Table Of ContentStudies
in Pure Mathematics
To the Memory of Paul Turan
Editorial Board
Editor-in-Chief: Paul Erdos
Associate Editors: Laszl4 Alpar, Gabor Halasz
and Andras Sarkozy
Springer Basel AG
Library of Congress Cataloging in Publication Data
Main entry under title :
Studies in pure mathematics.
Includes index.
1. Mathematics—Addresses, essays, lectures.
2. Turân, P. (Paul) 1910—1976. I. Turân, P.
(Paul), 1910—1976. IL Erdös, Paul, 1913-
QA7.S845 510 81-17016
AACR2
CI P Kurztitelaufnahme der Deutschen Bibliothek
Studies in pure Mathematics: to the memory
of Paul Turân/Paul Erdös, ed. ... — Basel ;
Boston; Stuttgart: Birkhäuser, 1983.
NE: Erdös, Paul [Hrsg.]
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.
© Springer Basel AG 1983
Originally published by Akadémiai Kiadô, Budapest in 1983
Softcover reprint of the hardcover 1st edition 1983
ISBN 978-3-7643-1288-6 ISBN 978-3-0348-5438-2 (eBook)
DOI 10.1007/978-3-0348-5438-2
Contents
Editors' Preface. . . . . . . . . . . . . . . . . . . 9
Preface, Personal reminiscences by P. ERD6s (Budapest) II
G. HALAsz (Budapest), Letter to Professor Paul Tunin. 13
ABBOT, H. L. and MEIR, A. (Edmonton), An extremal problem in
combinatorial number theory. . . . . . . . . . . . . . . . . . . 17
AJTAI, M., HAvAs, I. and KOMWS, J. (Budapest), Every group admits a bad
topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
ALPAR, L. (Budapest), Sur certains changements de variable des series de
puissances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
ANDERSON, 1. M. and CLUNlE, 1. (London), The spherical derivative of
meromorphic functions with relatively few poles . . . . . . . . . . 43
ASKEY, R. and ISMAIL, MOURAD E.-H. (Madison), A generalization of
ultraspherical polynomials . . . . . . . . . . . . . . . . . . . . 55
BELNA, C. (State Univ. Pennsylvania) and PIRANlAN, G. (Ann Arbor), A
Blaschke product with a level-set of infinite length . . . . . . . . . 79
BOLLOBAS, B. (Cambridge), CHUNG, F. R. K. (Murray Hi\I) and
GRAHAM, R. L. (Murray Hill), On complete bipartite subgraphs con-
tained in spanning tree complements. . . . . . . . . . . . . .. 83
BROWN, W. G. (Montreal), On an open problem of Paul Tunin concerning
3-graphs.. . . . . . . . . . . . . . . . . . . . . 91
CARLESON, L. see Appendix to the paper of J.-P. KAHANE and
Y. KATZNELSON ....................... .
CHUNG, F. R. K. (Murray Hi\I), ER06s, P. (Budapest) and SPENCER, J. (Stony
Brook), On the decomposition of graphs into complete bipartite
subgraphs. '.' . . . . . . . . . . 95
CHUNG, F. R. K. see also BOLLOBAS, B. ............. .
CLUNIE, 1. see ANDERSON, J. M. . . . . . . . . . . . . . . . . .
CsASZAR, A. (Budapest), Syntopogenous spaces and zero-set spaces. 103
DENES, 1. (Budapest), KIM, K. H. and ROUSH, F. W.(Montgomery),
Automata on one symbol. . . . . . . . . . . . . . . . . . . .. 127
DoBROWOLSKI, E. (WrocIaw), LAWTON, W. (Pasadena) and SCffiNZEL, A.
(Warszava), On a problem of Lehmer. . . . 135
ELBERT, A. (Budapest), On extremal polynomials . . . . . . . . . .. 145
6 Contents
ELUOTT, P. D. T. A. (Boulder), Subsequences of primes in residue classes to
prime moduli . . . . . . . . . . . . . . . . . . . . . . . . .. 157
ERD6s, P. and SARKOZY, A. (Budapest), Some asymptotic formulas on
generalized divisor functions I. . . . . . . . . . . . . . . . . .. 165
ERD6s, P. and T. S6s, V. (Budapest), On a generalization of Tunin's graph-
theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181
ERD6s, P. and SZALAY, M. (Budapest), On some problems of J. DENES and
P. TuRAN . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187
ERD6s, P. and SZEMEREDI, E. (Budapest), On sums and products of integers 213
ERD6s, P. see also CHUNG, F. R. K ................. .
