Bibliography

1: Christian Axler. On the sum of the first \(n\) prime numbers. J. Théor. Nombres Bordeaux, 31(2):293–311, 2019.
2: Sreerupa Bhattacharjee. A survey of Büthe’s method for estimating prime counting functions. Master’s thesis, University of Lethbridge, Lethbridge, Alberta, 2023.
3: Samuel Broadbent, Habiba Kadiri, Allysa Lumley, Nathan Ng, and Kirsten Wilk. Sharper bounds for the Chebyshev function \(\theta (x)\). Math. Comp., 90(331):2281–2315, 2021.
4: Jan Büthe. Estimating \(\pi (x)\) and related functions under partial RH assumptions. Math. Comp., 85(301):2483–2498, 2016.
5: Jan Büthe. An analytic method for bounding \(\psi (x)\). Math. Comp., 87(312):1991–2009, 2018.
6: Emanuel Carneiro, Micah B. Milinovich, and Kannan Soundararajan. Fourier optimization and prime gaps. Comment. Math. Helv., 94(3):533–568, 2019.
7: Yuanyou Cheng. An explicit upper bound for the Riemann zeta-function near the line \(\sigma =1\). Rocky Mountain J. Math., 29(1):115–140, 1999.
8: Andrés Chirre and Harald Andrés Helfgott. Optimal bounds for sums of non-negative arithmetic functions, 2025.
9: N. Costa Pereira. Estimates for the Chebyshev function \(\psi (x)-\theta (x)\). Math. Comp., 44(169):211–221, 1985.
10: Jean-Marie De Koninck and Patrick Letendre. New upper bounds for the number of divisors function. Colloq. Math., 162(1):23–52, 2020.
11: Juan Arias de Reyna. On the approximation of the zeta function by dirichlet polynomials. 2024.
12: Marc Deléglise and Jean-Louis Nicolas. The Landau function and the Riemann hypothesis. J. Comb. Number Theory, 11(2):45–95, 2019.
13: Harold G. Diamond. Elementary methods in the study of the distribution of prime numbers. Bull. Amer. Math. Soc. (N.S.), 7(3):553–589, 1982.
14: Daniele Dona, Harald A. Helfgott, and Sebastian Zuniga Alterman. Explicit \(L^2\) bounds for the Riemann \(\zeta \) function. J. Théor. Nombres Bordx., 34(1):91–133, 2022.
15: Adrian W. Dudek. An explicit result for primes between cubes. Funct. Approx. Comment. Math., 55(2):177–197, 2016.
16: Adrian W. Dudek and David J. Platt. On solving a curious inequality of Ramanujan. Exp. Math., 24(3):289–294, 2015.
17: Pierre Dusart. Inégalités explicites pour \(\psi (X)\), \(\theta (X)\), \(\pi (X)\) et les nombres premiers. C. R. Math. Acad. Sci. Soc. R. Can., 21(2):53–59, 1999.
18: Pierre Dusart. Explicit estimates of some functions over primes. Ramanujan J., 45(1):227–251, 2018.
19: Laura Faber and Habiba Kadiri. New bounds for \(\psi (x)\). Math. Comp., 84(293):1339–1357, 2015.
20: Laura Faber and Habiba Kadiri. Corrigendum to New bounds for \(\psi (x)\) [MR3315511]. Math. Comp., 87(311):1451–1455, 2018.
21: Andrew Fiori, Habiba Kadiri, and Joshua Swidinsky. Sharper bounds for the Chebyshev function \(\psi (x)\). J. Math. Anal. Appl., 527(2):Paper No. 127426, 28, 2023.
22: Andrew Fiori, Habiba Kadiri, and Joshua Swidinsky. Sharper bounds for the error term in the prime number theorem. Res. Number Theory, 9(3):Paper No. 63, 19, 2023.
23: G. H. Hardy and J. E. Littlewood. The zeros of Riemann’s zeta-function on the critical line. Math. Z., 10:283–317, 1921.
24: H. A. Helfgott. The ternary goldbach conjecture is true, 2013.
25: Daniel R. Johnston and Andrew Yang. Some explicit estimates for the error term in the prime number theorem. J. Math. Anal. Appl., 527(2):23, 2023. Id/No 127460.
26: Habiba Kadiri. A zero density result for the Riemann zeta function. Acta Arith., 160(2):185–200, 2013.
27: Habiba Kadiri and Allysa Lumley. Short effective intervals containing primes. Integers, 14:Paper No. A61, 18, 2014.
28: Habiba Kadiri, Allysa Lumley, and Nathan Ng. Explicit zero density for the Riemann zeta function. J. Math. Anal. Appl., 465(1):22–46, 2018.
29: Jean-Pierre Massias and Guy Robin. Bornes effectives pour certaines fonctions concernant les nombres premiers. J. Théor. Nombres Bordeaux, 8(1):215–242, 1996.
30: Hugh L. Montgomery and Robert C. Vaughan. Multiplicative number theory. I. Classical theory, volume 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007.
31: Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi. Empirical verification of the even Goldbach conjecture and computation of prime gaps up to \(4\cdot 10^{18}\). Math. Comp., 83(288):2033–2060, 2014.
32: David J. Platt and Timothy S. Trudgian. The error term in the prime number theorem. Math. Comp., 90(328):871–881, 2021.
33: O. Ramaré. An explicit density estimate for Dirichlet \(L\)-series. Math. Comp., 85(297):325–356, 2016.
34: Olivier Ramaré. Explicit estimates for the summatory function of \(\Lambda (n)/n\) from the one of \(\Lambda (n)\). Acta Arith., 159(2):113–122, 2013.
35: Olivier Ramaré and Yannick Saouter. Short effective intervals containing primes. J. Number Theory, 98(1):10–33, 2003.
36: Jörg Richstein. Verifying the Goldbach conjecture up to \(4\cdot 10^{14}\). Math. Comp., 70(236):1745–1749, 2001.
37: Guy Robin. Estimation de la fonction de Tchebychef \(\theta \) sur le \(k\)-ième nombre premier et grandes valeurs de la fonction \(\omega (n)\) nombre de diviseurs premiers de \(n\). Acta Arith., 42(4):367–389, 1983.
38: Barkley Rosser. The n-th Prime is Greater than nlogn. Proc. London Math. Soc. (2), 45(1):21–44, 1939.
39: Barkley Rosser. Explicit bounds for some functions of prime numbers. Amer. J. Math., 63:211–232, 1941.
40: J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois J. Math., 6:64–94, 1962.
41: Yannick Saouter, Timothy Trudgian, and Patrick Demichel. A still sharper region where \(\pi (x)-{\rm li}(x)\) is positive. Math. Comp., 84(295):2433–2446, 2015.
42: Aleksander Simonic. Explicit zero density estimate for the Riemann zeta-function near the critical line. J. Math. Anal. Appl., 491(1):124303, 41, 2020.
43: Gérald Tenenbaum. Introduction to analytic and probabilistic number theory. Transl. from the 3rd French edition by Patrick D. F. Ion, volume 163 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 3rd expanded ed. edition, 2015.
44: E. C. Titchmarsh. The theory of the Riemann zeta-function. The Clarendon Press, Oxford University Press, New York, second edition, 1986. Edited and with a preface by D. R. Heath-Brown.
45: Enrique Treviño. The least inert prime in a real quadratic field. Math. Comp., 81(279):1777–1797, 2012.
46: Tim Trudgian. Updating the error term in the prime number theorem. Ramanujan J., 39(2):225–234, 2016.
47: Ramdinmawia Vanlalngaia. Explicit Mertens sums. Integers, 17:Paper No. A11, 18, 2017.