

∣ ∑_(a<n≤b) f(n)/n^s∣ ≤ 2M a^(σ_0−σ) (1 + ∣s − s0∣/(σ − σ_0))

Question 6

If the series ∑ f(n)/n^s coverges for s = σ0 +it0 what can be said about convegrence of σ > σ0

Answer 6

It converges for all σ > σ0

Question 7

If the series ∑ f(n)/n^s diverges for s = σ0 +it0 what can be said about convegrence of σ < σ0

Answer 7

It diverges for these values

Question 8

What is the abscissa of convergence

Different to the abscissa of absolute convergence

Answer 8

If the series ∑f(n)/n^s does not converge everywhere or diverge everywhere, then there exists a real number σc, called the abscissa of convergence, such that the series converges
for all s in the half-plane σ > σ_c and diverges for all s in the half-pane σ < σ_c

Question 9

Give the bounds of inequality for σ_a − σ_c

Answer 9

For any Dirichlet series with σ_c finite, we have
0 ≤ σ_a − σ_c ≤ 1