Handbook of Number Theory I by József Sándor Babes-Bolyai University of Cluj, Cluj-Napoca, Romania Dragoslav S Mitrinović formerly of the University of Belgrade, Servia and Borislav Crstici formerly of the Technical University of Timisoara, Romania A C.I.P Catalogue record for this book is available from the Library of Congress ISBN-10 ISBN-13 ISBN-10 ISBN-13 1-4020-4215-9 (HB) 978-1-4020-4215-7 (HB) 1-4020-3658-2 (e-book) 978-1-4020-3658-3 (e-book) Published by Springer, P.O Box 17, 3300 AA Dordrecht, The Netherlands www.springeronline.com Printed on acid-free paper 1st ed 1995 2nd printing All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed in the Netherlands TABLE OF CONTENTS PREFACE xxv BASIC SYMBOLS BASIC NOTATIONS Chapter I EULER’S ϕ-FUNCTION § I Elementary inequalities for § I Inequalities for (mn) § I Relations connecting , , d § I Inequalities for Jk , k , k § I Unitary analogues of Jk , k , d § I Composition of , , § I Composition of , § I On the function n/(n) § I Minimum of (n)/n for consecutive values of n § I.10 On (n + 1)/(n) § I.11 On ((n + 1), (n)) § I.12 On (n, (n)) § I.13 The difference of consecutive totients § I.14 Nonmonotonicity of (A measure) § I.15 Nonmonotonicity of Jk § I.16 Number of solutions of (x) = n! § I.17 Number of solutions of (x) = m § I.18 Number of values of less than or equal to x § I.19 On composite n with (n)|(n − 1) (Lehmer’s conjecture) § I.20 Number of composite n ≤ x with (n)|(n − 1) § I.21 (n) 9 10 11 12 13 13 14 15 16 18 18 19 19 20 20 21 22 23 24 24 n≤x § I.22 f k≤n § · (k) 25 x 2 25 (n)/n 27 (n) − I.23 On n≤x § k n I.24 On n≤x vi Table of Contents Jk (n) − x k+1 /(k + 1) (k + 1) 28 An expansion of Jk On n≤x 1/(n) and related questions p≤x ( p − 1) for p prime On n≤x ( f (n)), f a polynomial ∗ n≤x (n), n≤x (n) (n + k) and related results Asymptotic formulae for generalized Euler functions On (x, n) = m≤x,(m,n)=1 and on Jacobstahl’s arithmetic function § I.33 On the iteration of § I.34 Iterates of and the order of (k) (n)/(k+1) (n) § I.35 Normal order of ((n)) 29 29 30 31 31 32 33 34 35 36 Chapter II THE ARITHMETICAL FUNCTION d(n), ITS GENERALIZATIONS AND ITS ANALOGUES § II The divisor functions at consecutive integers § II On d(n + i ) > · · · > d(n + ir ) § II Relations connecting d, , , dk § II On d(mn) § II An inequality for dk (n) § II Majorization for log d(n)/ log § II max d(n) and max(d(n), d(n + 1)) and generalizations 39 39 40 41 42 42 42 44 § I.25 On n≤x § I.26 I.27 § I.28 § I.29 § I.30 § I.31 § I.32 § n≤x § n≤x II Highly composite, superior highly composite, and largely composite numbers § II Congruence property of d(n) § II.10 (x) = d(n) − x log x − (2␥ − 1)x 45 47 47 n≤x § d( p − 1), p prime II.11 49 p≤x dk (n) − x · Pk−1 (log x), k ≥ 51 dk2 (n) 55 (g ∗ dk ) (n) 55 II.15 (x) II.16 The divisor problem in arithmetic progressions § II.17 On 1/dk (n) 56 57 59 § II.12 k (x) = n≤x § II.13 n≤x § § II.14 On n≤x § n≤x Table of Contents vii § II.18 Average order of dk (n) over integers free of large prime factors § II.19 On a sum on dk and Legendre’s symbol § II.20 A sum on dk , d and § II.21 On d(n) · d(n + N ) and related problems 60 60 61 61 n≤x dk (n) · d(n + 1) and related questions 63 II.23 Iteration of d II.24 On d( f (n)) and d(d( f (n))), f a polynomial § II.25 On d(n + a) and d(m + n ) 65 66 67 § II.22 On n≤x § § n≤x § § § II.26 m,n≤x d(| f (r, s)|), f (x, y) a binary cubic form 68 II.27 Weighted divisor problem II.28 On d(n − k t ) 68 69 | f (r,s)|≤N § II.29 § II.30 § II.31 § II.32 § II.33 k f (x) On (2 p − 1), (a n − 1)/n -highly composite, -largely composite and -interesting numbers On (n)/n On (n, (n)) = and (n, (n)) = On ((n, (n))) = k Gaussian law of errors for On the statistical property of prime factors of natural numbers in arithmetic progressions Distribution of values of in short intervals Distribution of in the sieve of Eratosthenes Number of n ≤ x with (n) = i Number of n ≤ x with (n) = i The functions (n; E) and S(x, y; E, ) Sumsets with many prime factors On the integers n for which (n) = k Chapter VI FUNCTION µ; k-FREE AND k-FULL NUMBERS § VI Average order of (n) § VI Estimates for M(x) Mertens’ conjecture § VI in short intervals § VI Sums involving (n) with p(n) > y or P(n) < y, n ≤ x Squarefree numbers with restricted prime factors § VI Oscillatory properties of M(x) and related results § VI The function M(n, T ) = (n) 171 172 172 173 173 174 174 175 176 177 177 177 180 183 184 185 187 187 187 189 189 190 192 d|n,d≤T § VI Măobius function of order k § VI Sums on (n)/n, (n)/n , 2 (n)/n § VI Sums on (n) log n/n, (n) log n/n § VI.10 Selberg’s formula x § VI.11 A sum on (n) n § VI.12 A sum on (n) f (n)/n, f -multiplicative, ≤ f ( p) ≤ § VI.13 Gandhi’s formula § VI.14 An extremal property of Đ VI.15 On a sum connected with the Mă obius function 2 (n) 