FUCHS, W. H. J. (Cornell Univ.), On the growth of meromo rphic functions
on rays ......................... '. . .. 219
GAIER, D. (Giessen) and KJELLBERG, B. (Stockholm), Entire functions and
their derivative on an asymptotic arc. . . . . . . . . 231
GANEUUS, T. (GOteborg), Orthogonal polynomials and rational
approximation of holomorphic functions. . . . . . . 237
GRAHAM, R. L. see BOLLOBAS, B.. . . . . . . . . . . .
Gyl)RY, K. (Debrecen) and PAPP, Z. Z. (Budapest), Norm form equations
and explicit lower bounds for linear forms with algebraic coefficients. 245
HALASZ, G. (Budapest), On the first and second main theorem in Turim's
theory of power sums. . . . . . . . . . . . . . . . . . . . . .. 259
HARARY, F. and MILLER, Z. (Ann Arbor), Generalized Ramsey theory VIII.
The size ramsey number of small graphs . . . . . . . . . . . .. 271
HARRIS, B. (Madison), The asymptotic distribution of the order of elements
in alternating semigroups and in partial transformation semigroups. 285
HAVAS, I. see AlTAI, M. . . . . . . . . . . . . . . . . . . . . . . .
HAYMAN, W. K. (London) and KJELLBERG, B. (Stockholm), On the mini-
mum of a subharmonic function on a connected set . . . . . . .. 291
HEPPNER, E. und ScHWARZ, W. (Frankfurt am Main), Benachbarte
multiplikative Funktionen. . . . . . . . . . . . . . . . . . . . 323
HLAWKA, E. (Wien), Eine Bemerkung zur Theorie der Gleichverteilung 337
IuEV, L. (Sofia), Laguerre entire functions and Turim inequalities.' . 347
INDLEKOFER, K.-H. (Paderborn), On Turan's equivalent power series 357
ISMALL, MOURAD, E.-H. see ASKEY, R .............. .
JAGER, H. (Amsterdam), The average order of Gaussian sums. . . . 381
JUTlLA, M. (Turku and Djursholm), Zeros of the zeta-function near the
critical line . . . . . . . . . . . . . . . . . . . . . . . . . .. 385
KAHANE, I-P. (paris) et KATZNELSON, Y. (Jerusalem), series de Fourier des
fonctions bornees . . . . . . . . . . . . . . . . . . . . . . .. 395
CARLESON, L. (Djursholm), Appendix to the paper of I-P. KAHANE and
Y. KATZ NELSON . . . . . . . . . . . . . 411
KATAI, I. (Budapest), Characterization of log n . . . . . . . . . . . 415
KATZNELSON, Y. see KAHANE, I-P. ................. .
KIM, K. H. and ROUSH, F. W. (Montgomery), On a problem of Turim 423
KIM, K. H. see also DENES, J. . . . . . . . . . . . . .
KIELLBERG, B. see GAIER, D. and HAYMAN, W. K. . . . . . . . . .
Contents 7
KOLESNIK, G. (Austin) and STRAUS, E. G. (Los Angeles), On the sum of
powers of complex numbers. . . 427
KOMWS, J. see AlTAI, M. ....... .
LAWTON, W. see DoBROWOLSKI,.E.. . .. ........ .
LoRCH, L. (Downsview) and NEWMAN, D. J. (Philadelphia), On a
monotonicity property of some Hausdorff transforms of certain Fourier
series. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 443
LoRENTZ, G. G. (Austin), Theorem of Budan-Fourier and BirkhofT
interpolation. . . . . . . . . . . . . . . . . . . . . .' . . . .. 455
LovAsz, L. and SIMONOVlTS, M. (Budapest), On the number of complete
subgraphs of a graph II. 459
MEIR, A. see ABBOT, H. L.. . . . . . . . . . . . . . . . . . . . . .
