Explicit bounds for primes in arithmetic progressions

Michael A. Bennett, Greg Martin, Kevin O'Bryant & Andrew Rechnitzer

Note as of 20th of November 2018

Updated manuscript

• Follow this link to the updated manuscript.

Note as of 21st of December 2018

• Most of the data files and code are updated - as indicated below.
• Hope to complete the remaining updates soon.

Theorems 1.1, 1.2 & 1.3 - bounds on psi(x;q,a), theta(x;q,a) and pi(x;q,a) for x≥ x_0(q)

• See Appendices A.4 & A.6 (additional notes in A.5).
• Table of constants c_psi(q), c_theta(q), c_pi(q)
• Subdirectory containing code and data for computing c_psi(q), c_theta(q), c_pi(q)
• Table of constants x_psi(q), x_theta(q), x_pi(q) and x_m(q), x_e(q)
• Table of constants x_m(q) and x_e(q)
• Subdirectory containing code and data for computing x_psi(q), x_theta(q), x_pi(q)

Corollary 1.6 - bounds on pi(x;q,a) and p_n(q,a)

• See Appendix A.7.
• Subdirectory containing data
• Subdirectory containing code

Corollary 1.7 - bounds on psi(x;q,a), theta(x;q,a) and pi(x;q,a) for x≥ 1000 and x≥ 10^6

• See Appendix A.8.
• Subdirectory containing data
• Subdirectory containing code

Corollary 1.8 - bounds on psi(x;q,a), theta(x;q,a) and pi(x;q,a) for all q

• See Appendix A.9.
• Subdirectory containing code and data

Proposition 1.10 and Lemma 6.3 - Lower bound on L(1,chi)

• See Appendix A.10.
• Subdirectory containing data and sage code
• Note that Prop 1.10 and Lemma 6.3 were Prop 1.9 and Lemma 5.3 in an earlier version of the paper.

Proposition 2.5 - bound on N_chi(T)

• See Appendix A.1.
• Subdirectory containing data and sage code
• Note that Prop 2.5 was Prop 2.6 in an earlier version of the paper.

Appendix A.2 - computation of nu_2(q)

• Subdirectory containing nu data, which is used to compute c_psi(q), c_theta(q), c_pi(q)

Appendix A.3 - computation of worst-case error bounds for small x

• Table of constants b_psi(q), b_theta(q), b_theta#(q) and b_pi(q)
• Subdirectory with code and data