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Scientific achievements of Anatolii Alekseevich Karatsuba. (English. Russian original) Zbl 1286.01026
Proc. Steklov Inst. Math. 280, Suppl. 2, S1-S22 (2013); translation from Sovrem. Probl. Mat. 16, 7-30 (2012).

MSC:
01A70 Biographies, obituaries, personalia, bibliographies
11-03 History of number theory
Biographic References:
Karatsuba, Anatolii Alekseevich
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[1] A. A. Karatsuba, ”Solution of a problem of the theory of finite automata,” Uspekhi Mat. Nauk 15(3), 157–159 (1960). · Zbl 0097.00503
[2] A. A. Karatsuba and Yu. P. Ofman, ”Multiplication of set-valued numbers by automata,” Dokl. Akad. Nauk SSSR 145(2), 293–294 (1962).
[3] J.-P. Delahaye, ”Mathematiques et philosophie,” Pour Sci. 277, 100–104 (2000).
[4] A. A. Karatsuba, List of Scientific Publications, http://www.mi.ras.ru/ karatsuba/list.html .
[5] A. A. Karatsuba, Foundations of Analytic Number Theory (Nauka, Moscow, 1975) [in Russian]. · Zbl 0315.10039
[6] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, Theory of Multiple Trigonometric Sums (Nauka, Moscow, 1987) [in Russian]. · Zbl 0638.10037
[7] S.M. Voronin and A. A. Karatsuba, The Riemann Zeta-Function (Fizmatlit, Moscow, 1994) [in Russian].
[8] A. A. Karatsuba, Complex Analysis in Number Theory (CRC Press, Boca Raton, Fl., 1995). · Zbl 0856.11038
[9] G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis (Walter de Gruyter, Berlin, 2004). · Zbl 1074.11043
[10] G. I. Arkhipov and V. N. Chubarikov, ”On professor A.A. Karatsuba’s works inmathematics” Tr.Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 218, 7–19 (1997) [Proc. Steklov Inst.Math. 218, 1–14 (1997)].
[11] E. F. Moore, ”Gedanken-experiments on sequential machines,” in Automata Studies (Princeton Univ. Press, Princeton, N.J., 1956), pp. 129–153.
[12] A. Karacuba, ”Berechnungen und die Kompliziertheit von Beziehungen,” Elektron. Inform.-verarb. Kybernetik 11, 603–606 (1975). · Zbl 0341.68033
[13] A. A. Karatsuba, ”The complexity of computations” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 211, 186–202 (1995) [Proc. Steklov Inst. Math. 211, 169–183 (1995)]. · Zbl 1342.11100
[14] D. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms (Addison-Wesley, Reading, Mass., 1969; Mir, Moscow, 2000). · Zbl 0191.18001
[15] A. Schönhage and V. Strassen, ”Schnelle Multiplikation großer Zahlen,” Computing 7(3–4), 281–292 (1971). · Zbl 0223.68007 · doi:10.1007/BF02242355
[16] V. Strassen, ”Gaussian elimination is not optimal,” Numer.Math. 13(4), 354–356 (1969). · Zbl 0185.40101 · doi:10.1007/BF02165411
[17] A. A. Karatsuba, ”Estimates of trigonometric sums of a special form and their applications,” Dokl. Akad. Nauk SSSR 137(3), 513–514 (1961).
[18] A. A. Karatsuba, ”An analogue of Waring’s problem,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 1, 38–46 (1962).
[19] A. A. Karatsuba, ”Distribution of fractional parts of polynomials of a special type,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 3, 34–39 (1962).
[20] A. A. Karatsuba, ”Waring’s problem for a congruence modulo the power of a prime,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 4, 28–38 (1962). · Zbl 0117.28206
[21] A. A. Karatsuba and N. M. Korobov, ”A mean-value theorem,” Dokl. Akad. Nauk SSSR 149(2), 245–248 (1963). · Zbl 0142.29302
[22] A. A. Karatsuba, ”Trigonometric sums of a special type and their applications,” Izv. Akad. Nauk SSSR, Ser. Mat. 28(1), 237–248 (1964). · Zbl 0152.03402
[23] A. A. Karatsuba, ”On estimation of the number of solutions of certain equations,” Dokl. Akad. Nauk SSSR 165(1), 31–32 (1965).
