 Exceptional-zero bound from a uniform upper bound in arithmetic progressions

ID: exceptional-zero-bound-from-a-uniform-upper-bound-in-arithmetic-progressions

If some

ϵ > 0

satisfies

ψ (x; q, a) \leq (2 - ϵ) \frac{x}{φ ( q )} (x \geq q^{2})

uniformly for sufficiently large

q

q

satisfies

β \leq 1 - \frac{c _{ϵ}}{( l o g q ) ^{2}} .

Choose

χ_{1} (a) = - 1

and

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

in the exceptional-zero asymptotic, with

A

large in terms of

ϵ

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