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Published online by Cambridge University Press:  10 September 2021

Kevin Broughan
Affiliation:
University of Waikato, New Zealand
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Bounded Gaps Between Primes
The Epic Breakthroughs of the Early Twenty-First Century
, pp. 555 - 566
Publisher: Cambridge University Press
Print publication year: 2021

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References

[1] Alfors, L., Complex Analysis, 2nd ed., McGraw-Hill, 1966.Google Scholar
[2] Alweiss, R. and Luo, S., Bounded Gaps Between Primes in Short Intervals (preprint).Google Scholar
[3] Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory, Springer, 1976.Google Scholar
[4] Apostol, T. M., Introduction to Analytic Number Theory, 2nd ed., Springer, 1990.Google Scholar
[5] Baker, R. C. and Pollack, P., Bounded Gaps Between Primes with a Given Primitive Root, Forum Math. 28 (2016), 675687.CrossRefGoogle Scholar
[6] Banks, W. D., Freiberg, T. and Maynard, J., On Limit Points of the Sequence of Normalized Prime Gaps, Proc. Lond. Math. Soc. (3) 113 (2016), 515–539.Google Scholar
[7] Barban, M. B., New Applications of the Great Sieve of Ju. V. Linnik. (Russian) Akad. Nauk Uzbek. SSR Trudy Inst. Mat. 22 (1961), 120.Google Scholar
[8] Bogaert, I. (private communication).Google Scholar
[9] Bombieri, E., On the Large Sieve, Mathematika 12 (1965), 201225.CrossRefGoogle Scholar
[10] Bombieri, E. and Davenport, H., Small Differences Between Prime Numbers, Proc. Roy. Soc. Ser. A 293 (1966), 118.Google Scholar
[11] Bombieri, E., Le Grand Crible dans la Théorie Analytique des Nombres, Astérisque, no 18 (1974) and 2nd ed. no 18 (1987).Google Scholar
[12] Bombieri, E., On Twin Almost Primes, Acta Arith. (1975) 28, 177193.Google Scholar
[13] Bombieri, E., The Asymptotic Sieve, Rend. Accad. Naz. XL 1 /2 (1975/76), 243269.Google Scholar
[14] Bombieri, E., Corrigendum to My Paper “On Twin Almost Primes” and an Addendum on Selberg’s Sieve, Acta Arith. 28 (1976), 457461.CrossRefGoogle Scholar
[15] Bombieri, E., Friedlander, J. B. and Iwaniec, H., Primes in Arithmetic Progressions to Large Moduli, Acta Math. 156 (1986), 203251.Google Scholar
[16] Bombieri, E., Friedlander, J. B. and Iwaniec, H., Primes in Arithmetic Progressions to Large Moduli, II, Math. Ann. 277 (1987), 361393.Google Scholar
[17] Bombieri, E., Selberg’s Sieve and Its Applications, in Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21 29–49, Academic Press.Google Scholar
[18] Bombieri, E., Friedlander, J. B. and Iwaniec, H., Primes in Arithmetic Progressions to Large Moduli, III, J. Amer. Math. Soc. 2 (2) (1989), 215224.Google Scholar
[19] Bombieri, E., J. B. Friedlander and H. Iwaniec, Some Corrections to an Old Paper, arXiv:1903.01371v1 [math.NT] 4 March 2019.Google Scholar
[20] Broughan, K. A., The gcd-sum function, Journal of Integer Sequences, 4 (2002) Article 01.2.2, p1–19.Google Scholar
[21] Broughan, K. A., Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents, Cambridge University Press, 2017.Google Scholar
[22] Broughan, K. A., Equivalents of the Riemann Hypothesis, Volume Two: Analytic Equivalents, Cambridge University Press, 2017.Google Scholar
[23] Brun, V., Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare, Arch. Mat. Natur. vol. 34, no. 8 (1915), 1–19. (1915).Google Scholar
[24] Brun, V., Le crible d’eratosthéne et le theorém de Goldbach, Videnskapssel-skapets Sknfter Kristiania, Mat. Naturv. Klasse., no 3 (1920), 1–36.Google Scholar
[25] Casselman, B., The Polyface of Polymath, Notices of the Amer. Math. Soc., June/July 2015, 659.Google Scholar