2 (n) 2 (n) (n) § VI.16 Sums over , , , (n) 2 (n) (n) nd(n) 193 194 195 196 197 197 197 198 199 199 xii Table of Contents § VI.17 The distribution of integers having a given number of prime factors § VI.18 Number of squarefree integers ≤ x § VI.19 On squarefree integers § VI.20 Intervals containing a squarefree integer § VI.21 Distribution of squarefree numbers § VI.22 On the frequency of pairs of squarefree numbers § VI.23 Smallest squarefree integer in an arithmetic progression § VI.24 The greatest squarefree divisor of n § VI.25 Estimates involving the greatest squarefree divisor of n § VI.26 Estimates for N (x, y) = card {n ≤ x : ␥ (n) ≤ y} § VI.27 Number of non-squarefree odd, positive integers ≤ x § VI.28 Number of squarefree numbers ≤ X which are quadratic residues (mod p) § VI.29 Squarefree integers in nonlinear sequences § VI.30 Sumsets containing squarefree and k-free integers § VI.31 On the Mă obius function § VI.32 Number of k-free integers ≤ x § VI.33 Number of k-free integers ≤ x, which are relatively prime to n § VI.34 Schnirelmann density of the k-free integers § VI.35 Powerfree integers represented by linear forms § VI.36 On the power-free value of a polynomial § VI.37 Number of r -free integers ≤ x that are in arithmetic progression § VI.38 Squarefree numbers as sums of two squares § VI.39 Distribution of unitary k-free integers § VI.40 Counting function of the (k, r )-integers § VI.41 Asymptotic formulae for powerful numbers § VI.42 Maximal k-full divisor of an integer § VI.43 Number of squarefull integers between successive squares Chapter VII FUNCTION π(x), ψ(x), θ(x), AND THE SEQUENCE OF PRIME NUMBERS § VII Estimates on (x) Chebyshev’s theorem The prime number theorem x dy § VII Approximation of (x) by log y § VII On (x) − li x Sign changes § VII On (x) − (x − y) for y = x § VII On (x + y) ≤ (x) + (y) § VII On ( ∗ (k) − (k)) q≤k≤n 200 201 202 202 204 205 206 208 209 210 210 211 211 212 213 213 216 217 218 218 220 221 221 222 222 226 226 227 227 228 229 232 235 237 608 Chapter XVI n≤x f (n)=0 = C1 x L (x) + · · · + C N x L N (x) log1−N x + O(x log−N x) f (n) where each L j (x) ( j = 1, , N ) is a slowly oscillating function asymptotic to 1/ log log x b) Let f (n) be a non-negative integer-valued additive arithmetical function such that for every prime p, f ( p) = 0, f ( p ) = and < f ( p k ) < Ck for every k ≥ and some fixed C > Then for every fixed integer N ≥ there exist computable constants e0 , e1 , , e N such that = e0 x + e1 x 1/2 · L (x) log−1 x + · · · + en x 1/2 · L N (x) log−N x + f (n) n≤x f (n)=0 + O(x 1/2 · log−N −1 x) where each L j (x)( j = 1, , N ) is a slowly oscillating function asymptotic to 1/ log log x c) Let f (n) and g(n) be two non-negative integer-valued additive arithmetical functions such that for every prime p, f ( p) = g( p) = 1, and f ( p k ) < Ck, g( p k ) < Ck for every k ≥ and some C > Then for every fixed integer N ≥ there exist computable constants a j , b j ( j = 1, , N ) such that g(n) a N + b N L N (x) a2 + b2 L (x) + = x · a1 + b1 L (x) + + ··· + f (n) log x log N −1 x n≤x f (n)=0 + O(x/ log N x) (See J.-M de Koninck and A Ivi´c, pp 133–134.) 4) Let f (n) be a non-negative integer-valued arithmetic function such that: (∗) (∗∗) f (n) is multiplicative and f ( p) = for all primes p or f (n) is additive and f ( p) = for all primes p If h = o(x) as x → ∞, then, uniformly on |z| ≤ 1, z f (n) = G(z) · h + O(hx −1/4 log x) + x 1/2 for which k f ( p ) · p −k < +∞ if this set is non-empty and p,k≥2 0 ( ) = +∞ otherwise Let E be the set of all ≥ for which 0 ( ) < and let R > be the supremum of the set E Then for every fixed integer N ≥ there exist functions A0 (z), A1 (z), , A N (z) analytic on |z| ≤ such that A0 (0) = A1 (0) = = A N (0) = 0, and N z f (n) = x(log x)z−1 · n≤x A j (z) log− j x + O(log−N −1 x) j=0 where the O-constant is uniform for |z| ≤ § XVI.50 Semigroup-valued multiplicative functions 1) An arithmetical function f is stable, if the set f −1 (g) posseses an asymptotic density for every g ∈ Im f If these sets posses logarithmic densities, f is called logarithmically stable a) Let G be a commutative semigroup Then every G-multiplicative function (i.e a G-valued multiplicative function) is logarithmically stable b) We call a commutative semigroup G stabilizing, if every G-multiplicative function is stable Call a semigroup almost-group, if G = G ∪ G , where G is a group and G is finite Any semigroup which is the direct product of finitely many almost group is stabilizing I.Z Ruzsa Semigroup-valued multiplicative functions Acta Arith 62 (1982), 79–90 610 Chapter XVI 2) Let G be an Abelian group and let f be G-multiplicative Let d(A) denote the (asymptotic) density of the set A Call f concentrated if there is a finite subgroup G of G such that 1/ p < ∞ and deconcentrated