[24] A. A. Karatsuba, ”On systems of congruences,” Izv. Akad. Nauk SSSR, Ser. Mat. 29(4), 935–944 (1965).
[25] A. A. Karatsuba, ”Theorems on the mean and complete trigonometric sums,” Izv. Akad. Nauk SSSR, Ser. Mat. 30(1), 183–206 (1966).
[26] A. A. Karatsuba, TheMethod of Trigonometric Sums andMean-Value Theorems, Doctoral Dissertation in Physics and Mathematics (SteklovMath. Inst., Ross. Akad. Sci., Moscow, 1966).
[27] A. A. Karatsuba, ”The method of trigonometric sums and theorems of the mean,” Mat. Zametki 1(1), 99–110 (1967) [Math. Notes, 1 (1), 64–71 (1967)].
[28] A. A. Karatsuba, ”The mean value of the modulus of a trigonometric sum,” Izv. Akad. Nauk SSSR, Ser. Mat. 37(6), 1203–1227 (1973) [Math. USSR-Izv., 37 (6), 1203–1227 (1973)].
[29] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, ”An upper bound of the modulus of a multiple trigonometric sum,” Tr.Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 143, 3–31 (1977) [Proc. Steklov Inst. Math. 143, 1–31 (1980)]. · Zbl 0432.10020
[30] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, ”Multiple trigonometric sums,” in Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 151, 3–128 (1980) [Proc. Steklov Inst. Math. 151 (2), 1–126 (1982)]. · Zbl 0441.10037
[31] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, ”Multiple trigonometric sums and their applications,” Izv. Akad. Nauk SSSR, Ser. Mat. 44(4), 723–781 (1980) [Math. USSR-Izv. 17 (1), 1–54 (1981)]. · Zbl 0446.10030
[32] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, ”Special cases of the theory of multiple trigonometric sums,” Izv. Akad. Nauk SSSR, Ser. Mat. 47(4), 707–784 (1983) [Math. USSR-Izv. 23 (1), 17–82 (1984)]. · Zbl 0536.10031
[33] I.M. Vinogradov and A. A. Karatsuba, ”Themethod of trigonometric sums in number theory,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 168, 4–30 (1984) [Proc. Steklov Inst. Math. 168, 3–30 (1986)]. · Zbl 0549.10027
[34] U. V. Linnik, ”On Weyl’s sums,” Mat. Sb. 12(1), 28–39 (1943). · Zbl 0063.03578
[35] G. I. Arkhipov, ”Mean-value theorem for the modulus of multiple trigonometric sums,” Mat. Zametki 17(1), 143–153 (1975) [Math. Notes 17 (1), 84–90 (1975)]. · Zbl 0315.10033
[36] G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, ”Trigonometric integrals,” Izv. Akad. Nauk SSSR, Ser. Mat. 43(5), 971–1003 (1979) [Math. USSR-Izv. 15 (2), 211–239 (1980)]. · Zbl 0493.10040
[37] I.M. Vinogradov, Selected Works (Akad. Nauk SSSR, Moscow, 1952) [in Russian]. · Zbl 0048.03104
[38] M. A. Chakhkiev, ”On the convergence exponent of the singular integral in the multi-dimensional analogue of Tarry’s problem,” Izv. Ross. Akad. Nauk, Ser. Mat. 67(2), 211–224 (2003) [Izv.: Math. 67 (2), 405–418 (2003)]. · Zbl 1112.11016 · doi:10.4213/im432
[39] A. A. Karatsuba, ”On the function G(n) in Waring’s problem,” Izv. Akad. Nauk SSSR, Ser. Mat. 49(5), 935–947 (1985) [Math. USSR-Izv. 27 (2), 239–249 (1986)]. · Zbl 0594.10041
[40] G. I. Arkhipov and A. A. Karatsuba, ”On local representation of zero by a form,” Izv. Akad. Nauk SSSR, Ser. Mat. 45(5), 948–961 (1981) [Math. USSR-Izv. 19 (2), 231–240 (1982)]. · Zbl 0476.10018
[41] A. A. Karatsuba, ”Distribution of inverse values in a residue ring modulo a given number,” Dokl. Akad. Nauk 333(2), 138–139 (1993) [Dokl. Math. 48 (3), 452–454 (1994)].