[26] Castryck, W., Fouvry, E., Harcos, G., et al., New Equidistribution Estimates of Zhang Type, Algebra and Number Theory, 8, (2014), 20672199.Google Scholar
[27] Castillo, A., Hall, C., Lemke Oliver, R. J., Pollack, P. and Thompson, L., Bounded Gaps Between Primes in Number Fields and Function Fields, Proc. Amer. Math. Soc. 143 (2015), 28412856.Google Scholar
[28] Chen, J. R., On the Representation of a Larger Even Integer as the Sum of a Prime and the Product of at Most Two Primes, Sci. Sinica 16 (1973), 157176.Google Scholar
[29] Chen, J. R., On the Representation of a Larger Even Integer as the Sum of a Prime and the Product of at Most Two Primes, Sci. Sinica 16 (1973), 157–176, in The Goldbach Conjecture, 2nd ed., Y. Wang, Ed. Chapter 20, 275–294, World Scientific, 2002.Google Scholar
[30] Cipra, B., Proof Promises Progress in Prime Progressions, Science 304 (2004), 1095.Google Scholar
[31] Cipra, B., Third Time Proves Charm for Prime-Gap Theorem, Science 308 (2005), 1238.Google Scholar
[32] Cochrane, T. and Plinner, C., Using Stepanov’s Method for Exponential Sums Involving Rational Functions, J. Number Theory 116 (2006), 270292.Google Scholar
[33] Cojocaru, A. C. and Ram Murty, M., An Introuction to Sieve Methods and Their Applications, Cambridge University Press, 2005.Google Scholar
[34] Csicsery, G., Counting from Infinity: Yitang Zhang and the Twin Primes Conjecture, Zala films, Oakland, CA, 2015, www.zalafilms.com/films/countingindex.html.Google Scholar
[35] Davenport, H., Multiplicative Number Theory, 3rd ed., Springer, 2000.Google Scholar
[36] Deligne, P., La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137252.Google Scholar
[37] Deshouillers, J. -M. and Iwaniec, H., Kloosterman Sums and Fourier Coefficients of Cusp Forms, Invent. Math. 70 (1982), 219288.CrossRefGoogle Scholar
[38] Diamond, H. and Halberstam, H., Some Applications of Sieves of Dimensions Exceeding 1, 101–107, in Sieve Methods, Exponential Sums, and Their Applications in Number Theory G. R. H. Greaves, G. Harman, M. N. Huxley, Eds., (Cardiff, 1995), (London Mathematical Society Lecture Note Series 237), Cambridge University Press, 1997.CrossRefGoogle Scholar
[39] Dickson, L. E., A New Extension of Dirichlet’s Theorem on Prime Numbers, Messenger of Mathematics, 33 (1904), 155161.Google Scholar
[40] Edwards, H. M., Riemann’s Zeta Function, Academic Press, 1974. Reprinted by Dover, 2001.Google Scholar
[41] Elliott, P. D. T. A. and Halberstam, H., A Conjecture in Prime Number Theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, 1970, 5972.Google Scholar
[42] Ellison, W. and F., Prime Numbers, Wiley, 1985.Google Scholar
[43] Engelsma, T., Narrow Admissible Ktuples, http://math.mit.edu/∼primegaps/.Google Scholar
[44] Erdős, P., On the Difference of Consecutive Primes, Quart. J. Math. Oxford Ser. 6 (1935), 124128.Google Scholar
[45] Erdős, P., The Difference of Consecutive Primes, Duke Math. J. 6 (1940), 438441.Google Scholar
[46] Erdős, P., On Some Problems on the Distribution of Prime Numbers, C. I. M. E. Teoria dei numeri Math. Congr. Varenna 1954 (1955), 8.Google Scholar
[47] Erdős, P. and Straus, E. G., Remarks on the Differences Between Consecutive Primes, Elem. Math. 35 (1980), 115118.Google Scholar
[48] Farkas, B., Pintz, J. and Révész, S., On the Optimal Weight Function in the Goldston–Pintz-Yildirim Method for Finding Small Gaps Between Consecutive Primes, in 75–104, Paul Turán Memorial Volume: Number Theory, Analysis, and Combinatorics, 75–104, J. Pintz, A. Biró, K. Győry, G. Harcos, M. Simonovits, and J. Szabados, Eds., De Gruyter Proc. Math., De Gruyter, Berlin, 2014.Google Scholar
[49] Folland, G., Fourier Analysis and Its Applications. Wadsworth and Brooks, 1992.Google Scholar