otherwise f ( p)∈G / Then either: (i) f is deconcentrated, d( f −1 (g)) = for all g ∈ G or (ii) f is concentrated, d( f −1 (g)) = d( f −1 (g)) > for every g ∈ Im f and g∈G I.Z Ruzsa General multiplicative functions Acta Arith 32 (1977), 313–347 3) A class R of subsets of a semigroup G is called divisible if K ∈ R, g ∈ G imply g −1 K ∈ R Let G be a semigroup, f, f be G-multiplicative functions and R a divisible class of subsets of G Suppose moreover that 1/ p < ∞ and that either f ( p)= f ( p) f (2k ) = f (2k ) for all k or f (2k ) = f (2)k for all k If the sequences f 0−1 (K ) have densities for all K ∈ R, then so the sequences f −1 (K ) The same assertion holds for logarithmic density (See I.Z Ruzsa (1977).) Index of authors Abbott, H.L.: II 31; III 38; XII Ablyalimov, S.B.: II 32 Adhikari, S.D.: I 25 Afuwape, A.U.: XII 29 Agarwala, B.K.: XIV Aiello, W.: III 48 Alaoglu, L.: I 7; III 15, 41 Alexander, L.B.: III 37 Alladi, K.: IV 24, 25, 27; V 25, 30; VI 4; XII 20; XVI 43 Allakov, I.: X 18; XI 25 Almkvist, G.: XIV 1, Anderson, D.R.: X 12 Anderson, I.: V 26 Anderson, R.J.: V 8; VI Andrews, G.E.: XIV Ankeny, N.C.: IX 17; XV Annapurna, V.: III Apostol, T.M.: VI 7; XI Archibald, R.: VII 14 Arhipov, G.I.: X Artin, E.: XV 23 ◦ Aslund, N.: XII Atanassov, K.T.: III 13; XII 10 Atkinson, F.V.: II 15 Aull, C.E.: III 38 Auluck, F.C.: XIV 7, 12 Ax, J.: XV 34 Axer, A.: VI 32 Ayoub, R.: V 8; XV 17 Babaev, G.: II 21; III Babu, G.J.: V 29; IX 20 Backlund, R.J.: VII 25 Badarăev, A.S.: III 20 Badea, C.: VII 14; IX 26 Bagchi, H.D.: I Bager, A.: III Ba˘ıbulatov, R.S.: XVI Baker, R.C.: IV 16; VI 32; IX 28, 32; X 22, 30, 31, 34; XV 16 Bakshtys, A.: XVI 38 Balakrishnan, U.: II 21; XII 25 Balasubramanian, R.: II 31; VI 19, 41; IX 25 Balazard, M.: I 18; V 31, 33 Balog, A.: IV 14, 16, 22, 30; VII 4, 8; VIII 19, 24, 29; IX 21, 24, 28, 33; X 1, 10, 18; XI 13; XIII 21 Bang, T.: XII Barban, M.B.: II 11; III 11; V 19; VII 8; VIII 10, 15, 20; IX 14; XVI 21 Bareiss, E.: XII Barlaz, J.: XII Bartoˇs, P.: XII Bateman, P.T.: I 18; III 51; V 7; VI 41; VIII 24; IX 9; XI 2; XIV 5, 21 Bear, R.: III 44 Beasley, L.B.: XIII Beck, W.E.: III 37 Beesack, P.R.: XII 24 Behrend, F.: III 31 Bellman, R.: II 10, 21, 23; VI 20 Bencze, M.: III 34 Benkoski, S.J.: III 23, 43 Bentz, H.-J.: VII 39, XV Berndt, B.C.: XI 12 Bertram, E.A.: XIII 26 Bertrand, J.: VII 14 Beurling, A.: VIII 38 Bhramarambica, M.V.S.: VI 34 Birch, B.J.: IV 33, X Bode, D.: III 40 Bohman, J.: XIV 19 Bojani´c, R.: III 21 Bojarincev, A.E.: VII 20 Bombieri, E.: II 11; VII 8, 25, 30; VIII 6, 12; X 5, 35; XI 8, 23 Bonse, H.: VII 17 Boreviˇc, Z.I.: XV 34 Borho, W.: III 43 612 Borozdkin, K.G.: IX Bottorf, G.A.: II 23 Bovey, J.D.: XIII 3, 4, Boyd, A.V.: XII 29 Boyd, D.W.: XIV 27 Brauer, A.: VII 25; XIV 21; XV 3, 4, 11 Bredikhin (Bredihin), B.M.: II 11, 22; VIII 33; IX 14, 16; XIII 31 Brenner, J.L.: XIII Brent, R.P.: III 30 Breusch, R.: VII 30; VIII 2, Brindza, B.: XII 22 Brinitzer, E.: III 26; VI 24; XVI 49 Browkin, J.: IX, Brown, E.: III 38 Brouwer, A.E.: IV Bruckman, P.S.: III 12 Brăudern, J.: X 22 Bruijn, N.G de: IV 2, 21, 32; VI 6, 25; XII 14; XIII 21; XIV 19, 20 Brun, V.: VII 8; VIII 12 Brăunner, R.: IX Buchmann, J.: IV Buchner, P.: XII 9, 24 Buchstab, A.A.: IV 22, 32; IX Buell, D.A.: XV Bugulov, E.A.: III 37 Burde, K.: XV Burgess, D.A.: XI 1, 3, 4, 7, 15, 18, 20, 21, 33; XV 3, 4, 14, 15, 21, 31, 39 Buriev, K.: IX 22 Burr, S.A.: II 12 Bykovskiˇı, V.A.: II 22 Cai, T.: I 27 Canfield, E.R.: XIV 26 Cao, H.Z.: XIV 26 Carlitz, L.: VI 22 Carmichael, R.D.: I 17 Cellini, P.: XIII 34 Ces`aro, E.: XII 24, 26 Chalk, J.H.H.: XV 32, 38 Chebyshev (Tchebytcheff), P.L.: VII 1, 14, 28, 29, 30, 35 Cheer, A.Y.: IV 32; VII 15, 23 Chein, J.E.: III 30 Index of Authors Chen, J.-R.: II 15; VII 9; VIII 5; IX 2, 4, 5, 8; X 17 Chen, W.: XIV 26 Chevalley, C.: XV 34 Chidambaraswamy, J.: I 15, 29 Chin, T.: II 10 Chowdhury, M.R.: XII 19 Chowla, I.: XV 36 Chowla, P.: XV 11 Chowla, S.D.: IV 5, 20; VIII 23; X 28; XIII 6; XV 11, 12, 17, 36 Chu, J.T.: XII 25 ˇ Chudakov (Cudakov), N.G.: VII 4; VIII 14, 19; IX 8; X 10 Cillernelo, J.: X 39 Cochrane, T.: XV 40 Cohen, E.: I 31; II 9, 29; V 14; VI 24, 37; IX 13; XIII 29 Cohen, G.L.: I 19; II 14; III 30, 33, 35, 37, 38; Cohen, H.: VI 18 Cohen, S.D.: XV 21 Coleman, M.D.: IX Collisohn, J.E.: XVI Condict, J.T.: III 30 Cook, R.J.: VII 23; X 14; XV 17 Cook, T.J.: III 45 Cooper, C.N.: V 24 Cordoba, A.: X 39 Corput, J.G van der: VII 30; IX 8, 13; X 3, Corr´adi, K.: II 10 Cram´er, H.: II 12; VII 34; VIII 27 Crews, Ph.L.: III 31 Croker, R.: IX 13; XV ˇ Cudakov (Chudakov), N.G.: VII 4; VIII 14, 19; IX 8; X 10 ˇ Culanovski, I.V.: VIII 12 Daboussi, H.: VII 30; X 32; XVI 27 Dar´oczy, Z.: XVI Davenport, H.: III 31; VII 25; VIII 7; X 6, 21, 30, 35, 36 Davis, J.A.: XIII 11 Delange, H.: V 3, 14, 31, 32, 35; X 32; XVI 14, 22, 27, 28, 30, 49 Deligne, P X 38 Delmar, F.: II 24 D´enes, J.: XIII Index of Authors Deshouillers, J.