## Explicit bounds for primes in arithmetic progressions

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

### Note as of 20th of November 2018

### 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)

Updated 7 December 2018.

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

Updated 7 December 2018.

### 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

Updated 7 December 2018.

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

Still needs updating as of 21 December 2018.

### 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.

Updated 21 December 2018.

### 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.

Updated 21 December 2018.

### 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)

Updated 21 December 2018.

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

Unchanged - 7 December 2018.

Last updated = 21 December, 2018.