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Explicit bounds for primes in arithmetic progressions. (arXiv:1802.00085v1 [math.NT])
来源于:arXiv
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 $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{3600}
\frac q{\phi(q)} \frac{x}{\log x}, $$ for all $x \geq 7.94 \cdot 10^9$ (with
sharper constants obtained for individual such moduli $q$). We establish
inequalities of the same shape for the other standard prime-counting functions
$\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime
congruent to $a\pmod q$ when $q\le4500$. For moduli $q>10^5$, we find even
stronger explicit inequalities, but only for much larger values of $x$. Along
the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for
quadratic characters $\chi$, and an improved explicit upper bound for
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