-M.: II 22; IV 8, 12, 14; VIII 28; X 40 Dette, W.: VIII Diamond, H.G.: I 14; VII 30; VIII 38 Diandana, P.H.: VI 34 Dickman, K.: XIV 20, 21 Dickson, L.E.: III 35, XII 17 Ding, P.: X 12 Ding, X.X.: VIII Dirichlet, G.L.: I 21; II 10; III 4; VIII Dixmier, J.: XIV 8, 10, 21 Dixon, J.D.: XIII Dixon, R.D.: II 12 Dodd, F.W.: XIV 26 Dodson, M.: XV 36 Dornhoff, L.: XIII 25, 28 Dos Reis, M.: VIII 30 Dress, F.: VI 2, 15, 18 Dressler, R.E.: I 12; III 19; V 8; VII 18; IX Drozdova, A.A.: XIII 12 Dudley, U.: VIII 36 Duncan, R.L.: III 1; V 1, 3, 4; VI 34 Dupain, Y.: V 31 Duttlinger, J.: XIII 12, 32 Ebbenhorst, C.T van: VI 16 Ecklund, jr., E.F.: XII 3, 7, Eda, Y.: VIII Edgorov, Z.: II 16 Eggleton, R.B.: XII Elistratov, I.V.: XI 31; XVI 27 Elliott, P.D.T.A.: VIII 6, 10; IX 26; X 34, 35; XI 23, 27; XV 5, 8, 10, 21; XVI 1, 3, 6, 11, 13, 15, 16, 27, 32, 36, 38 Ennola, V.: IV 21 Entringer, R.C.: XIII 11 Erber, T.: XII 29 Erd˝os, P.: I 7, 10, 12, 13, 14, 16, 17, 18, 23, 24, 32, 34, 35; II 1, 2, 7, 8, 23, 24, 35; III 2, 5, 7, 14, 15, 18, 19, 21, 23, 33, 35, 41, 43, 51; IV 1, 3, 6, 7, 9, 12, 17, 18, 21, 24, 25, 27, 28, 29, 30; V 9, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 31, 34; VI 6, 22, 23, 30, 34, 41; VII 1, 8, 14, 22, 24, 25, 26, 40; VIII 2, 5, 613 13, 32, 35; IX 6, 9, 10, 13, 20, 23, 25; XI 5; XII 1, 2, 3, 4, 6, 7, 8, 12, 13, 14, 15, 16, 17, 20, 21, 22; XIII 2, 3, 4, 6, 8, 11, 12, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29; XIV 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 17, 18, 21, 25, 26, 28; XV 1, 3, 10, 21; XVI 1, 4, 5, 14, 20, 23, 29, 35, 43 Erhart, E.: VII Estermann, T.: II 22; IX Euler, L.: VII 1; XV Evans, R.J.: XI 12 Evelyn, C.J.A.: VI 32 Fabrikowski, J.: III 28, 29, 48 Fa˘ınle˘ıb, A.S.: I 18, 24; XIV 22 Faulkner, M.: XII Feh´er, J.: XVI 46 Fein, M.B.: XVI 12 Filaseta, M.: VI 20, 32, 41; XV Fine, N.J.: III 36; XV 17 Finsler, P.: VII 14 Fischer, K.-H.: VI 38 Fjellstedt, L.: XV 35 Fluch, W.: VI 23 Fogels, E.: VIII Fomenko, O.M.: IX 28 Forman, W.: VIII 36 ´ Fouvry, E.: II 11, 22; IV 14, 22; VII 8, 9, 10, 11; VIII 11, 12, 16; IX 8; X 26; XI 22 Følner, E.: XI 30 Freiman, G.A.: XIII 12 Freiman, G.: XIV Freud, R.: XIII 11 Friedlander, J.B.: II 11; IV 14, 16, 21, 22, 32; VI 10; VII 4, 8, 11; VIII 6, 12, 21, 24; IX 21, 23; XI 23; XV 31 Frăoberg, C.-E.: XIV 19 Fuglede, B.: XII Fujii, A.: II 11; VII 31, 39; IV 24; XI 13 Gafurov, N: II 21, 25; III Galambos, J.: II 34; IV 17; V 27; XVI 17, 19, 31, 38 Gallagher, P.X.: X 10, 35; XI 13, 23, 25; XV 26; XVI 36 614 Gandhi, J.M.: VI 13 Garrison, B.: VIII 30 Gauss, C.F.: XV Gautschi, W.: XII 29 Gegenbauer, L.: VI 18, 32 Gelfond, A.O.: VIII 34; XI 10; XV Geluk, J.L.: XIV 22 Ghosh, A.: V 27; X 18 Giota, A.A.: II 29; VI 8, Giordano, G.: VII 5, 18 Glazkov, V.V.: V 16 Goetgheluck, P.: XII Goh, W.M.Y.: XIII Goldfeld, D.M.: VIII 12 Goldner, F.: XII Goldstein, L.J.: XI 32 Goldstein, R.L.: I Goldston, D.A.: IV 32; VII 15, 23, 33, 36 Golfeld, M.: XV 23 Golomb, S.W.: VI 13 Golubeva, E.P.: IX 28 Gonˇcarov, V.I.: XIII Gonek, S.M.: VII 36 Gonkale, D.: XII 25 Gordon, B.: II 29; XII Goryunov, Yu.Yu.: II 20 Găottsch, G.: XV 28 Gradstein, I.S.: III 30, 35 Graham, R.L.: III 33; XII 4, 8, 21, XIII 11; XIV 21; XV 11 Graham, S.W.: IV 16; VI 20, VIII 5, 6, 21 Granville, A.: I 34; III 18; IV 21, 32; VIII 5, 6, 21 Greaves, G.: II 26; IX 15 Greenwell, R.N.: XIV 26 Grimm, C.A.: XII 15 Grimmett, G.: VI 21 Grinstead, C.: XII 20 Gritsenko, S.A.: IX Gronwall, T.H.: III 2, 3, Grosswald, E.: V 7; VI 41; VII VIII 3, 26 Grăun, O.: III 31 Grupp, F.: IV 14; VII Gupta, A.K.: XII 24 Index of Authors 31, 22; 32; 22, 30; Gupta, H.: I 16; VI 11; XII 3, 9, 20; XIV 2, 3, 5, 12, 24 Gupta, M.: I Gupta, R.: XV 23 Gurland, J.: XII 25 Guy, R.K.: XII 16, 21; XIV 21; XV Gyapjas, F.: V 28 Gy˝ory, K.: I 10; II 2; III 14; IV 8, 10 Hadamard, J.: VII 1, 30, 35 Hafner, J.L.: II 10, 12, 27; IV 21 Hagis, jr., P.: I 19; III 30, 31, 33, 37, 38, 42, 47, 48; XIV Hahn, S.G.: XV 42 Hajela, D.: X 21, 28 Hal´asz, G.: V 31, 33; XVI 14, 28, 37 Halberstam, H.: II 11; III 9; V 6; VII 4, 10, 11; VIII 6, 7, 30; IX 4; X 35; XVI 44 Hall, R.: I 18, 34; II 7; IV 23; V 31; VI 6, 21, 22; XII 12; XIII 22; XIV 13; XV 41; XVI 37, 44 Hanson, D.: VII 14; XII 2, Harborth, H.: XII Hardy, G.E.: III 48 Hardy, G.H.: I 3; II 10, 12; V 1, 2, 19, 31, 32; VI 17; VII 5; X 28; XIV 1, 14 Harg, G.H.: V 15 Harman, G.: I 14; IV 16; VI 20, 31; VII 10, 13; IX 28, 29, 30, 32, 33, 34, 35; X 10, 17, 28, 29, 31, 33; XV 16 Harrington, W.J.: XII 20 Haselgrove, C.B.: V 8; IX Hatalov´a, H.: I Hausman, M.: I 8, 9; III 14, 50, 52; XIII 30 Heath-Brown, D.R.: II 1, 10, 12, 15, 22; III 30; VI 20, 22, 23; VII 4, 11, 12, 23; VIII 5, 20, 23, 28; IX 7; X 2, 14; XIII 12; XV 23, 40 Hecke, E.: XI 32 Hegyv´ary, N.: XIII 11 Heilbronn, H.: IX 8; XV 1, 20 Hensley, D.: V 7, 32; VII 5; XIV 26 Heppner, E.: II 23, 32; V 12; VIII 32; XIII 12, 27; XVI 27, 30, 42 Hering, F.: XII Index of Authors Herrmann, O.: XIII Herstein, I.N.: XIII Herzog, J.: XIII 16; XIV 8, 9, 32 Hildebrand, A.: II 1; III 14; IV 21, 32; V 32; VII 25, 30; VIII 21; XI 2, 3; XV 5, 11; XVI 1, 6, 15, 35, 36 Hille, E.: XIV 26 Hinz, J.G.: IX 4; XI Hoheisel, G.: VII Hong, Y.: X 12 Hooley, C.: II 22, 25; IV 8, 12, 13, 14; VI 36; VIII 12, 20, 32; X 20, 28; XV 23, 27, 33 Horbowitz, J.: X 37 Hornfeck, B.: III 32, 33 Hua, L.