[50] Ford, K., A Simple Proof of Gallagher’s Singluar Series Sum Estimate, arXiv:1108.3861v2 [math.NT] 6 December 2016.Google Scholar
[51] Ford, K., Long gaps between primes, J. Amer. Math. Soc. 31 (2018), 65105.Google Scholar
[52] Ford, K., Green, B., Konyagin, S. and Tao, T., Large Gaps Between Consecutive Primes, arXiv:1408.4505v2 [math.NT] 9 November 2015.Google Scholar
[53] Fouvry, E., Autour du theoreme de Bombieri–Vinogradov. Acta Math. 152 (1984), 219244.Google Scholar
[54] Fouvry, E. and Iwaniec, H., On a Theorem of Bombieri–Vinogradov Type, Mathematika, 27 (1980), 135152.Google Scholar
[55] Fouvry, E. and Iwaniec, H., Primes in Arithmetic Progressions, Acta Arith. 42 (1983), 197218.Google Scholar
[56] Fouvry, E. and Grupp, F., On the Switching Principle in Sieve Theory, J. reine angew Math. 370, (1986), 101126.Google Scholar
[57] Fouvry, E. and Iwaniec, H., The Divisor Function over Arithmetic Progressions (with an appendix by N. Katz), Acta Arith. 61 (1992), 271–287.Google Scholar
[58] Fouvry, E., Kowalski, E. and Michel, P., On the Exponent of Distribution of the Ternary Divisor Function, Mathematika, 61 (2015), 121144.Google Scholar
[59] Friedlander, J. B., Prime Numbers: A Much Needed Gap Is Finally Found, Notices Amer. Math. Soc. 62 (2015), 660664.Google Scholar
[60] Friedlander, J. B. and Granville, A., Limitations to the Equi-Distribution of Primes I, Ann. of Math. 129 (1989), 363382.Google Scholar
[61] Friedlander, J. B. and Iwaniec, H., Incomplete Kloosterman Sums and a Divisor Problem, with an Appendix by B. J. Birch and E. Bombieri, On Some Exponential Sums, Ann. of Math., 121 (1985), 319–350.Google Scholar
[62] Friedlander, J. B. and Iwaniec, H., The Polynomial x2 + y4 Captures Its Primes, Ann. of Math. 148 (1998), 9451040.Google Scholar
[63] Friedlander, J. B. and Iwaniec, H., What Is the Parity Phenomenon?, Notices Amer. Math. Soc. 56 (2009), 817818.Google Scholar
[64] Friedlander, J. B. and Iwaniec, H., Opera de Cribro, American Mathematical Society, 2010.Google Scholar
[65] Gallagher, P. X., The Large Sieve, Mathematika 14 (1967), 1420.CrossRefGoogle Scholar
[66] Gallagher, P. X., A Larger Sieve, Acta Arith. 53 (1971), 7781.Google Scholar
[67] Gallagher, P. X., On the Distribution of Primes in Short Intervals, Mathematika 23 (1976), 49.Google Scholar
[68] Gallagher, P. X., Corrigendum: “On the Distribution of Primes in Short Intervals”, Mathematika 28 (1981), 86.Google Scholar
[69] Goldston, D. A., Are There Infinitely Many Twin Primes?, (preprint).Google Scholar
[70] Goldston, D. A., Motohashi, Y., Pintz, J. and Yildirim, C. Y., Small Gaps Between Primes Exist, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 6165.Google Scholar
[71] Goldston, D. A. and Yildirim, C. Y., Higher Correlations of Divisor Sums Related to Primes. III. Small Gaps Between Primes, Proc. Lond. Math. Soc. (3) 95 (2007), 653–686.Google Scholar
[72] Goldston, D. A., Pintz, J. and Yildirim, C. Y., Primes in Tuples I, Ann. of Math. (2) 170 (2009), 819–862.Google Scholar
[73] Goldston, D. A. and Ledoan, A. H., Jumping Champions and Gaps Between Consecutive Primes, Int. J. Number Theory 7 (2011), 14131421.Google Scholar
[74] Goldston, D. A. and Ledoan, A. H., On the Differences Between Consecutive Prime Numbers, I, Integers 12B (2012/13), #A3, 8 pages.Google Scholar
[75] Goldston, D. A. and Ledoan, A. H., The Jumping Champion Conjecture, Mathematicka 61 (2015), 719740.Google Scholar
[76] Goldston, D. A. and Ledoan, A. H., Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers, 115–125 in Analytic Number Theory. In Honor of Helmut Maier’s 60th Birthday. Eds. C. Pomerance and M. Th. Rassias, Springer, 2015.Google Scholar
