 Prime number theorem in an arithmetic progression with an exceptional zero

ID: prime-number-theorem-in-an-arithmetic-progression-with-an-exceptional-zero

Uniformly for

x \geq 2

and

(a, q) = 1

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

where the second term occurs only when a real character

χ_{1}

β

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