-K.: X 9, 13, 14; XI 7; XV 21, 32 Hudson, R.H.: XV 3, 5, 10 Hunsucker, J.L.: II 12, 40 Huxley, M.N.: II 10; VI 36; VII 4, 13, 25; VIII 4, 8; X 5; XV 30 Ikehara, S.: XIV 26 Il’jasov, I.I.: I 24; XVI 41 Imoru, C.O.: XII 29 Indlekofer, K.-H.: II 34, XVI 11, 18, 25, 33, 34, 35, 47 Ingham, A.E.: I 30; II 22; III 9; VII 3, 4, 30; XIV 1, 18 Iseki, K.: XIV Iseki, S.: XIV 17 Ishikawa, H.: VII 18 Ismoilov, D.: II 21; III 7; XI 17 Israilov, M.I.: X 18 Iv´anyi, A.: XVI Ivi´c, A.: I 27; II 12, 17, 30, 32, 33; III 2, 25, 26, 35; IV 1, 3, 24, 25, 26, 28; V 4, 7, 9, 10, 11, 12, 13, 14; VI 4, 10, 22, 41; VII 7, 37, 38; XIII 12, 13, 14, 16, 17, 18, 19, 20, 29, 31, 32; XIV 28; XVI 42, 49 Iwaniec, H.: II 10, 11, 20; IV 8, 12, 14; VI 15, VII 4, 8, 10, 11, 23; VIII 6, 8, 12, 16, 30; X 5, 28; XI 23; XV Jacobsthal, E.: I 32 Jager, H.: XI 29 James, R.D.: XIV 26 Jia, C.H.: IV 1, 16; IX 2, 32 615 Jia, R.Q.: VI 8, 32, 41 Johnson, D.B.: III 31 Johnson, E.L.: XII Jo´o, I.: XIV 30; XVI 46 Jordan, C.: XIII Jordan, J.H.: XV 11 Joshi, V.S.: VI 32; XV 21 Jurkat, W.B.: IV 32; VI 2; VII 34 Juˇskis, Z.: XVI Jutila, M.: II 10; IV 7, 16; VII 4; VIII 5, 8; X 17, 24; XI 16 Kac, M.: II 36; V 27; XVI Kaczorowski, J.: VII 3, 34, 35, 37; VIII 40; IX Kalajdˇzi´c, G.: XII Kalecki, M.: IV 1, 24 Kalinka, V.: II 13 Kalm´ar, L.: XIV 26 Kamilov, M.Kh.: X Kamke, E.: XV 30 Kampen, E.R van: XVI 23 Kan, J.: IX Kanemitsu, S.: X 27; XI Kanold, H.-J.: I 32; II 37; III 17, 30, 32, 33, 34, 38, 51; V 15; XII 4, Kar, K.: VII Karacuba (Karatsuba), A.A.: II 19; X 4; XI 8, 9, 19, 35; XV 31, 34 Karanikolov, C.: VII Karˇsiev, A.K.: II 11; IX 16 K´atai, I.: I 9, 35; II 9, 10, 11, 23, 24, 32; III 17, 22; IV 31; V 18, 28, 32; VI 5, 32; VII 40; XVI 1, 2, 18, 19, 20, 24, 25, 47 Katz, N.M.: XI 33, 34 Kaufman, R.M.: 1X 28 Kawada, K.: VIII 19 Kazarinoff, D.K.: XII 25 Keates, M.: IV 12 Keˇcki´c, J.D.: XII 29 Keller, J.B.: XIV Kemeny, J.G.: IV Kendall, D.G.: I 1; II 33; XIII 12, 13 Kennedy, R.E.: V 24 Kerawala, S.M.: IV 24; IX 10 Kershaw, D.: XII 29 Kesava Menon, P.: XII 29 Kessler, I.: XIV 616 ` XI 25 Khamzaev, E.: Khare, S.P.: XII 3, Kishore, M.: I 19; III 30, 31, 37, 39 Klimov, N.I.: VIII 17 Knapowski, S.: IV 11; VI 5; VII 3, 37, 39; VIII 3, 39 Knessl, Ch.: XIV Knopfmacher, J.: XIII 32 Knopp, K.: XII 25 Knăodel, W.: VII 23 Knuth, D.E.: XIV 19 Kocarev, B.G.: XV 34 Koch, H von: VII 34 Kolesnik, G.A.: II 10, 15; VI VIII 28; IX 28; X 3; XIII 12 Koninck, J.-M de: I 27; II 17, 30, IV 24, 25, 28, 31; V 4, 14; VI 25; VII 7, 38; XIII 19, 32; XVI 49 Konjagin, S.V.: XV 30 Kopetzky, H.G.: II 16 Korobov, H.M.: VII 2, 35; X 6, 12 Korobov, N.M.: XV 31, 34, 35 Koshiba, Z.: VIII Kotov, S.V.: IV 12 Kozma, G.: XIII 23 Krăatzel, E.: I 26; II 27; III 3; VI 41; X 3, 7, 8; XIII 12, 14, 15, 32; XIV Krishnajah, P.V.: I 15 Kristhal, R.G.: XVI 27 Kruyswijk, D.: VI Kubilius (Kubilyus) I.P.,: V 27, IX 17; XVI 3, 4, 14 Kuhn, P.: VII 30 Kuhnel U.: III 30 Kunoff S.: XII 23 Kurepa, ¯D.: XII 30 Kuznetsov, N.V.: II 21 Laborde, P.: VII 10, 11 Lacampagne, C.B.: VI 27; XII Laffy, T.J.: VIII 22 Laforgia, A.: XII 29 Lagarias, J.C.: IV 16, 21 Lagrange, J.L.: XV 32 Laguerre, E.: XIV 21 Landau, E.: I 8, 27; II 10, 12; IV V 14, 32; VI 1, 8, 9, 17, 18, Index of Authors 38, 34, 20; 32; 16, 38, 29, 33; 12; 37; VII 1, 5, 27; VIII 3, 9; X 2, 11; XI 2; XIII 1; XIV 17; XV 33 Landman, B.M.: XIV 26 Landreau, B.: II 24; XVI 43 Lang, S.: XV 34 Langevin, M.: IV 7, 19 Langford, E.S.: III 1, 10 Laub, M.: III 14 Lavrik, A.F.: VII 8, 30; VIII 19; X 18 Lee, J.: XVI Lehmer, D.H.: I 19; X 15; XIV 1, XV 11 Lehmer, D.N.: VII Lehmer, E.: XV 11 Lehner, J.: XIV 2, 3, 12 Leitman, D.: VIII 28, 31; IX 36 Lenstra, H.W.: III 18 Lev, A.: XIII 23 Leveque, W.J.: II 36, V 27 Levin, B.V.: XIV 22; V 10; VIII 10, 25, 30; IX 14; XVI 6, 38 Lewin, M.: XIV 21 Lewis, D.J.: XI 33; XV 14 Li H.Z.: VI 32 Li Z.F.: II 15 Lieuwens, E.: I 19 Ligh, S.: III 47 Lindner, C.C.: III Linfoot, E.H.: VI 32 Linnik, Ju.V (Yu.V.): II 11, 22; III 11, VII 11, VIII 5; IX 14, 17; X 16, 35; XV 3, 4, Lint, J.H van: IV 1, 21; VI 16; VIII 12; XIV Littlewood, J.E.: II 12; VII 2, 3, 5, 34 Liu, J.: IV 1; VIII Liu, H.