[77] Goldston, D. A., Graham, S. W., Pintz, J. and Yildirim, C. Y., Small Gaps Between Products of Two Primes, Proc. Lond. Math. Soc, (3) 98 (2009), 741–774.Google Scholar
[78] Goldston, D. A., Graham, S. W., Pintz, J. and Yildirim, C. Y., Small Gaps Between Primes or Almost Primes, Trans. Amer. Math. Soc., 361 (2010), 52855330.Google Scholar
[79] Goldston, D. A., Graham, S. W., Pintz, J. and Yildirim, C. Y., Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers, Int. Math. Res. Not. IRMN, (2011), 14391450.CrossRefGoogle Scholar
[80] Gowers, T., (Ed.) The Princeton Companion to Mathematics, Princeton University Press, 2008.Google Scholar
[81] Granville, A., Least Primes in Arithmetic Progressions, Théorie des Nombres (Quebec, PQ, 1987), Walter de Gruyter, 1989, 306321.Google Scholar
[82] Granville, A., Primes in Intervals of Bounded Length, Bull. Amer. Math. Soc. 52 (2015), 171222.Google Scholar
[83] Granville, A., About the Cover: A New Mathematical Celebrity, Bull. Amer. Math. Soc. 52 (2015), 335337.Google Scholar
[84] Granville, A. and Soundararajan, K., An Uncertainty Principle for Arithmetic Sequences, Annals Math. 165 (2007), 593635.Google Scholar
[85] Greaves, G., Sieves in Number Theory, Springer-Verlag, 2001.Google Scholar
[86] Green, B. and Tao, T., The Primes Contain Arbitrarily Long Arithmetic Progressions, Ann. of Math. (2) 167 (2008), 481–547.Google Scholar
[87] Gupta, R. and Murty, M. R., A Remark on Artin’s Conjecture, Invent. Math. 78 (1984), 127130.Google Scholar
[88] Halberstam, H. and Richert, H. -E., Sieve Methods, Academic Press, 1974.Google Scholar
[89] Hardy, G. H. and Littlewood, J. E., Some Problems of “Partitio Numerorum”; III: On the Expression of a Number as a Sum of Primes, Acta Math. 44 (1923), 170.Google Scholar
[90] Hardy, G. H. and Wright, J. M., An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008.Google Scholar
[91] Harman, G., Prime Detecting Sieves, Princeton University Press, 2007.Google Scholar
[92] Harris, J., Algebraic Geometry: A First Course, Springer, 1992.Google Scholar
[93] Hartshorne, R., Algebraic Geometry, 8th printing, Springer, 1997.Google Scholar
[94] Haugland, J. K., Application of Sieve Methods to Prime Numbers, Ph.D. thesis, Oxford University, 1999.Google Scholar
[95] Heath-Brown, D. R., Prime Numbers in Short Intervals and a Generalized Vaughan Identity, Canad. J. Math. 34 (1982), 13651377.Google Scholar
[96] Heath-Brown, D. R., Prime Twins and Siegel Zeros, Proc. London Math. Soc. 47 (1983), 193224.Google Scholar
[97] Heath-Brown, D. R., The Divisor Function d3 (n) in Arithmetic Progressions, Acta Arith. 47 (1986), 13651377.Google Scholar
[98] Heath-Brown, D. R., Artin’s Conjecture for Primitive Roots, Quart. J. Math. Oxford (2) 37 (1986), 27–38.Google Scholar
[99] Heath-Brown, D. R., Primes Represented by x3 + 2y3, Acta Math. 186 (2001), 184.Google Scholar
[100] Heath-Brown, D. R., Obituary Atle Selberg, Bull. London Math. Soc, 42 (2010), 949955.Google Scholar
[101] Heilbronn, H., On the Class Number of Imaginary Quadratic Fields, Quart. J. Math., 5 (1934), 150160.CrossRefGoogle Scholar
[102] Hensley, D. and Richards, I., Primes in Intervals, Acta Arith. 25 (1973/74), 375391.Google Scholar
[103] Hethcote, H. W., Asymptotic Approximations with Error Bounds for Zeros of Airy and Cylindrical Functions, Ph.D. thesis, University of Michigan, Ann Arbor, 1968.Google Scholar
[104] Hethcote, H. W., Error Bounds for Asymptotic Approximations Zeros of Transcendental Functions, SIAMJ. Math. Anal. 1 (1970), 147152.Google Scholar
[105] Hilderbrand, A., Erdős’ Problems on Consecutive Integers, Paul Erdős and His Mathematics I (Bolyai Society Mathematical Studies, 11) Budapest, 2002, 305–317.Google Scholar
