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Analytic number theory in India during 2001-2010. (English) Zbl 1431.11003

Summary: In this article we summarize the contribution of Indian mathematicians to analytic number theory during 2001–2010.

MSC:

11-03 History of number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11Nxx Multiplicative number theory
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[1] S. D. Adhikari, A. A. Ambily, and B. Sury, Zero-sum problems with subgroup weights, Proc. Indian Acad. Sci. Math. Sci., 120(3) (2010), 259-266. · Zbl 1207.11029 · doi:10.1007/s12044-010-0035-y
[2] S. D. Adhikari, S. Gun, and P. Rath, Remarks on some zero-sum theorems, Proc. Indian Acad. Sci. Math. Sci., 119(3) (2009), 275-281. · Zbl 1189.11014 · doi:10.1007/s12044-009-0027-y
[3] S. D. Adhikari and A. Granville, Visibility in the plane, J. Number Theory, 129(10) (2009), 2335-2345. · Zbl 1176.11027 · doi:10.1016/j.jnt.2009.02.019
[4] S. D. Adhikari, C. David, and J. Jiménez Urroz, Generalizations of some zero-sum theorems, Integers, 8(A52) (2008), 11pp. · Zbl 1202.11018
[5] S. D. Adhikari, M. N. Chintamani, B. K. Moriya, and P. Paul, Weighted sums in finite abelian groups, Unif. Distrib. Theory, 3(1) (2008), 105-110. · Zbl 1217.11019
[6] S. D. Adhikari, R. Balasubramanian, F. Pappalardi, and P. Rath, Some zero-sum constants with weights, Proc. Indian Acad. Sci. Math. Sci., 118(2) (2008), 183-188. · Zbl 1207.11030 · doi:10.1007/s12044-008-0010-z
[7] S. D. Adhikari and Y-G. Chen, Davenport constant with weights and some related questions, II, J. Combin. Theory Ser. A, 115(1) (2008), 178-184. · Zbl 1210.11031 · doi:10.1016/j.jcta.2007.03.004
[8] S. D. Adhikari, R. Balasubramanian, and P. Rath, Some combinatorial group invariants and their generalizations with weights, Additive combinatorics, 327-335, CRM Proc. Lecture Notes, 43, Amer. Math. Soc., Providence, RI, 2007 · Zbl 1173.11010 · doi:10.1090/crmp/043/18
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[12] S. D. Adhikari, G. Coppola, and A. Mukhopadhyay, On the average of the sum-of-p-prime-divisors function, Acta Arith., 101(4) (2002), 333-338. · Zbl 0997.11073 · doi:10.4064/aa101-4-3
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[17] R. Balasubramanian and K. Ramachandra, Some problems of analytic number theory, IV, Hardy-Ramanujan J., 25 (2002), 5-21. · Zbl 1027.11071
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[19] R. Balasubramanian and K. Ramachandra, Mean square of the Hurwitz zeta-function and other remarks, Hardy-Ramanujan J., 27 (2004), 8-27 (2005). · Zbl 1113.11054
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[22] R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math., 175(3) (2006), 285-304. · Zbl 1110.30001 · doi:10.4064/sm175-3-7
[23] Balasubramanian, R.; Bhowmik, G., Upper bounds for the Davenport constant, 61-69 (2007), Berlin · Zbl 1173.11011
[24] R. Balasubramanian and G. Prakash, Asymptotic formula for sum-free sets in abelian groups, Acta Arith., 127(2) (2007), 115-124. · Zbl 1127.11016 · doi:10.4064/aa127-2-2
[25] R. Balasubramanian, S. Laishram, T. N. Shorey, and R. Thangadurai, The number of prime divisors of a product of consecutive integers, J. Comb. Number Theory, 1(3) (2009), 253-261. · Zbl 1256.11027
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[27] Y. Bugeaud and T. N. Shorey, On the Diophantine equation \[\frac{x^m1}{x-1}=\frac{y^n1}{y-1}\] xm1x−1=yn1y−1, Pacific J. Math., 207(1) (2002), 61-75. · Zbl 1047.11028 · doi:10.2140/pjm.2002.207.61