MILLER, Z. see HARARV, F. . . . . . . . . . . . . . . . . . . . . .
MONTGOMER'Y, H. L. (Ann Arbor), Zeros of approximations to the zeta
function . . . . . . . . . . . . . . . . . . . . . . . . . .. 497
MOTOHASHI, Y. (Tokyo), Large sieve extensions of the Brun-Titchmarsch
theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 507
NARKlEWlCZ, W. (Wroclaw), On a question of Alladi and Erdos on sums of
squares . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 517
NEWMAN, D. J. see LORCH, L. . . . . . . . . . . . . . . . . . . . .
NIEDERREITER, H. (Wien), A quasi-Monte Carlo method for the approximate
computation of the extreme values of a function . . . . . . . . .. 523
PALFV, P. P. and SZALAY, M. (Budapest), On a problem of P. TuRAN
concerning Sylow subgroups ............. '. . . . .. 531
PAPP, Z. Z. see GvlBtv, K.. . . . . . . . . . . . . . . . . . . . . .
PIERRE, R. (Sherbrooke) and RAHMAN, Q. I. (Montreal), On polynomials
with curved majorants . . . . . . . . . . . . . . . . . . . . .. 543
PlNTz, J. (Budapest), Oscillatory properties of the remainder term of the
prime number formula. . . . . . . . . . . . . . . . . . . . .. 551
PIRANlAN, G. see BELNA, CH.. . . . . . . . . . . . . . . . . . . . .
POMMERENKE, CH. (Herlin-Minneapolis) and PuRzITSKV, N. (Toronto-
Berlin), <1n some universal bounds for Fuchsian groups. 561
PURZITSKV, N. see POMMERENKE, CH. . . . .
RAHMAN, Q. I. see PIERRE,"R. ............. .
ROUSH, F. W. see DENES, J. and KIM, K. H.. . . . . . . .
RUZSA, I. Z. (Budapest), On the variance of additive functions 577
SACHS, H. and STIEBITZ, M. (Ilmenau), Automorphism group and spectrum
of a graph. . . . . . . '. . . . 587
SARKOZV, A. see ERD&, P. ..................... .
SeHlNZEL, A. see DoBROWOLSKI, E.. . . . . . . . . . . . . . . . . .
SCHMIDT, W. M. (Boulder-Tokyo), The joint distribution of the digits of
certain integer s-tulpes . . . . . . . . . . . . . . . . . . . . .. 605
ScHOENBERG, I. J. (West Point), On Jacobi-Bertrand's proof of a theorem of
Poncelet. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 623
ScHWARZ, W. see HEPPNER, E.. . . . . . . . . . . . . . . . . . . .
SHAH, S. M. (Lexington), Entire functions of bounded value distribution and
gap power series. . . . . . . . . . . . . . . . . . . . . . . .. 629
8 Contents
SmDLOVSKY, A. B. (Moscow), Estimates for the moduli of polynomials with
algebraic coefficients at the value of E-functions . . . . . . . . .. 635
SIEBERT, H. (Mannheim), Sieve methods and Siegel's zeros. . . . . .. 659
SIMONOVITS, M. (Budapest), Extremal graph problems and graph products 669
SIMONOVITS, M. see also LovAsz, L. ................. .
SOMORJAI, G. t (Budapest), On the coefficients of rational functions. .. 681
T. S6s, V. (Budapest), On stroung irregularities of the distribution of
{nIX} sequences . . . . . . . . . . . . . . . . 685
T. S6s, V. see also Ean(}s, P.. . . . . . . . . . . . . . . . . . . . .
SPENCER, J. see CHUNG, F. R. K ................... .
STEW ART, C. L. (Waterloo) and TIJDEMAN, R. (Leiden), On density-difference
sets of sets of integers . . . . . . . . . . . 70 I
STIEBITZ, M. see SACHS, M. . . . . . . . . . . . . . . . . . . . . .
STRAUS, E. G. see KOLESNIK, G ................... .