-Q.: IV 16; VI 24, 41; VII 9; VIII 28; XIII 12, 24 Liu, M.C.: IX 2, 5; XI 15 Livingston, M.: XIV London, H.: XV 12 Lorch, L.: XII 29 Lord, G.: III 40, 43, 51 Lorentz, G.G.: XII Lou, S.: VII 4, 16 Loxton, J.H.: X 13, XIV 18 Lu, H.W.: II 28 Lu, M.G.: VII 23; X 12, 35; XIII 24 Index of Authors Lucht, L.: III 49; VI 12; XVI 27, 29 Lune, J van de: V Lupa¸s, A.: XII 24, 26, 28 Lupa¸s, L.: XII 26, 28 Luthra, S.M.: XIV Mac Donald, J.C.L.: XII Mac Leod, R.A.: III 3, 4; VI 2, 18 Mahler, K.: IV 15; XII 6, 8; XIV 20 Maier, H.: I 18, 34; III 15; IV V 27; VII 4, 22, 23, 25; VIII 1, 21, 23 Makai, E.: XII Mamangakis, S.E.: VII 17 Mangoldt, H von: VI 1, Manstaviˇcius, E.: XVI 10 Manzur, H.S.: XIV Margenstern, M.: III 50 Marko, F.: XII Masai, P.: I 17 Masser, D.W.: I 14 Massias, J.-P.: XIII Matsumoto, K.: II 16 Mattics, L.E.: III 14, XIV 26 Mauclaire, J.-L.: XVI Maxsein, Th.: VIII 35 Mays, M.E.: XIII 27 Makowski, A.: I 3, 4, 7; III 13, IX McCarthy, P.J.: III 30, 34 McDaniel, W.L.: III 30, 34, 37, 52 McDonagh, S.: II 28 McIver, A.: XIII 24, 28 Meijer H.G.: VIII 22 Meissner, O.: III Mejer, J.: VII Mel’nik, V.I.: II 12 Mendes France, M.: V 15 Menzer, H.: VI 41 Mercier, A.: II 22, 32; III 4; IV 31 Mertens, F.: I 21; II 29; VII 28, 37; VIII Metsăankylăa, R.: XIII 10 Meyer, J.: IX 26 Meyer, U.: VII 10 Miech, R.J.: IX 13 Mientka, W.E.: IV 16 Mignotte, M.: IV 8; XII 617 32; 19, 15; 29, Mikawa, H.: VII 10; VIII 12, 19, 25; IX Mills, W.H.: VIII 36 Mil’uolo, A.: VIII 32 Minc, H.: XII 29 Minoli, D.: III 44 Mirsky, L.: I 30; II 3, 5; III 9; VI 22, 35; XIV 16 Mit‘kin, D.A.: X 12, 13; XV 31 Mitrinovi´c, D.S.: III 10; XII 9, 24, 25, 29 Mitsui, T.: IX 10 Molsen, K.: VIII Montgomery, H.L.: I 13, 23, 27; VI 32; VII 4, 5, 13, 31, 33; VIII 6, 7, 17; IX 6, 7, 8; X 32, 35; XI 13, 21, 23; XIV 27; XV 16, 18, 19; XVI Moore, K.: XIII Mordell, L.J.: XV 35, 37, 38 Moser, L.: I 3; VI 18; IX 9; X 10; XII 4; XIII Motohashi, Y.: II 21, 22, 34; V 5; VI 3, 20; VIII 5, 12, 25, 33; XI 24 Motzkin, T.: IX 25 Mozzochi, C.J.: II 10; VII 16 Murata, L.: XV 27 Mureddu, M.: IV Murty, G.S.R.Ch.: VII 23 Murty, M Ram: I 35; III 22; V 26; XIII 24, 28; XV 23 Murty, V Kumar: I 35; III 22; V 26; XIII 24, 28 Muskat, J.B.: III 30 Nagell, T.: IV 12; XV 3, 30 Nagura, J.: VII 14 Naimi, M.: VI Nair, M.: II 24, 35; VI 36; VII 32; IX 28 Najar, R.M.: III 37 Nanda, V.S.: XIV 12 Narkiewicz, W.: XVI 45 Narlikar, M.J.: XIII 27 Nathanson, M.B.: VI 30; IX 1; XIII 33 Nebb, J.: III 12 Negmatova, G.D.: XI 19 Netto, E.: XIV 21 Neumann, P.M.: XIII 24, 28 618 Newberry, R.S.: III 45 Newman, D.: XII Newman, D.J.: XIV Ng, E.K.-S.: IX Nicol, C.A.: I 3; VI 27 Nicolas, J.-L.: I 8; II 6, 7, 8; III 2, 41; V 17, 21, 23, 31; XII 16, 20; XIII 6, 8, 9, 10, 11, 25; XVI 42 Niederreiter, H.: X 37; XI 34 Nieland, L.W.: X Nisnevi´c, L.B.: XV 34 Niven, I.: I 12, 33 Nordon, D.: VIII 31 Norton, K.K.: II 12; V 7, 32; XI Nowak, W.G.: I 27; II 16; XIII 19 Obl´ath, R.: XII 22 Odlyzko, A.M.: VI 2; VIII 32; XIV Odoni, R.W.K.: XV 21 Olsen, J.E.: XV Oppenheim, A.: I Orazov, M.: VIII 18; IX 6, 13 Ore, O.: XV 30 Osborn, R.: I Ouellet, M.: II 12 P´alfy, P.P.: XIII 28, 29 Pan, C.B.: IX 2; X 18 Pan, C.D.: VIII 6, 16; IX 2; X 18 Pan, C.T.: I 23, IX Panaitopol, L.: VI 24; VII 5, 21, 25; XII Panteleeva, E.I.: XI 13 Papp, Z.: I 10; II 2; III 14 Parker, T.: IX Parnami, J.C.: I 34 Parson, A.: XVI 28 Parson, L.A.: X 24 Pavlov, A.I.: XIII Peˇcari´c, J.E.: XII 29 Pennington, W.B.: XIV 20 Pereira, N Costa: VI 2; VII 14, 23, 30 Perelli, A.: VI 14; VII 4, 31, 33; VIII 8; IX 6, 8, 28; X 18; XI 13 Perel’muter, G.I.: XI 33; XV 29 Perron, O.: XV P´etermann, Y.-F.S.: I 23; III 45; VI 14 Petersson, H.: XIV Phillips, E.: X 3, Phong, B.M.: XVI 46 Index of Authors Phragm´en, E.: VII 34 Pigno, L.: VII 18; IX Pillai, S.: XII 11; XV 21 Pil’tja˘ı, G.Z.: VII 25 Pintz, J.: IV 16; VI 2, 5, 31, 32; VII 3, 4, 31, 34, 35; VIII 8, 24, 39; IX 6, 8; XII 20; XV Pitman, J.: XII 1; XIV ˇ Pjatecki˘ı-Sapiro, I.I.: VIII 28; IX 21; XIII 12 Plaksin, V.A.: V 3; IX Pleasants, P.A.B.: XII Plăunnecke, H.: IX 27 Pollak, R.M.: XI 11 Pollington, A.: X 28 P´olya, G.: I 3, IV 15; VII 34; XI 1, 2, 3; XII 9, 26, 28; XV 15 Polyakov, I.V.: IX 13 Pomerance, C.: I 7, 17, 18, 20, 34, 35; II 1; III 17, 18, 21, 30, 31, 34, 40, 42, 51; IV 1, 3, 17, 22; V 13, 16, 32, 34; VII 8, 14, 22, 24, 28; VIII 5, 19, 23, 24; XII 15; XIII 24; XIV 26; XV 21 Popadi´c, M.S.: III 10 Popov, O.V.: VI 28; XV Popoviciu, T.: I 2, 34 Porubski, S.: I 3; 12 P´osa, L.: VII 17 Postnikov, A.G.: X 19; XIII 12, 31; XVI 27 Potocki˘ı, V.V.: X 25 Prachar, K.: I 28; II 6; V 6; VI 23, 37; VII 8, 25, 26; VIII 4, 5; IX 3, 7, 20 Prasad, V.S.R.: II 29; III 46; VI 34 Preissmann, E.: II 10 Pustil’nikov, L.D.: X Putz, R.: XIII Qiu, Z.: XI Rademacher, H.: VII 17; XIV Radoux, Ch.: I 22; III 16 Rakhmonov, Z.Kh.: IX 13 Ramachandra, K.: II 31; IV 7, 16; VI 19, 41; IX 8, 20; XII 14, 15 Ramanujan, S.: II 7, 8; III 2, 6; VI 2, 19, 31, 32; VIII 14; XIV 1, 14 Ramaswami, V.: IV 21, 22 Index of Authors Rameswar, Rao D.: III 36 Rankin, R.A.: II 33; III 17; IV 21; V 12; VI 20; VII 22, 25; X 5; XII 13; XIII 12, 13 Recknagel, W.: VI 38 Reddy, D.R.: III 46 R´edei, L.: XV Redmond, D.: XI 28 Rech, S.: VII 17 Reidlinger, H: III 37 Reisel, H.: IX Rendall, D.G.: V 12 R´enyi, A.: V 14, 27; VII 23, 24; IX 3; X 35; XIII 22; XVI 17, 29 R´enyi, K.: VI 36 R´ev´esz, Sz.Gy.: VII 34 Ricei, G.: VI 35; VII 25; VIII 2, Richards, I.: VII Richert, H.