[28] Choie, Y. J.; Sankaranarayanan, A.; Sengupta, J., On the sign changes of Hecke eigenvalues, 25-32 (2009), New Delhi · Zbl 1247.11077 · doi:10.1007/978-93-86279-46-0_3
[29] K. Chakraborty and A. Mukhopadhyay, Exponents of class groups of real quadratic function fields, Proc. Amer. Math. Soc., 132(7) (2004), 1951-1955. · Zbl 1042.11079 · doi:10.1090/S0002-9939-04-07269-7
[30] K. Chakraborty and A. Mukhopadhyay, Exponents of class groups of real quadratic function fields, II, Proc. Amer. Math. Soc., 134(1) (2006), 51-54. · Zbl 1137.11348 · doi:10.1090/S0002-9939-05-07953-0
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[32] K. Chakraborty, F. Luca, and A. Mukhopadhyay, Class numbers with many prime factors, J. Number Theory, 128(9) (2008), 2559-2572. · Zbl 1236.11095 · doi:10.1016/j.jnt.2008.03.010
[33] K. Chakraborty and M. Ram Murty, On the number of real quadratic fields with class number divisible by 3, Proc. Amer. Math. Soc., 131(1) (2003), 41-44. · Zbl 1024.11073 · doi:10.1090/S0002-9939-02-06603-0
[34] K. Chakraborty, S. Kanemitsu, and T. Kuzumaki, Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function, Hardy-Ramanujan J., 32 (2009), 38-53. · Zbl 1203.11057
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[36] K. Chakraborty, S. Kanemitsu, and H-L. Li, On the values of a class of Dirichlet series at rational arguments, Proc. Amer. Math. Soc., 138(4) (2010), 1223-1230. · Zbl 1185.11052 · doi:10.1090/S0002-9939-09-10171-5
[37] Y-G. Chen and A. Mukhopadhyay, The view-obstruction problem for polygons, Publ. Math. Debrecen, 60(1-2) (2002), 101-105. · Zbl 0996.11047
[38] J. Cilleruelo, D. S. Ramana, and O. Ramaré, The number of rational numbers determined by large sets of integers, Bull. Lond. Math. Soc., 42(3) (2010), 517-526. · Zbl 1205.11016 · doi:10.1112/blms/bdq021
[39] S. Cooper, S. Gun, and B. Ramakrishnan, On the lacunarity of two-eta-products, Georgian Math. J., 13(4) (2006), 659-673. · Zbl 1163.11033
[40] S. Das, Note on Hermitian Jacobi forms, Tsukuba J. Math., 34(1) (2010), 59-78. · Zbl 1223.11064 · doi:10.21099/tkbjm/1283967408
[41] S. Das, Some aspects of Hermitian Jacobi forms, Arch. Math. (Basel), 95(5) (2010), 423-437. · Zbl 1211.11062 · doi:10.1007/s00013-010-0176-3
[42] S. Das, Nonvanishing of Jacobi Poincaré series, J. Aust. Math. Soc., 89(2) (2010), 165-179. · Zbl 1248.11035 · doi:10.1017/S1446788710001576
[43] J-M. De Koninck, F. Luca, and A. Sankaranarayanan, Positive integers whose Euler function is a power of their kernel function, Rocky Mountain J. Math., 36(1) (2006), 81-96. · Zbl 1141.11045 · doi:10.1216/rmjm/1181069489
[44] S. Ganguly, J. Hoffstein, and J. Sengupta, Determining modular forms on SL2(Z) by central values of convolution L-functions, Math. Ann., 345(4) (2009), 843-857. · Zbl 1234.11065 · doi:10.1007/s00208-009-0380-2
[45] S. Ganguly, On the dimension of the space of cusp forms of octahedral type, Int. J. Number Theory, 6(4) (2010), 767-783. · Zbl 1208.11058 · doi:10.1142/S1793042110003198
[46] S. Ganguly and G. Pal, Integers without large prime factors in short intervals, conditional results, Proc. Indian Acad. Sci. Math. Sci., 120(5) (2010), 515-524. · Zbl 1276.11156 · doi:10.1007/s12044-010-0049-5
[47] W. D. Gao and R. Thangadurai, On the structure of sequences with forbidden zero-sum subsequences, Colloq. Math., 98(2) (2003), 213-222. · Zbl 1057.11011 · doi:10.4064/cm98-2-7
[48] W. D. Gao, I. Z. Ruzsa, and R. Thangadurai, Olsons constant for the group ℚp p ℚp, J. Combin. Theory Ser. A, 107(1) (2004), 49-67. · Zbl 1107.11014 · doi:10.1016/j.jcta.2004.03.007