[106] Hildebrand, A. and Maier, H., Gaps Between Prime Numbers, Proc. Amer. Math. Soc, 104 (1988), 19.Google Scholar
[107] Ho, K. -H. and Tsang, K. -M., On Almost Prime K-tuples, J. Number Theory 120 (2006), 3346.Google Scholar
[108] Hooley, C., On Artin’s Conjecture, J. Reine Angew. Math. 225 (1967), 209220.Google Scholar
[109] Huxley, M. N., Small Differences Between Consecutive Primes, Mathematika 20 (1973), 229232.Google Scholar
[110] Huxley, M. N., Small Differences Between Consecutive Primes II, Mathematika 24 (1977), 142152.Google Scholar
[111] Huxley, M. N., An Application of the Fouvry–Iwaniec Theorem, Acta Arith. 43 (1984), 441443.Google Scholar
[112] Huxley, M. N. and Iwaniec, H., Bombieri’s Theorem in Short Intervals, Mathematika 22 (1975), 188194.Google Scholar
[113] Ingham, A. E., The Distribution of Prime Numbers, Cambridge University Press, 1932.Google Scholar
[114] Iwaniec, H., Primes Represented by Quadratic Polynomials in Two Variables, Acta Arith. 24 (1973/74), 435459.Google Scholar
[115] Iwaniec, H., A New Form of the Error Term in the Linear Sieve, Acta Arith. 37 (1980), 307320.Google Scholar
[116] Iwaniec, H., Conversations on the Exceptional Character, 97–132, in Analytic Number Theory: Lecture Notes in Mathematics, vol. 1891, Springer, 2006.Google Scholar
[117] Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society, 2004.Google Scholar
[118] Jurkat, W. B. and Richert, H. -E., An Improvement of Selberg’s Sieve Method, I, Acta Arith. 11 (1965), 217240.Google Scholar
[119] Jutila, M., A Statistical Density Theorem for L-Functions with Applications, Acta Arith. 16 (1969/70), 207216.Google Scholar
[120] Klarreich, E., Unheralded Mathematician Bridges the Prime Gap, Quanta Magazine, 2013, www.quantamagazine.org/20130519Google Scholar
[121] De Koninck, J. -M. and Luca, F., Analytic Number Theory: Exploring the Anatomy of Integers, American Mathematical Society, 2012.Google Scholar
[122] Landau, E., Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel, Göttinger Nachrichten (1924), 202206.Google Scholar
[123] Landau, E., Handbuch der lehre von der Verteilung der Primzahlen, 2nd ed., volumes 1 and 2, Chelsea, 1953.Google Scholar
[124] Lang, T. and Wong, R., ”Best Possible” Upper Bounds for the First Two Positive Zeros of the Bessel Function J v (v): The Infinite Case, J. Comp. and App. Math. 71 (1996), 311–329.Google Scholar
[125] Legendre, A. M., Théorie des Nombres, 2nd ed., Paris, 1808.Google Scholar
[126] Linnik, U. V., The Large Sieve, C. R. (Doklady) Acad. Sci. URSS 30 (1941), 292294.Google Scholar
[127] Linnik, U. V., The Dispersion Method in Binary Additive Problems, Translated from the Russian. Amer. Math. Soc., 1963.Google Scholar
[128] Lorch, L., Some Inequalities for the First Positive Zeros of Bessel Functions, SIAM J. Math. Anal. 24 (1993), 814823.Google Scholar
[129] Lorch, L. and Uberti, R., ”Best Possible” Upper Bounds for the First Positive Zeros of the Bessel Functions – the Finite Part, J. Comp. and Appl. Math. 72 (1996), 249–258.Google Scholar
[130] Mackenzie, D., Prime Proof Helps Mathematicians Mind the Gaps, Science 300(2003), 32.Google Scholar
[131] Mackenzie, D., Prime-Number Proof’s Attempt Falls Short, Science 300 (2003), 1066.Google Scholar
[132] Maier, H., Small Differences Between Prime Numbers, Michigan Math. J. 35 (1988), 323344.Google Scholar
[133] Maier, H. and Pomerance, C., Unusually Large Gaps Between Consecutive Primes, Trans. Amer. Math. Soc. 322 (1990), 201237.Google Scholar
[135] Maynard, J., On the Brun–Titchmarsh Theorem, Acta Arith. 157 (2013), 249296.Google Scholar
[136] Maynard, J., 3-Tuples Have at Most 7 Prime Factors Infinitely Often, Math. Proc. Camb. Phil. Soc. 155 (2013), 443457.Google Scholar
