Explicit bounds for primes in arithmetic progressions

27 Nov 2018 Bennett Michael A. Martin Greg O'Bryant Kevin Rechnitzer Andrew

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{160} \frac{x}{\log x}, $$ for all $x \ge 8 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$)... (read more)

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