SURANYI, J. (Budapest), Some notes on the power sums of complex numbers
o. . . . . . . . . . . . . . . . . . . . . . . . ..
whose sum is 711
SWINNERTON-DYER, H. P. F. (Cambridge), The field of definition of the
Neron-Severi group. . . . . . . . . . . . . . . 719
SZALAY, M. see ERD6s, P. and PALFI, P. P ...... .
SZEMEREDI, E. see ERD6s, P. . . . . . . . . . . . . .
Sziisz, P. (Stony Brook), On Hadamard's gap theorem. 733
TIJDEMAN, R. see STEWART,C. L. .......... .
VAUGHAN, R. C. (London), A remark on Freud's tauberian theorem 737
VERTESI, P. (Budapest), Two problems of P. TURAN . . . . . . . . 743
WALDSCHMIDT, M. (Paris), Un lemme de Schwarz pour des intersections
d'hyperplans. . . . . . . . . . . . . . . . . . . . . . . . . 751
WIERTELAK, K. (poman), On the density of some sets of primes III . .. 761
Editors' Preface
This volume, written by his friends, collaborators and students, is offered to the
memory of Paul Tunin.
Most of the papers they contributed discuss subjects related to his own fields of
research. The wide range of topics reflects the versatility of his mathematical activity.
His work has inspired many mathematicians in analytic number theory, theory of
functions of a complex variable, interpolation and approximation theory, numerical
algebra, differential equations, statistical group theory and theory of graphs.
Beyond the influence of his deep and important results he had the exceptional
ability to communicate to others his enthusiasm for mathematics. One of the strengths
of Turan was to ask unusual questions that became starting points of many further
results, sometimes opening up new fields of research. We hope that this volume will
illustrate this aspect of his work adequately.
Born in Budapest, on August 28, 1910, Paul Turan obtained his Ph. D. under
L. Fejer in 1935. His love for mathematies enabled him to work even under inhuman
circumstances during the darkest years of the Second World War. One of his major
achievements, his power sum method originated in this period. After the war he was
visiting professor in Denmark and in Princeton. In 1949 he became professor at the
Eotvos Lorand University of Budapest, a member of the Hungarian Academy of
Sciences and a leading figure of the Hungarian mathematical community. His
untimely death on September 26, 1976 prevented him from working out a great many
of the ideas he had so abundantly.
Even the relatively large extent of this volume did not permit to invite and include
all desirable contributions.
We express our gratitude to the Akademiai Kiad6 for having made it possible for
the many friends of Paul Turan to pay their tribute to a great mathematician.
The Editors
Studies in Pure Mathematics
To the Memory of Paul Turim
Preface
Personal reminiscences
I met Paul Turan first in September 1930 at the University of Budapest though we
knew of each others existence since we both worked for the mathematical journal for
high school students and our first joint paper appeared there, i.e. a solution of a
problem which we obtained independently. At our first meeting I asked him if the sum
of the reciprocals of the primes diverges or converges. He informed me that it diverges
and he told me about the prime number theorem. (Seven or eight years earlier I learned
from my father that the number of primes is infinite.)
We met and discussed mathematics nearly every day until I went to Manchester in
October 1934, after that date until early September 1938 I spent half my time in
England half in Hungary but when I was in England we corresponded a great deal. Our
first joint paper dealt with elementary number theory, in which we deal with a problem
ofG. Griinwald and D. Lazar: Let f(n) be the largest integer so that if 1 ~ al < ... < an
n
are any set of n integers then (ai + aj) has at least f(n) distinct prime factors.
We proved I~i;::;j;:;;;n
n
(1) cllog n<f(n)<c2-
logn
and we conjectured that f(n)flog n-+oo, which is still open.
Very soon we started our collaboration on interpolation which produced many
more "serious" results.
From England I returned to Hungary for Christmas, Easter and summer vacations.
In the spring of 1938 Hitler succeeded in disturbing my plans, but I could return to
Hungary precariously in the summer of 1938. On September 3, 1938, I did not like the
news and in the evening I was on my way to England and three and one-half weeks later
to the USA. We corresponded until 1941, then there was an enforced gap offour years.
As soon as possible we started our first postwar joint paper on the difference of
consecutive primes which we wrote in correspondence. In this paper we stated a few
problems which I find very interesting: Put d,,=Pk+1 -p", Is it true that