[137] Maynard, J., Almost Prime K-tuples, Mathematica 60 (2014), 108138.Google Scholar
[138] Maynard, J., Small Gaps Between Primes, Ann. of Math. (2) 181 (2015), 383–413.Google Scholar
[139] Maynard, J., Dense Clusters of Primes in Subsets, Compos. Math. 152 (2016), 15171554.Google Scholar
[140] Maynard, J., Large Gaps Between Primes, Ann. of Math. (2) 183 (2016), 915–933.Google Scholar
[141] Moh, T. T., Zhang, Yitang’s Life at Purdue (Jan 1985–1991) (Revised in Bold Face. 2018) www.math.purdue.edu/∼ttm/ZhangYt.pdf.Google Scholar
[142] Montgomery, H. L., Primes in Arithmetic Progression,, Mich. Math. J. 17 (1970), 3339.Google Scholar
[143] Montgomery, H. L. and Vaughan, R. C., The Large Sieve, Mathematika 20 (1973), 119134.Google Scholar
[144] Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory I: Classical Theory, Cambridge University Press, 2007.Google Scholar
[145] Motohashi, Y., On Some Improvements of the Brun–Titchmarsh Theorem, J. Math. Soc. Japan 26 (1974), 306-323.Google Scholar
[146] Motohashi, Y., An Induction Principle for the Generalization of Bombieri’s Prime Number Theorem, Proc. Japan Acad. 52 (1976), 273275.Google Scholar
[147] Motohashi, Y., Sieve Methods and Prime Number Theory, Tata IFR and Springer-Verlag, 1983.Google Scholar
[148] Motohashi, Y., The Remainder Term in the Selberg Sieve, Number Theory in Progress, 2, de Gruyter (1999), 1053–1064.Google Scholar
[149] Motohashi, Y., An Overview of the Sieve Method and Its History, arXiv:math/0505521v2 [math.NT] 27 Dec. 2006.Google Scholar
[150] Motohashi, Y., The Twin Primes Conjecture, arXiv:1401.6614v2 [math.NT] (16 March 2014).Google Scholar
[151] Motohashi, Y. and Pintz, J., A Smoothed GPY Sieve, Bull. Lond. Math. Soc. 40 (2008), 298310.Google Scholar
[152] Nathanson, M. B., Additive Number Theory: The Classical Bases, Springer, 1996.Google Scholar
[153] Nathanson, M. B., Elementary Methods in Number Theory, Springer, 2000.Google Scholar
[154] Neal, V., Closing the Gap: The Quest to Understand Prime Numbers, Oxford, 2017.Google Scholar
[155] Niederreiter, R. L. H., Finite Fields, Addison-Wesley, 1983.Google Scholar
[156] Oberhettinger, F., Tables of Mellin Transforms, Springer-Verlag, 1974.CrossRefGoogle Scholar
[157] Odlyzko, A., Rubinstein, M. and Wolf, M., Jumping Champions, Experiment. Math. 8 (1999), 107118.Google Scholar
[158] Olver, F. W. J., The Asymptotic Expansion of Bessel Functions of Large Order, Philos. Trans. Roy. Soc. London Ser A 247 (1954), 328368.Google Scholar
[159] Olver, F. W. J., Error Bounds for First Approximations in Turning Point Problems, J. Soc. Ind. Appl. Math. 11 (1963), 748772.Google Scholar
[160] Olver, F. W. J., Error Bounds for Asymptotic Expansions in Turning Point Problems, J. Soc. Ind. Appl. Math. 12 (1964), 200214.Google Scholar
[161] Olver, F. W. J., Asymptotics and Special Functions, Academic Press, 1974.Google Scholar
[162] Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (Ed.), NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, 2010.Google Scholar
[163] Pan, C. D., Ding, X. X. and Wang, Y., On the Representation of Every Large Even Integer as a Sum of a Prime an Almost Prime, Sci. Sinica 18 (1975), 599610.Google Scholar
[164] Pil’tjai, G. Z., The Magnitude of the Difference Between Consecutive Primes, (Russian) Studies in Number Theory, Izdat. Saratov. Univ., Saratov, 4 (1972), 73–79.Google Scholar
[165] Pintz, J., Very Large Gaps Between Consecutive Primes, J. Number Theory 63 (1997), 286301.Google Scholar
[166] Pintz, J., On the Singlar Series in the Prime K-tuple Conjecture, arXiv:1004.1084v1 [math.NT] 7 April 2010.Google Scholar
[167] Pintz, J., A Note on Bounded Gaps Between Primes, arXiv1303.1497v4 [math.NT] 17 July 2013.Google Scholar