[49] W. D. Gao, A. Panigrahi, and R. Thangadurai, On the structure of p-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem, Proc. Indian Acad. Sci. Math. Sci., 115(1) (2005), 67-77. · Zbl 1113.11012 · doi:10.1007/BF02829840
[50] W. D. Gao and R. Thangadurai, On zero-sum sequences of prescribed length, Aequationes Math., 72(3) (2006), 201-212. · Zbl 1111.11014 · doi:10.1007/s00010-006-2841-y
[51] W. D. Gao, Q. H. Hou, W. A. Schmid, and R. Thangadurai, On short zero-sum subsequences, II, Integers, 7(A21) (2007), 22 pp. · Zbl 1201.11030
[52] M. Z. Garaev and A. Sankaranarayanan, The sum involving derivative of ζ(s) over simple zeros, J. Number Theory, 117(1) (2006), 122-130. · Zbl 1168.11318 · doi:10.1016/j.jnt.2005.05.016
[53] S. Gun, On the zeros of certain cusp forms, Math. Proc. Cambridge Philos. Soc., 141(2) (2006), 191-195. · Zbl 1153.11018 · doi:10.1017/S0305004106009522
[54] S. Gun, Transcendental zeros of certain modular forms, Int. J. Number Theory, 2(4) (2006), 549-553. · Zbl 1201.11045 · doi:10.1142/S1793042106000711
[55] S. Gun, M. Manickam, and B. Ramakrishnan, A canonical subspace of modular forms of half-integral weight, Math. Ann., 347(4) (2010), 899-916. · Zbl 1219.11070 · doi:10.1007/s00208-009-0455-0
[56] S. Gun and B. Ramakrishnan, On special values of certain Dirichlet L-functions, Ramanujan J., 15(2) (2008), 275-280. · Zbl 1194.11050 · doi:10.1007/s11139-007-9077-x
[57] S. Gun and B. Ramakrishnan, On the representation of integers as sums of an odd number of squares, Ramanujan J., 15(3) (2008), 367-376. · Zbl 1241.11045 · doi:10.1007/s11139-007-9082-0
[58] S. Gun, B. Ramakrishnan, B. Sahu, and R. Thangadurai, Distribution of quadratic non-residues which are not primitive roots, Math. Bohem., 130(4) (2005), 387-396. · Zbl 1105.11034
[59] S. Gun, F. Luca, P. Rath, B. Sahu, and R. Thangadurai, Distribution of residues modulo p, Acta Arith., 129(4) (2007), 325-333. · Zbl 1133.11056 · doi:10.4064/aa129-4-3
[60] K. Györy, L. Hajdu, and N. Saradha, On the Diophantine equation n(n + d)(n + (k1)d) = byℓ, Canad. Math. Bull., 47(3) (2004), 373-388. · Zbl 1115.11020 · doi:10.4153/CMB-2004-037-1
[61] N. Hirata-Kohno, S. Laishram, T. N. Shorey, and R. Tijdeman, An extension of a theorem of Euler, Acta Arith., 129(1) (2007), 71-102. · Zbl 1137.11022 · doi:10.4064/aa129-1-6
[62] H. Iwaniec, W. Kohnen, and J. Sengupta, The first negative Hecke eigenvalue, Int. J. Number Theory, 3 (2007), 355-363. · Zbl 1219.11066 · doi:10.1142/S1793042107001024
[63] T. Jagathesan and M. Manickam, On Shimura correspondence for non-cusp forms of half-integral weight, J. Ramanujan Math. Soc., 23(3) (2008), 211-222. · Zbl 1168.11012
[64] M. Jutila and K. Srinivas, Gaps between the zeros of Epsteins zeta-functions on the critical line, Bull. London Math. Soc., 37(1) (2005), 45-53. · Zbl 1166.11313 · doi:10.1112/S0024609304003716
[65] W. Kohnen and J. Sengupta, Nonvanishing of symmetric square L-functions of cusp forms inside the critical strip, Proc. Amer. Math. Soc., 128(6) (2000), 1641-1646. · Zbl 1029.11015 · doi:10.1090/S0002-9939-99-05419-2
[66] W. Kohnen and J. Sengupta, On quadratic character twists of Hecke L-functions attached to cusp forms of varying weights at the central point, Acta Arith., 99(1) (2001), 61-66. · Zbl 0982.11026 · doi:10.4064/aa99-1-5
[67] W. Kohnen and J. Sengupta, A certain class of Poincaré series on Spn, II, Tohoku Math. J. (2), 54(1) (2002), 61-69. · Zbl 1039.11027 · doi:10.2748/tmj/1113247179
[68] W. Kohnen and J. Sengupta, On the average of central values of symmetric square L-functions in weight aspect, Nagoya Math. J., 167 (2002), 95-100. · Zbl 1048.11040 · doi:10.1017/S0027763000025447