[42] A. A. Karatsuba, ”Fractional parts of functions of a special form,” Izv. Ross. Akad. Nauk, Ser. Mat. 59(4), 61–80 (1995) [Izv.: Math. 59 (4), 721–740 (1995)]. · Zbl 0874.11050
[43] A. A. Karatsuba, ”Analogues of Kloosterman sums,” Izv. Ross. Akad. Nauk, Ser. Mat. 59(5), 93–102 (1995) [Izv.: Math. 59 (5), 971–981 (1995)].
[44] A. A. Karatsuba, ”Analogues of incomplete Kloosterman sums and their applications,” Tatra Mt. Math. Publ. 11, 89–120 (1997). · Zbl 0978.11037
[45] A. A. Karatsuba, ”Kloosterman double sums,” Mat. Zametki 66(5), 682–687 (1999) [Math. Notes 66 (5), 565–569 (1999)]. · Zbl 0971.11045 · doi:10.4213/mzm1212
[46] J. Friedlander and H. Iwaniec, ”The Brun-Titchmarsh theorem,” in Analytic Number Theory (Cambridge Univ. Press, Cambridge, 1997), pp. 85–93. · Zbl 0910.11036
[47] J. Friedlander and H. Iwaniec, ”The polynomial X 2 + Y 4 captures its primes,” Ann. Math. (2) 148(3), 945–1040 (1998). · Zbl 0926.11068 · doi:10.2307/121034
[48] D. R. Heath-Brown, ”The largest prime factor of x 3 + 2,” Proc. London Math. Soc. (3) 82(3), 554–596 (2001). · Zbl 1023.11048 · doi:10.1112/plms/82.3.554
[49] A. A. Glibichuk, ”Combinational properties of sets of residues modulo a prime and the Erdos-Graham problem,” Mat. Zametki 79(3), 384–395 (2006) [Math. Notes 79 (3), 356–365 (2006)]. · Zbl 1129.11004 · doi:10.4213/mzm2708
[50] A. A. Karatsuba, ”On the zeros of the function \(\zeta\)(s) on short intervals of the critical line,” Izv. Akad. Nauk SSSR, Ser.Mat. 48(3), 569–584 (1984) [Math. USSR-Izv. 24 (3), 523–537 (1984)]. · Zbl 0545.10026
[51] A. A. Karatsuba, ”The distribution of zeros of the function \(\zeta\)(1/2 + it),” Izv. Akad. Nauk SSSR, Ser. Mat. 48(6), 1214–1224 (1984) [Math. USSR-Izv. 25 (3), 519–529 (1985)]. · Zbl 0558.10033
[52] A. A. Karatsuba, ”Zeros of the Riemann zeta function on the critical line,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 167, 167–178 (1985) [Proc. Steklov Inst. Math. 167, 187–200 (1986)]. · Zbl 0566.10030
[53] A. Selberg, ”On the zeros of Riemann’s zeta function,” Skr. Norske Vid. Akad. Oslo I 10, 1–59 (1942). · Zbl 0027.20201
[54] A. A. Karatsuba, ”On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line,” Izv. Ross. Akad. Nauk, Ser. Mat. 56(2), 372–397 (1992) [Izv.: Math. 40 (2), 353–376 (1993)]. · Zbl 0761.11034
[55] G. H. Hardy and J. E. Littlewood, ”Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes,” Acta Math. 41(1), 119–196 (1916). · JFM 46.0498.01 · doi:10.1007/BF02422942