-E.: II 10, 2, 27; IV 32; VI 16, 20; VII 10, 11; VIII 12, 30; IX 9; XIII 12; XVI 44 Richman, D.R.: XV Richmond, B.: XIV 20 Richmond, L.B.: IX 10; XIV 15, 17 Riddel, J.: IX Rieger, G.J.: II 32; III 42; IV 2; V 28; VI 10, 25, 29; VIII 28; IX 6, 18, 21; XIV 2, 26 Riele, H.J.J te: III 18, 30, 44; VII Rigge, V.: XII 17 Rivat, J.: VIII 28 Robbins, N.: III 30 Roberts, J.B.: XIV 21 Robertson, M.M.: XIV 12, 16 Robin, G.: II 6, 7; III 2; V 15; VII 3, 35; VIII 38; XIII Robinson, D.F.: III 50 Robinson, R.: VI 37 Rodosskiˇı, K.A.: VII 33 Rodriguez, G.: II 11 Roessler, F.: VI 29 Rogers, K.: II 29; VI 18, 34 Rohrbach, H.: VII 14 Romanoff, N.P.: IX 13 Rosser, J.B.: I 8; VI 16; VII 1, 5, 19, 28, 29, 30, 35 Rossum-Wijsmuller, M van: XVI 19 Roth, K.F.: VI 20, IX 10, 15; X 35 619 Rousselet, B.: VIII 12 Ruben, H.: XII 25 Russ, S.: III Rutkowski, J.: II Ruzsa, I.Z.: IV 30; IX 27; XII 4; XV 41; XVI 1, 3, 13, 14, 50 Ryavec, C.: XVI Ryzhova, N.P.: II 11; V Sabirov, P.: XVI 30 Saffari, B.: II 29 Sahib, S.: XV 11 Sahu, H.: VII Saias, E.: IV 21 Salerno, S.: VII 4, 33; VIII 8, 39; IX Sali´e, H.: III 31; X 28 Sander, J.W.: VII 28; XII 1, 4, 5; XV 28 S´andor, Gy.: XV 32 S´andor, J.: I 3, 4, 5, 6, 7, 8, 31; II 3, 4; III 1, 2, 10, 12, 15, 27, 36; VII 14, 17, 21; XII 24, 27, 28, 29 Sankaranarayanan, A.: I 25 Saradha, N.: I 35 Sarkăozy, A.: II 1, 7, 38; IV 30; V 16, 27, 31, 33, 34; VI 30; VII 8; IX 24, 31; X 1; XI 6; XII 3, 5, 14; XIV 6, 8, 10, 11, 25; XV 1; XVI Sath´e, L.G.: II 9; V 31, 32; VI 17 Sathre, L.: XII 29; Satyanarayana, B.: III Satyanarayana, M.: III Scheid, H.: XII Scheidler, R.: XII Schierwagen, A.: II 27 Schinzel, A.: I 7, 10; II 1; III 14, 15; IV 12; V 17; VIII 5; IX 26; XIV 28; XV 40 Schlickewei, H.P.: XV 40 Schmidt, E.: VII 34 Schmidt, P.G.: VI 41; XIII 12, 29 Schmidt, W.M.: XV 40 Schmutz, E.: XIII 2; XIV 17 Schnirelman, L.: VII 8; IX Schoenberg, I.J.: I 24 Schoenfeld, L.: I 8; VI 2, 16; VII 1, 5, 19, 28, 29, 30, 35 Schonheim, J.: XIV 23 Schur, I.: XII 7, 13; XIV 23 620 Schwarz, W.: V 8; VI 25; VII 28; XIII 12, 16; XVI 27, 30 Scott, W.R.: XIII Scourfield, E.J.: I 11; II 16; III 17; IV 33 Seelbinder, B.M.: XIV 21 Segal, B.: XI 17 Segal, S.L.: I 19; II 10; VI 8; VII 5; XVI 17 Selberg, A.: II 12; V 7, 31, 32; VI 10, 16; VII 1, 5, 13, 31; VIII 1; X 35; XI 24 Selfridge, J.L.: VI 27; XII 3, 6, 7, 8, 15, 17, 20, 21; XV Selmer, E.S.: XII Sengupta, J.: II 21 Servais, Cl.: III 31 Shan, Z.: I 20; IX Shanks, D.: XV Shao, P.C.: II Shao, P.Z.: X 13 Shapiro, H.N.: I 23, 29, 33; II 23; III 8, 14, 35, 50; V 2, 3; VI 10, 20, 35; VII 35, 37; VIII 1, 3, 11, 36; IX 1, 25; XI 5, 11; XIII 30; XV 13, 21, 24, 25; XVI 4, 12 Sheingorn, M.: X 24 Sherman Lehman, R.: VII Shiu, P.: I 14; II 35; VI 41, 43 Shiratan, K.: XI Shirokov, B.M.: XVI 29 Shockley, J.E.: XIV 21 Shorey, T.N.: IV 7, 9, 10, 12, 15, 19; V 22; IX 23, 25; XII 7, 13, 14, 15, 17, 18 Shparlinskij, I.E.: X 15; XV 29, 30 Siegel, C.L.: IV 12; VIII Sierpi´nski, W.: I 1, 10; II 1; III 1, 43; VII 23; VIII 32; IX 26 Simmons, G.J.: XIII 11 Singh, K.: XII 20 Singmaster, D.: XII Singwi, K.S.: XIV Sismondi, S.: XII 29 Sitaramachandrarao, R.: I 23, 27, 30; II 31; III 26; IV 2; VI 25, 32, 39 Sitaramaiah, V.: II 29; III 6; XVI 39 Sivaramakrishnan, R.: I 1, 3; III 10 Index of Authors Sivaramasarma, A.: I 31 Skewes, S.: VII Slavi´c, D.W.: XII 29 Smati, A.: I 18 Smida, H.: II 18 Smith, B.: X 21; XIII 23 Smith, H.W.: XII Smith, P.R.: VIII 35 Smith, R.A.: II 16; III 6; X 13; XV 38 Sobirov, A.S.: VII 30 Sokolovski˘ı, A.V.: XI Somasundaram, D.: II Somayajulu, B.S.K.R.: I 3, 10; VII Soundararajan, K.: VIII 37; XVI 43 Săohne, P.: XI Sparer, G.N.: VIII 36 Spira, R.: III 52 Spiro, C.A.: I 34; II 1; III 18; XIII 24 Spitznagel, jr., E.E.: XIII 25 Squalli, H.: VI 24, 25 Srinivasan, A.C.: III 50 Srinivasan, B.R.: VII 30, XIII 12 Srinivasan, S.: XIII 24, 27, 28; XV 23 Stahl, W.: XII Stakenas, V.: XVI Stark, H.M.: V 8; VI 2, 32, 34 St´as, W.: VII 34, 37 Statuleviˇcius, V.: IX Steˇckin (Stechkin), S.B.: X 12, 15; XV 30 Stein, A.H.: XII Steining, J.: VII 30 Stevens, H.: I 31 Stewart, B.M.: III 50 Stewart, C.L.: II 38; IV 6, 9, 10, 30; V 34; VI 29; VII 8; IX 31 Stiffler, J.J.: X 12 Stirling, J.: XII 24 Stothers, W.W.: XV 21 Straus (Strauss), E.G.: III 28, 48, 51; XII 4, 20 Stux, I.: IX 36 Subbarao, M.W.: II 30, 31; III 15, 28, 29, 45, 48; VI 34, 40, 41 Subhankulov, M.A.: IX 14 Subrahamanyam, P.: VI 42 Sugunamma, M.: III