Bibliography
- 1
Harald Andrés Helfgott, The ternary Goldbach problem, Snapshots of Modern Mathematics from Oberwolfach 2014-03. 2014 English snapshot, DOI 10.14760/SNAP-2014-003-EN. Primary locators: https://doi.org/10.14760/SNAP-2014-003-EN; https://publications.mfo.de/handle/mfo/429.
- 2
G. H. Hardy and J. E. Littlewood, Some problems of Partitio numerorum; III: On the expression of a number as a sum of primes, Acta Math. 44, 1–70. version of record. Primary locators: https://doi.org/10.1007/BF02403921; https://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203038474495327.
- 3
I. M. Vinogradov, Representation of an odd number as a sum of three primes, Doklady Akademii Nauk SSSR 15, 291–294. original 1937 Russian publication. Primary locator: https://www.mathnet.ru/eng/person26537#bib65.
- 4
Andrés Chirre and Harald Andrés Helfgott, Optimal bounds for sums of non-negative arithmetic functions. arXiv:2512.15709v1. Primary locator: https://arxiv.org/abs/2512.15709v1.
- 5
LMFDB Collaboration, Riemann zeta zero data: source, completeness, and rigor statements. public knowledge pages accessed 2026-07-19. Primary locators: https://www.lmfdb.org/knowledge/show/rcs.source.zeros.zeta; https://www.lmfdb.org/knowledge/show/rcs.cande.zeros.zeta; https://www.lmfdb.org/knowledge/show/rcs.rigor.zeros.zeta.
- 6
FLINT developers, acb_dirichlet Riemann-zeta zero-counting and isolation routines. FLINT 3.6.0. Primary locators: https://flintlib.org/doc/acb_dirichlet.html#riemann-zeta-function-zeros; https://github.com/flintlib/flint/tree/v3.6.0/src/acb_dirichlet.
- 7
Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to 3 times 10 to the 12th power, BLMS 53, 792–797. arXiv:2004.09765v1 and version of record. Primary locators: https://arxiv.org/abs/2004.09765v1; https://doi.org/10.1112/blms.12460.
- 8
Harald Andrés Helfgott, Minor arcs for Goldbach’s problem. arXiv:1205.5252v4. Primary locator: https://arxiv.org/abs/1205.5252v4.
- 9
Harald Andrés Helfgott, Major arcs for Goldbach’s problem. arXiv:1305.2897v4. Primary locator: https://arxiv.org/abs/1305.2897v4.
- 10
Harald Andrés Helfgott, The ternary Goldbach problem. arXiv:1501.05438v2. Primary locator: https://arxiv.org/abs/1501.05438v2.
- 11
Harald Andrés Helfgott, The ternary Goldbach conjecture is true. arXiv:1312.7748v2. Primary locator: https://arxiv.org/abs/1312.7748v2.
- 12
Harald Andrés Helfgott and David J. Platt, Numerical verification of the ternary Goldbach conjecture up to 8.875 times 10 to the 30th power. arXiv:1305.3062v2. Primary locator: https://arxiv.org/abs/1305.3062v2.
- 13
Henri Cohen, François Dress, and Mohamed El Marraki, Explicit estimates for summatory functions linked to the Möbius mu-function, FACM 37.1, 51–63. version of record. Primary locators: https://doi.org/10.7169/facm/1229618741; https://www.math.u-bordeaux.fr/~hecohen/artdm4.dvi.
- 14
Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, Math. Comp. 87, 1013–1028. arXiv:1610.08551v2 and version of record. Primary locators: https://arxiv.org/abs/1610.08551v2; https://doi.org/10.1090/mcom/3275.
- 15
Ethan Simpson Lee and Nicol Leong, New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function. arXiv:2208.06141v4. Primary locator: https://arxiv.org/abs/2208.06141v4.
- 16
Olivier Ramaré, Explicit estimates on several summatory functions involving the Moebius function, Math. Comp. 84, 1359–1387. version of record, read together with the 2019 corrigendum. Primary locators: https://doi.org/10.1090/S0025-5718-2014-02914-1; https://ramare-olivier.github.io/Maths/mcom2914.pdf.
- 17
Olivier Ramaré, Corrigendum to Explicit estimates on several summatory functions involving the Moebius function, Math. Comp. 88, 2383–2388. version of record. Primary locator: https://doi.org/10.1090/mcom/3449.
- 18
Olivier Ramaré, From explicit estimates for primes to explicit estimates for the Möbius function, Acta Arith. 157.4, 365–379. version of record. Primary locator: https://doi.org/10.4064/aa157-4-4.
- 19
Olivier Ramaré and Sebastián Zúñiga-Alterman, From explicit estimates for the primes to explicit estimates for the Möbius function – II. arXiv:2408.05969v2 and version of record. Primary locators: https://arxiv.org/abs/2408.05969v2; https://doi.org/10.7169/facm/250121-19-5.