[168] Pintz, J., Polignac Numbers, Conjectures of Erdős on Gaps Between Primes, Arithmetic Progressions in Primes and the Bounded Gap Conjecture, 367– 384, in J. Sander, J. Steuding and R. Steuding (Eds.), in From Arithmetic to Zeta-Functions, Springer, 2016.Google Scholar
[169] Pollack, P., Bounded Gaps Between Primes with a Given Primitive Root, Algebra Number Theory 8 (2014), 17691786.Google Scholar
[170] de Polignac, A., Six propositions arithmologiques déduites du crible d’Ératosthène, Nouvelles annales de mathématiques (1), 8 (1849), 23–429.Google Scholar
[171] Polymath, D. H. J., A New Proof of the Density Hales–Jewett Theorem, Ann. of Math. (2) 175 (2012), 1283–1327.Google Scholar
[172] Polymath, D. H. J., The “Bounded Gaps Between Primes” Polymath Project, a Retrospective Analysis, EMS Newletter, December 2014, 1323.Google Scholar
[173] Polymath, D. H. J., New Equidistribution Estimates of Zhang Type, and Bounded Gaps Between Primes, Algebra and Number Theory (8) 9 (2014), 2067–2199.Google Scholar
[174] Polymath, D. H. J., New Equidistribution Estimates of Zhang Type, and Bounded Gaps Between Primes, axXiv:1402.0811v2 [math.NT] 12 July 2014.Google Scholar
[175] Polymath, D. H. J., Variants of the Selberg Sieve,and Bounded Intervals Containing Many Primes, Polymath Research in the Mathematical Sciences, 1 (2014), 183.Google Scholar
[176] Porter, J. W., Some Numerical Results in the Selberg Sieve Method, Acta Arith. 20 (1972), 417421.Google Scholar
[177] Ramaré, O. with D. S. Ramana, Arithmetical Aspects of the Large Sieve Inequality, Hindustan, 2009.Google Scholar
[178] Rankin, R. A., The Difference Between Consecutive Prime Numbers, J. London Math. Soc. 13 (1938), 242247.Google Scholar
[179] Rankin, R. A., The Difference Between Consecutive Prime Numbers, II. Proc. Cambridge Philos. Soc. 36 (1940), 255266.Google Scholar
[180] Rankin, R. A., The Difference Between Consecutive Prime Numbers, V, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 331–332.Google Scholar
[181] Rényi, A., On the Representation of an Even Number as the Sum of a Prime and of an Almost Prime, Amer. Math. Soc. Transl. (2) 19 (1962), 299–321. Translated from the Russian Izv. Akad. Nauk SSSR ser. Mat. 12 (1948), 57–78.Google Scholar
[182] Ricci, G., Sull’andamento della differenza di numeri primi consecutivi, Riv. Mat. Univ. Parma 5 (1954), 354.Google Scholar
[183] Ricci, G., Recherches sur l’allure de la suite ((pn+1pn )/ log pn ), Colloque sur la Théorie des Nombres, Bruxelles, 1955 (G. Thone, 1956), 93–106.Google Scholar
[184] Rosser, J. B. and Schoenfeld, L., Approximate Formulas for Some Functions of Prime Numbers, Illinois J. Math. 6 (1962), 6494.Google Scholar
[185] Rudin, W., Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.Google Scholar
[186] Rudin, W., Functional Analysis, 2nd ed., McGraw-Hill, 1991.Google Scholar
[187] Schönhage, A., Eine Bemerkung zur Konstruktion grosser Primzahllücken, Arch. Math. 14 (1963), 2930.Google Scholar
[188] Segal, M., The Twin Primes Hero, Nautilus Magazine, September 2013, http://nautil.us/issue/5/fame/the-twin-primes-hero.Google Scholar
[189] Selberg, A., On an Elementary Method in the Theory of Primes, Norske Vid. Selsk. Forhdl. 19 (1947), 6467.Google Scholar
[190] Selberg, A., The General Sieve Method and Its Place in Prime Number Theory, Proc. Intern. Cong. Math., 1 (1950), 286292.Google Scholar
[191] Selberg, A., Sieve Methods, Proc. Symp. Pure Math. 20 (1971), 311351.Google Scholar
[192] Selberg, A., Remarks on Sieves, Proc. 1972 Number Theory conf., Boulder 1972, 205–216.Google Scholar
[193] Selberg, A., Collected Papers. Vol. I. With a Foreword by K. Chandrasekharan, Springer-Verlag, Berlin, 1989.Google Scholar