Index of Authors Suryanarayana, D.: I 23, 24, 30, 32; II 29, 31; III 16, 26, 31, 38; VI 8, 9, 24, 32, 33, 39, 40, 42; XVI 39 Sved, M.: XII Sylvester, J.J.: III 30, IV 7; XII 7, 13; XIV 21 Szalay, M.: XIII 1, 6, 7, 8; XIV 3, 4, 5, 6, 9, 14; XV 22 Szaltikov, A.I.: I 23 Szeg˝o, G.: I 3; XII 9, 26, 28 Szekeres, G.: VI 41; IX 10; XII 11; XIII 12; XIV 2, Szemeredi, E.: XII Szăusz, P.: XVI 38 Szydto, B.: VII 34 ˇ Safareviˇ c I.R.: XV 34 ˇ Salat, T.: I 1; VII 27 ˇ Siaulys, J.: XVI Takaku, A.: III 24 Tamba, M.: XIV 31 Tanaka, M.: VI 7; IX 12 Tanny S.M.: XII 9, 10 Tatuzawa, T.: VIII Tchebychef (Chebyshev), P.L.: VII 1, 14, 28, 29, 30, 35 Tenenbaum, G.: II 11, 22; IV 2, 12, 21, 22; V 14, 15, 27, 31, 32; VI 4, 15; X 23, 26; XI 22; XII 22; XVI 27 Tichy, R.: X 31 Tietăavăainen, A.: XV 36, 38 Tijdeman, R.: IV 7, 12; IX 23; XII 7, 14, 15, 18 Timofeev, N.M.: V 10; VIII 6, 8; XVI 6, 10, 38 Titchmarsh, E.C.: II 11, 12; VIII 12; X 3, 5, Todd, J.: IV Toeplitz, O.: VII 17 Toffin, Ph.: IX 21 Tolev, L.I.: II 16; IX 21 Tomescu, I.: XII 9, 10 Tong, K.C.: II 10 T´oth, L.: I 5, 30, 31; III 6, 26, 27; XII 25, 28 Tourigny, J.M.: III Toyoizumi, M.: XI 14 Tran, T.H.: II 621 Trifonov, O.: VI 20, 32 Trost, E.: I Tsang, K.M.: IX 2, Tuliagonov, S.T.: XVI 30, 38 Tull, J.P.: XVI 28, 40 Tur´an, P.: II 1; IV 30; V 3, 8, 20, 26; VII 3, 24, 34, 35, 39, 40; VIII 3, 39; X 10; XIII 2, 3, 4, 8, 21; XIV 13, 14; XV 21; XVI 3, 17 Turk, J.: IV 19 Turnwald, G.: X 31 Turull, A.: V 6, 22 Tuttas, F.: XVI 27 Tzanakis, N.: IV Uchiyama, S.: VI 9, 16, 24, 36; VII 11; VIII 3, 5, 20; IX 19 Udrescu, N.S.: VII 5, 14 Usol’cev, L.P.: II 3; X 19 Uspensky, J.V.: XIV Vaaler, J.D.: XVI 43 Vaidya, A.M.: I 1; II 29; VI 8, 9, 18, 19 Valdez, J.: VII 28 Valette, A.: I 17 Vall´ee-Poussin, Ch de la: VI 8; VII 1, 2, 30, 35; VIII Vangipuram, S.: III Vannucci, L.: XII Varbanec, P.D.: II 16 Vasi´c, P.M.: XII 29 Vaughan, R.C.: I 13; II 11; VI 32; VII 5; VIII 17, 32; IX 1, 7, 8, 32, 34; X 17, 18, 29, 30, 32, 35; XI 21, 27; XV 19, 26 Vegh, E.: XV 22 Venkataraman, C.S.: III 10 Vernescu, A.: XII 25 Vijayaraghavan, T.: IV 20 Vinogradov, A.I.: VIII 6, 10; IX 6; XI 15; XV Vinogradov, I.M.: VII 2, 29, 35; IX 1, 2, 32, 36; X 4, 7, 9, 10, 12, 16, 17; XI; XV 3, 10, 13, 15, 20, 21, 38 Vitolo, A.: IX Volinets, L.M.: XIII Vol’koviˇc, V.E.: V 25 Voronoi, G.: II 10, 12 622 Index of Authors Vose, M.D.: XII 23 Wagstaff, S.S.: VIII Waldschmidt, M.: IX 25 Walfisz, A.: I 21, 23, 24; III 4, 5; VI 1, 32; VII 35; VIII 6, 9; X 2, 11 Wall, Ch.R.: 4; III 26, 27, 31, 36, 45, 47 Wallis, J.: XII 25 Walum, H.: XV 17 Wang, T.: IX Wang, W.: VIII Wang, Y.: I 10; VII 8; VIII 30; IX 7; XV 21, 40 Ward, D.R.: VI 16 Warga, J.: IX Warlimont, R.: IV 20, 32; VI 23, 37; VII 23; XIII 6, 27; XV 21, 23; XVI 48 Warning, E.: XV 34 Warren, L.J.: III 45 Washington, L.: VIII 27 Watson, G.N.: XII 24, 29 Watt, N.: X Webb, W.A.: III 11 Weber, G.C.: III 30 Weber, J.M.: III 45 Weil, A.: X 10, 12, 28; XI 18; XV 31 34 Weis, J.: VII 14 Weisman, C.S.: XII Westzynthius, E.: VII 25 Weyl, H.: X 1, 2, Wheeler, F.S.: IV Whittaker, E.: XII 24 Whyburn, C.: XV 14 Wielandt, H.: XIII Wigert, S.: II Wild, K.: XIV Wilf, H.S.: XIII Williams, H.C.: XII Williams, K.S.: VIII 3; XV 3, 6, 38 Williamson, A.: XIII 4, 5, Wilson, B.M.: II 13, 17 Wilton, J.R.: X 24 Winston, K.: XII Wintner, A.: XVI 5, 23, 29 колхоз 11/15/06 Wirsing, E.: III 32, 33; VII 30, 37; IX 6; XIII 12; XIV 28; XVI 1, 2, 24, 28, 29 Wolke, D.: II 21; V 2, 14, 32; VI 20, 31; VII 23, 37; VIII 16, 28, 31; IX 2, 36; X 18, 35; XI 26; XV 16; XVI Woolridge, K.: I 17 Wright, E.M.: I 3; V 15; VI 17; X 28 Wu, D.H.: IX Wu, J.: VII 10 Wyman, M.: XIII Xie, S.G.: VII 11 Xuan T.Z.: II 18; IV 21, 24, 28; V 9, 11, 13 Yang, Z.H.: XV 21 Yao, Q.: VII, 4, 16 Yeung, C.N.: XIII 29 Yin W.-L.: II 15 Young, R.: VII 18 Yu, X.: VII 28, 37; XVI 26 Yăuh, M.T.: II 15 Zaccagnini, A.: VII 22; IX Zame, A.: V 6, 22 Zannier, U.: VI 14 Zarzycki, P.: II 16 Zeitz, H.: VII 25 Zeller, K.: XII Zhan, T.: V 32; VI 41; VIII 8; X 21 Zhang, G.: VIII 20 Zhang, M.: IX Zhang, M.Y.: X 12 Zhang, W.: VIII 38, XI 29 Zhang, W.-B.: VIII 38 Zhang, W.P.: II 12; IV Zheng, Z.: XV ˇ VII 27; XII Zn´am S.: Zsygmondy, K.: IV Zuker, M.: XII 9, 10 Zulauf A.: IX 11, 14 ˇ Zeltonogov, V.M.: III 16 ... (a | b) Legendre? ?s symbol (n) number of squarefree divisors of n d e (n) number of exponential divisors of n e (n) sum of exponential divisors of n d ∗∗ (n) number of bi-unitary divisors of. .. progressions § IV.23 On (n, (n)) = B( n) § IV.24 Sums over k (n), Bk (n), B( n) − (n), , (n) B( n) − (n) P(n) 11 4 11 4 11 5 11 6 11 6 11 7 11 8 11 9 12 0 12 1 12 1 12 2 12 3 12 3 12 4 12 5... record for this book is available from the Library of Congress ISBN -10 ISBN -13 ISBN -10 ISBN -13 1- 4020-4 215 -9 (HB) 978 -1- 4020-4 215 -7 (HB) 1- 4020-3658-2 (e-book) 978 -1- 4020-3658-3 (e-book) Published