[69] W. Kohnen and J. Sengupta, Waldspurgers formula and central critical values of L-functions of newforms in weight aspect, Number theoretic methods (Iizuka, 2001), 213-217, Dev. Math., 8, Kluwer Acad. Publ., Dordrecht, 2002. · Zbl 1132.11326
[70] W. Kohnen and J. Sengupta, On Koecher-Maass series of Siegel modular forms, Math. Z., 242(1) (2002), 149-157. · Zbl 1041.11036 · doi:10.1007/s002090100311
[71] Kohnen, W.; Sankaranarayanan, A.; Sengupta, J., The quadratic mean of automorphic L-functions, 262-279 (2006), Hackensack, NJ · Zbl 1109.11030 · doi:10.1142/9789812774415_0012
[72] W. Kohnen and J. Sengupta, On the first sign change of Hecke eigenvalues of newforms, Math. Z., 254(1) (2006), 173-184. · Zbl 1220.11060 · doi:10.1007/s00209-006-0940-z
[73] W. Kohnen and J. Sengupta, The first negative Hecke eigenvalue of a Siegel cusp form of genus two, Acta Arith., 129(1) (2007), 53-62. · Zbl 1134.11019 · doi:10.4064/aa129-1-4
[74] W. Kohnen and J. Sengupta, Signs of Fourier coefficients of two cusp forms of different weights, Proc. Amer. Math. Soc., 137(11) (2009), 3563-3567. · Zbl 1259.11047 · doi:10.1090/S0002-9939-09-09982-1
[75] K. Kumarasamy and M. Manickam, On Taylor expansion of Jacobi forms of half-integral weight, J. Ramanujan Math. Soc., 23(2) (2008), 167-182. · Zbl 1221.11119
[76] S. Laishram and T. N. Shorey, Number of prime divisors in a product of consecutive integers, Acta Arith., 113(4) (2004), 327-341. · Zbl 1046.11004 · doi:10.4064/aa113-4-3
[77] S. Laishram and T. N. Shorey, Number of prime divisors in a product of terms of an arithmetic progression, Indag. Math. (N.S.), 15(4) (2004), 505-521. · Zbl 1142.11356 · doi:10.1016/S0019-3577(04)80015-6
[78] S. Laishram and T. N. Shorey, The greatest prime divisor of a product of consecutive integers, Acta Arith., 120(3) (2005), 299-306. · Zbl 1165.11334 · doi:10.4064/aa120-3-5
[79] S. Laishram, An estimate for the length of an arithmetic progression the product of whose terms is almost square, Publ. Math. Debrecen, 68(3-4) (2006), 451-475. · Zbl 1111.11018
[80] S. Laishram and T. N. Shorey, Grimms conjecture on consecutive integers, (English summary), Int. J. Number Theory, 2(2) (2006), 207-211. · Zbl 1116.11008 · doi:10.1142/S1793042106000498
[81] S. Laishram and T. N. Shorey, The greatest prime divisor of a product of terms in an arithmetic progression, Indag. Math. (N.S.), 17(3) (2006), 425-436. · Zbl 1165.11014 · doi:10.1016/S0019-3577(06)80042-X
[82] S. Laishram and T. N. Shorey, The equation n(n + d)(n + (k-1)d) = by2 with ω(d)6 or d1010, Acta Arith., 129(3) (2007), 249-305. · Zbl 1140.11020 · doi:10.4064/aa129-3-2
[83] S. Laishram and T. N. Shorey, Squares in arithmetic progression with at most two terms omitted, Acta Arith., 134(4) (2008), 299-316. · Zbl 1177.11031 · doi:10.4064/aa134-4-1
[84] S. Laishram, T. N. Shorey, and S. Tengely, Squares in products in arithmetic progression with at most one term omitted and common difference a prime power, Acta Arith., 135(2) (2008), 143-158. · Zbl 1158.11018 · doi:10.4064/aa135-2-4
[85] S. Laishram and T. N. Shorey, Irreducibility of generalized Hermite-Laguerre polynomials, II, Indag. Math. (N.S.), 20(3) (2009), 427-434. · Zbl 1196.33009 · doi:10.1016/S0019-3577(09)80016-5
[86] S. Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory, 6(8) (2010), 1869-1873. · Zbl 1230.11011 · doi:10.1142/S1793042110003848
[87] H. Lao and A. Sankaranarayanan, The average behavior of Fourier coefficients of cusp forms over sparse sequences, Proc. Amer. Math. Soc., 137(8) (2009), 2557-2565. · Zbl 1225.11059 · doi:10.1090/S0002-9939-09-09855-4