[56] J. Moser, ”A certain sum in the theory of the Riemann zeta-function,” Acta Arith. 31(1), 31–43 (1976); ”On a Hardy-Littlewood theorem in the theory of the Riemann zeta-function,” Acta Arith. 31 (1), 45–51 (1976); ”Supplement: ’On a Hardy-Littlewood theorem in the theory of the Riemann zeta-function’,” Acta Arith. 35 (4), 403–404 (1979); ”Corrections to: ’A certain sum in the theory of the Riemann zeta-function’ [Acta Arith. 31 (1976), no. 1, 31–43]; ’On a Hardy-Littlewood theorem in the theory of the Riemann zetafunction” [ibid. 31 (1976), no. 1, 4551]; Supplement [ibid. 35 (1979), no. 4, 403404],” Acta Arith. 40 (1), 97–107 (1981). · Zbl 0296.10018 · doi:10.4064/aa-31-1-31-43
[57] A. A. Karatsuba, ”On the distance between consecutive zeros of the Riemann zeta function that lie on the critical line,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 157, 49–63 (1981). · Zbl 0472.10040
[58] A. A. Karatsuba, ”The Riemann zeta function and its zeros,” Uspekhi Mat. Nauk 40(5), 19–70 (1985) [Russ. Math. Surveys 40 (5), 23–82 (1985)]. · Zbl 0589.10040
[59] A. A. Lavrik, ”Uniform approximations and zeros in short intervals of the derivatives of the Hardy Z-function,” Anal. Math. 17(4), 257–279 (1991). · Zbl 0774.11049 · doi:10.1007/BF01905933
[60] A. Selberg, ”Contributions to the theory of the Riemann zeta-function,” Arch. Math. Naturvid. 48(5), 89–155 (1946). · Zbl 0061.08402
[61] A. A. Karatsuba, ”Density theorem and the behavior of the argument of the Riemann zeta function,” Mat. Zametki 60(3), 448–449 (1996) [Math. Notes 60 (3), 333–334 (1996)]. · doi:10.4213/mzm1847
[62] A. A. Karatsuba, ”On the function S(t),” Izv. Ross. Akad. Nauk, Ser. Mat. 60(5), 27–56 (1996) [Izv.: Math. 60 (5), 901–931 (1996)]. · Zbl 0898.11032 · doi:10.4213/im86
[63] A. A. Karatsuba, ”Omega theorems for zeta sums,” Mat. Zametki 73(2), 228–233 (2003) [Math. Notes 73 (2), 212–217 (2003)]. · Zbl 1035.11035 · doi:10.4213/mzm181
[64] A. A. Karatsuba, ”On the zeros of the Davenport-Heilbronn function lying on the critical line,” Izv. Akad. Nauk SSSR, Ser. Mat. 54(2), 303–315 (1990) [Math. USSR-Izv. 36 (2), 311–324 (1991)]. · Zbl 0701.11028
[65] H. Davenport and H. Heilbronn, ”On the zeros of certain Dirichlet series,” J. London Math. Soc. 11(3), 181–185 (1936); ”On the zeros of certain Dirichlet series. II,” J. London Math. Soc. 11 (4), 307–312 (1936). · JFM 62.0138.01 · doi:10.1112/jlms/s1-11.3.181
[66] S.M. Voronin, ”On the zeros of some Dirichlet series lying on the critical line,” Izv. Akad. Nauk SSSR, Ser. Mat. 44(1), 63–91 (1980) [Math. USSR-Izv. 16 (1), 55–82 (1981)]. · Zbl 0441.10038
[67] A. A. Karatsuba, ”On zeros of the Davenport-Heilbronn function,” in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, Italy, 1989) (Univ. Salerno, Salerno, 1992), pp. 271–293. · Zbl 0796.11035
[68] A. A. Karatsuba, ”A refinement of theorems on the number of zeros lying on intervals of the critical line of certain Dirichlet series,” Uspekhi Mat. Nauk 47(2), 193–194 (1992) [Russ. Math. Surveys 47 (2), 219–220 (1992)]. · Zbl 0798.11033
