Explicit bounds for primes in arithmetic progressions
| Autor: | Bennett, Michael A., Martin, Greg, O'Bryant, Kevin, Rechnitzer, Andrew |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $q$). 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 $q\le1200$. 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 exceptional zeros. Comment: 103 pages. We implemented an improvement in the method in Section 2 (which produces the bound nu(q)), resulting in a change to most of the constants in our theorems. To appear in Illinois J. Math. Results of computations, and the code used for those computations, can be found at: http://www.nt.math.ubc.ca/BeMaObRe/ |
| Databáze: | arXiv |
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