Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 4 / e

Solution

Assume

β

exists and choose a reduced residue class

a

with

χ_{1} (a) = - 1

; such a class exists because

χ_{1}

is nonprincipal. Fix a sufficiently large constant

A = A (ϵ)

and take

x = exp (A (lo g q)^{2}) .

Then

x \geq q^{2}

for large

q

. Multiplying the formula from part (d) by

φ (q) / x

gives

\frac{φ ( q )}{x} ψ (x; q, a) = 1 + \frac{x ^{β - 1}}{β} + O (φ (q) (lo g q)^{2} exp [- \frac{c A l o g q}{1 + A}]) .

Choose

A

so large that the error is at most

ϵ /4

for all sufficiently large

q

. The assumed upper bound then implies

\frac{x ^{β - 1}}{β} \leq 1 - \frac{3 ϵ}{4} .

Since

β < 1

, this gives

x^{β - 1} \leq 1 - \frac{3 ϵ}{4} .

(1 - β) A (lo g q)^{2} \geq - lo g (1 - \frac{3 ϵ}{4}) .

Therefore, with a positive constant depending only on

ϵ

β \leq 1 - \frac{c}{( lo g q ) ^{2}} .

There are no discussions about this article yet.

There are currently no matching articles.