- 20
David J. Platt, Numerical computations concerning the GRH. arXiv:1305.3087v1. Primary locator: https://arxiv.org/abs/1305.3087v1.
- 21
Timothy S. Trudgian, An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions, Math. Comp. 84, 1439–1450. arXiv:1206.1844v4 for the mapped 0.317/6.401 claim; version of record separately checked and non-identical. Primary locators: https://arxiv.org/abs/1206.1844v4; https://doi.org/10.1090/S0025-5718-2014-02898-6.
- 22
Andrew Fiori, A Note on the Phragmén–Lindelöf Theorem. arXiv:2502.13282v3. Primary locator: https://arxiv.org/abs/2502.13282v3.
- 23
H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, JLMS 8, 73–82. version of record. Primary locator: https://doi.org/10.1112/jlms/s2-8.1.73.
- 24
Wijit Yangjit, On the Montgomery–Vaughan weighted generalization of Hilbert’s inequality. arXiv:2203.14950v1. Primary locator: https://arxiv.org/abs/2203.14950v1.
- 25
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20.2, 119–134. version of record. Primary locators: https://doi.org/10.1112/S0025579300004708; https://personal.science.psu.edu/rcv4/personal/Publications/large_sieve.pdf.
- 26
Olivier Ramaré, Explicit estimates on the summatory functions of the Möbius function with coprimality restrictions, Acta Arith. 165.1, 1–10. version of record. Primary locator: https://doi.org/10.4064/aa165-1-1.
- 27
Olivier Ramaré, Some elementary explicit bounds for two mollifications of the Möbius function, FACM 49.2, 229–240. version of record and author offprint. Primary locators: https://doi.org/10.7169/facm/2013.49.2.3; https://ramare-olivier.github.io/Maths/MuLog-4.pdf.
- 28
Olivier Ramaré, On Šnirel’man’s constant, Annali SNS Pisa 22.4, 645–706. version of record scan. Primary locator: https://www.numdam.org/item/ASNSP_1995_4_22_4_645_0/.
- 29
Andrew Granville and Olivier Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43.1, 73–107. version of record. Primary locators: https://doi.org/10.1112/S0025579300011608; https://ramare-olivier.github.io/Maths/granvilleramare.pdf.
- 30
Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comp. 65.213, 397–425. version of record and author-hosted scan. Primary locators: https://doi.org/10.1090/S0025-5718-96-00669-2; https://ramare-olivier.github.io/Maths/rumely.pdf.
- 31
Olivier Ramaré and Yannick Saouter, Short effective intervals containing primes, JNT 98.1, 10–33. version of record. Primary locators: https://doi.org/10.1016/S0022-314X(02)00029-X; https://ramare-olivier.github.io/Maths/gap.pdf.
- 32
J. Barkley Rosser, Explicit bounds for some functions of prime numbers, AJM 63.1, 211–232. version of record. Primary locators: https://doi.org/10.2307/2371291; https://archive.org/details/sim_american-journal-of-mathematics_1941-01_63_1.
- 33
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6.1, 64–94. version of record. Primary locator: https://doi.org/10.1215/ijm/1255631807.
- 34
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta and psi, Math. Comp. 29, 243–269. version of record. Primary locator: https://doi.org/10.1090/S0025-5718-1975-0457373-7.
- 35
N. M. Temme, DLMF Chapter 12: Parabolic Cylinder Functions. DLMF Version 1.2.7, released 2026-06-15; numbered equations accessed 2026-07-19. Primary locators: https://dlmf.nist.gov/12; https://dlmf.nist.gov/12.2.E2; https://dlmf.nist.gov/12.2.E11; https://dlmf.nist.gov/12.2.E16; https://dlmf.nist.gov/12.5.E1; https://dlmf.nist.gov/12.5.E6; https://dlmf.nist.gov/12.9.E1; https://dlmf.nist.gov/12.9.E2; https://dlmf.nist.gov/12.9.E3.
- 36
F. W. J. Olver, Asymptotics and Special Functions, 1997 reprint of the 1974 edition. AKP Classics 1997 reprint; pagination checked against the scanned edition. Primary locator: https://www.routledge.com/Asymptotics-and-Special-Functions-1st-Edition/Olver/p/book/9780429064616.
- 37
Harold Davenport, Multiplicative Number Theory, 3rd ed., revised by Hugh L. Montgomery, GTM 74. Springer third edition, 2000. Primary locator: https://link.springer.com/book/9780387950976.
- 38
Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, AMS Colloquium Publications 53. AMS 2004 edition. Primary locator: https://bookstore.ams.org/coll-53.
- 39
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I: Classical Theory, CSAM 97. Cambridge first edition, first published in print 2006. Primary locator: https://doi.org/10.1017/CBO9780511618314.
- 40
E. C. Titchmarsh, revised by D. R. Heath-Brown, The Theory of the Riemann Zeta-Function, 2nd ed. Oxford second edition, 1986. Primary locator: https://global.oup.com/academic/product/the-theory-of-the-riemann-zeta-function-9780198533696.