[69] A. A. Karatsuba, ”On the zeros of a special type of function connected with Dirichlet series,” Izv. Akad. Nauk SSSR, Ser. Mat. 55(3), 483–514 (1991) [Math. USSR-Izv. 38 (3), 471–502 (1992)]. · Zbl 0737.11021
[70] A. A. Karatsuba, ”Uniform approximation of the remainder term in the Dirichlet divisor problem,” Izv. Akad. Nauk SSSR, Ser. Mat. 36(3), 475–483 (1972) [Math. USSR-Izv. 6 (3), 467–475 (1972)]. · Zbl 0253.10035
[71] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Clarendon, Oxford, 1951; Inostrannaya Literatura, Moscow, 1953). · Zbl 0042.07901
[72] H.-E. Richert, ”Einführung in die Theorie der starken Rieszschen Summierbarkeit von Dirichletreihen,” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1960, 17–75 (1960). · Zbl 0095.05701
[73] A. A. Karatsuba, ”The multidimensional Dirichlet divisor problem and zero free regions for the Riemann zeta function,” Funct. Approx. Comment. Math. 28, 131–140 (2000). · Zbl 0977.11042
[74] A. A. Karatsuba, ”On the relationship between the multidimensional Dirichlet divisor problem and the boundary of zeros of \(\zeta\)(s),” Mat. Zametki 70(3), 477–480 (2001) [Math. Notes 70 (3), 432–435 (2001)]. · doi:10.4213/mzm760
[75] A. A. Karatsuba, ”Lower bounds for the maximum modulus of \(\zeta\)(s) in small domains of the critical strip,” Mat. Zametki 70(5), 796–797 (2001) [Math. Notes 70 (5), 724–726 (2001)]. · Zbl 1137.11332 · doi:10.4213/mzm792
[76] A. A. Karatsuba, ”Lower bounds for the maximummodulus of the Riemann zeta function on short segments of the critical line,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(6), 99–104 (2004) [Izv.: Math. 68 (6), 1157–1163 (2004)]. · doi:10.4213/im513
[77] R. Balasubramanian, ”On the frequency of Titchmarsh’s phenomenon for \(\zeta\)(s), IV,” Hardy-Ramanujan J. 9, 1–10 (1986). · Zbl 0662.10030
[78] A. A. Karatsuba, ”Sums of characters and primitive roots in finite fields,” Dokl. Akad. Nauk SSSR 180(6), 1287–1289 (1968).
[79] A. A. Karatsuba, ”Estimates of character sums,” Izv. Akad. Nauk SSSR, Ser. Mat. 34(1), 20–30 (1970) [Math. USSR-Izv. 4 (1), 19–29 (1970)].
[80] A. A. Karatsuba, ”Sums of characters over prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 34(2), 299–321 (1970) [Math. USSR-Izv. 4 (2), 303–326 (1970)].
[81] I. M. Vinogradov, ”An Estimate for a certain sum extended over the primes of an arithmetic progression,” Izv. Akad. Nauk SSSR, Ser. Mat. 30(3), 481–496 (1966). · Zbl 0144.28303
[82] A. A. Karatsuba, ”Sums of characters in sequences of shifted prime numbers, with applications,” Mat. Zametki 17(1), 155–159 (1975) [Math. Notes 17 (1), 91–93 (1975)]. · Zbl 0315.10034
[83] A. A. Karatsuba, ”The distribution of products of shifted prime numbers in arithmetic progressions,” Dokl. Akad. Nauk SSSR 192(4), 724–727 (1970).
[84] A. A. Karatsuba, ”Lower estimates of sums of polynomial characters,” Mat. Zametki 14(1), 67–72 (1973) [Math. Notes 14 (1), 593–596 (1973)].