[194] Selberg, A., Collected Papers. Vol. II. With a Foreword by K. Chandrasekharan, Springer-Verlag, Berlin, 1991.Google Scholar
[195] Shiu, P., A Brun–Titchmarsh Theorem for Multiplicative Functions, J. Reine Angew. Math. 313 (1980), 161170.Google Scholar
[196] Sitaramachandrarao, R., On an Error Term of Landau, II, Rocky Mt. J. Math. 15 (1985), 579588.Google Scholar
[197] Soundararajan, K., Nonvanishing of Quadratic Dirichlet L-Functions at s = 2,1 Ann. of Math. (2) 152 (2000), 447–488.Google Scholar
[198] Soundararajan, K., Small Gaps Between Prime Numbers: The Work of Goldston–Pintz–Yildirim, Bull. Amer. Math. Soc. (N.S.) 44 (2007), 1–18.Google Scholar
[199] Strichartz, R., A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994.Google Scholar
[200] Tao, T., Structure and Randomness: Pages from Year One of a Mathematical Blog, Amer. Math. Soc, 2008.Google Scholar
[201] Tao, T., Every Odd Number Greater Than 1 Is the Sum of at Most Five Primes, Math. Comp. 83 (2012), 9971038.Google Scholar
[202] Tao, T., Notes on Zhang’s Prime Gaps Paper, 1 June 2013, Accessible from the Polymath8 home page in section 9, “Recent Papers and Notes”.Google Scholar
[203] Tao, T., Web Based Lecture Notes on the Bombieri–Vinogradov Theorem, 2016.Google Scholar
[204] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.Google Scholar
[205] Thorner, J., Bounded Gaps Between Primes in Chebotarev Sets, Res. Math. Sci. 1 (2014), Art. 4, 16.Google Scholar
[206] Titchmarsh, E. C., A Divisor Problem, Rendiconti del Circolo matematico di Palermo, 54 (1930), 414429.Google Scholar
[207] Titchmarsh, E. C. and D. R. Heath-Brown, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, 1986.Google Scholar
[208] Trudgian, T. S., A Poor Man’s Improvement on Zhang’s Result: There Are Infinitely Many Prime Gaps Less Than 60 Million, arXiv 1305.6369v2 [math.NT] 4 June 2013.Google Scholar
[209] Uchiyama, S., On the Difference Between Consecutive prime Numbers, Acta Arith. 27 (1975), 153157.Google Scholar
[210] Vaughan, R. C., An Elementary Method in Prime Number Theory, Acta Arith. 37 (1980), 111115.Google Scholar
[211] Vinogradov, A. I., The Density Hypothesis for Dirichlet L-Series, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903934.Google Scholar
[212] Ward, D. R., Some Series Involving Euler’s Function, J. Lond. Math. Soc. 2 (4) (1927), 210214.Google Scholar
[213] Westzynthius, E., Über die verteilung der zahlen, die zu den n ersten primzahlen teilerfremd sind, Commentationes Physico Mathematicae, Societas Scientarium Fennica, Helsingfors 5 (1931), 137.Google Scholar
[214] Wu, J., Sur la suite des nombres premiers jumeaux, Acta Arith. 55 (1990) 365394.Google Scholar
[215] Zhang, Y., Bounded Gaps Between Primes, Ann. of Math. (2) 1979 (2014), 1121–1174.Google Scholar

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  • References
  • Kevin Broughan, University of Waikato, New Zealand
  • Book: Bounded Gaps Between Primes
  • Online publication: 10 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781108872201.021
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  • References
  • Kevin Broughan, University of Waikato, New Zealand
  • Book: Bounded Gaps Between Primes
  • Online publication: 10 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781108872201.021
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  • References
  • Kevin Broughan, University of Waikato, New Zealand
  • Book: Bounded Gaps Between Primes
  • Online publication: 10 September 2021
  • Chapter DOI: https://doi.org/10.1017/9781108872201.021
Available formats
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