[88] F. Luca, A. Mukhopadhyay, and K. Srinivas, Some results on Oppenheims factorisatio numerorum function, Acta Arith., 142(1) (2010), 41-50. · Zbl 1213.11020 · doi:10.4064/aa142-1-3
[89] F. Luca and A. Sankaranarayanan, On the moments of the Carmichael λ function, Acta Arith., 123(4) (2006), 389-398. · Zbl 1158.11040 · doi:10.4064/aa123-4-7
[90] F. Luca and A. Sankaranarayanan, On positive integers n such that ϕ(1) + ϕ(2) + + ϕ(n) is a square, Bol. Soc. Mat. Mexicana (3), 14(1) (2008), 1-6. · Zbl 1211.11005
[91] F. Luca and T. N. Shorey, Diophantine equations with products of consecutive terms in Lucas sequences, J. Number Theory, 114(2) (2005), 298-311. · Zbl 1081.11023 · doi:10.1016/j.jnt.2004.08.007
[92] F. Luca and T. N. Shorey, Diophantine equations with products of consecutive terms in Lucas sequences, II, Acta Arith,. 133(1) (2008), 53-71. · Zbl 1230.11041 · doi:10.4064/aa133-1-4
[93] F. Luca, I. E. Shparlinski, and R. Thangadurai, Quadratic non-residues versus primitive roots modulo p, J. Ramanujan Math. Soc., 23(1) (2008), 97-104. · Zbl 1202.11045
[94] F. Luca and R. Thangadurai, On an arithmetic function considered by Pillai, J. Théor. Nombres Bordeaux, 21(3) (2009), 693-699. · Zbl 1201.11092 · doi:10.5802/jtnb.695
[95] H. Maier and A. Sankaranarayanan, On a certain general exponential sum, Int. J. Number Theory, 1(2) (2005), 183-192. · Zbl 1107.11031 · doi:10.1142/S1793042105000224
[96] H. Maier and A. Sankaranarayanan, On an exponential sum involving the Möbius function, Hardy-Ramanujan J., 28 (2005), 10-29. · Zbl 1115.11053
[97] H. Maier and A. Sankaranarayanan, On certain exponential sums over primes, J. Number Theory, 129(7) (2009), 1669-1677. · Zbl 1192.11052 · doi:10.1016/j.jnt.2009.01.018
[98] H. Maier and A. Sankaranarayanan, The behaviour in short intervals of exponential sums over sifted integers, Illinois J. Math., 53(1) (2009), 111-133. · Zbl 1279.11101 · doi:10.1215/ijm/1264170842
[99] H. Maier and A. Sankaranarayanan, Exponential sums over primes in residue classes, Int. J. Number Theory, 6(4) (2010), 905-918. · Zbl 1204.11134 · doi:10.1142/S1793042110003319
[100] M. Manickam and B. Ramakrishnan, On Shimura, Shintani and Eichler-Zagier correspondences, Trans. Amer. Math. Soc., 352(6) (2000), 2601-2617. · Zbl 0985.11020 · doi:10.1090/S0002-9947-00-02423-5
[101] M. Manickam and B. Ramakrishnan, On Saito-Kurokawa correspondence of degree two for arbitrary level, J. Ramanujan Math. Soc., 17(3) (2002), 149-160. · Zbl 1068.11028
[102] M. Manickam and B. Ramakrishnan, An estimate for a certain average of the special values of character twists of modular L-functions, Proc. Amer. Math. Soc., 133(9) (2005), 2515-2517. · Zbl 1166.11327 · doi:10.1090/S0002-9939-05-08109-8
[103] M. Manickam and B. Ramakrishnan, An Eichler-Zagier map for Jacobi forms of half-integral weight, Pacific J. Math., 227(1) (2006), 143-150. · Zbl 1148.11023 · doi:10.2140/pjm.2006.227.143
[104] K. Matsumoto and A. Sankaranarayanan, On the mean square of standard L-functions attached to Ikeda lifts, Math. Z., 253(3) (2006), 607-622. · Zbl 1191.11016 · doi:10.1007/s00209-005-0926-2
[105] A. Mukhopadhyay and T. N. Shorey, Almost squares in arithmetic progression, II, Acta Arith., 110(1) (2003), 1-14. · Zbl 1030.11010 · doi:10.4064/aa110-1-1
[106] A. Mukhopadhyay and T. N. Shorey, Square free part of products of consecutive integers, Publ. Math. Debrecen, 64(1-2) (2004), 79-99. · Zbl 1049.11037
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