[85] A. A. Karatsuba, ”Sums of Legendre symbols of polynomials of second degree over prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 42(2), 315–324 (1978) [Math. USSR-Izv. 12 (2), 299–308 (1978)]. · Zbl 0403.10016
[86] A. A. Karatsuba, ”Distribution of values of nonprincipal characters,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 142, 156–164 (1976) [Proc. Steklov Inst. Math. 142, 165–174 (1979)]. · Zbl 0413.10034
[87] A. A. Karatsuba, ”Weighted character sums,” Izv. Ross. Akad. Nauk, Ser. Mat. 64(2), 29–42 (2000) [Izv.: Math. 64 (2), 249–263 (2000)]. · Zbl 0963.11046 · doi:10.4213/im283
[88] A. A. Karatsuba, ”Distribution of values of Dirichlet characters on additive sequences,” Dokl. Akad. Nauk SSSR 319(3), 543–545 (1991) [Dokl. Math. 44 (1), 145–148 (1992)]. · Zbl 0772.11028
[89] A. A. Karatsuba, ”Sums of characters with prime numbers and their applications,” Tatra Mt. Math. Publ. 20, 155–162 (2000). · Zbl 0992.11051
[90] A. A. Karatsuba, ”Arithmetic problems in the theory of Dirichlet characters,” Uspekhi Mat. Nauk 63(4), 43–92 (2008) [Russ. Math. Surveys 63 (4), 641–690 (2008)]. · doi:10.4213/rm9234
[91] A. A. Karatsuba, ”Approximation of exponential sums by shorter ones,” Proc. Indian Acad. Sci. Math. Sci. 97(1–3), 167–178 (1987). · Zbl 0648.10023 · doi:10.1007/BF02837821
[92] I. M. Vinogradov, ”On the mean value of the number of classes of purely root forms of a negative determinant,” Soobshch. Kharkov. Mat. O-va 16(1–2), 10–38 (1918).
[93] J. G. van der Corput, ”Verschärfung der Abschätzung beim Teilerproblem,” Math. Ann. 87(1–2), 39–65 (1922). · JFM 48.0181.01 · doi:10.1007/BF01458035
[94] A. A. Karatsuba and M. A. Korolev, ”A theorem on the approximation of a trigonometric sum by a shorter one,” Izv. Ross. Akad. Nauk, Ser. Mat. 71(2), 123–150 (2007) [Izv.: Math. 71 (2), 341–370 (2007)]. · Zbl 1168.11030 · doi:10.4213/im1136
[95] A. A. Karatsuba and E. A. Karatsuba, ”Applications of ATS in a quantum-optical model,” in Analysis and Mathematical Physics (Birkhäuser, Basel, 2009), pp. 211–232. · Zbl 1297.81193
[96] A. A. Karatsuba and E. A. Karatsuba, ”A resummation formula for collapse and revival in the Jaynes-Cummings model,” J. Phys. A: Math. Theor. 42(19), 195304 (2009). · Zbl 1163.81016 · doi:10.1088/1751-8113/42/19/195304
[97] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen (Teubner, Leipzig, 1909), Vol. 2. · JFM 40.0232.08
[98] M. E. Changa, ”Numbers whose prime divisors lie in special intervals,” Izv. Ross. Akad. Nauk, Ser. Mat. 67(4), 213–224 (2003) [Izv.: Math. 67 (4), 837–848 (2003)]. · Zbl 1066.11041 · doi:10.4213/im447
[99] A. A. Karatsuba, ”On a property of the set of primes as a multiplicative basis of natural numbers,” Dokl. Akad. Nauk 439(2), 159–162 (2011) [Dokl. Math. 84 (1), 467–470 (2011)]. · Zbl 1252.11072
[100] A. A. Karatsuba, ”A property of the set of prime numbers,” Uspekhi Mat. Nauk 66(2), 3–14 (2011) [Russ. Math. Surveys 66 (2), 209–220 (2011)]. · doi:10.4213/rm9419
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