REMEMBERING PAUL ERDŐS
Aleksandar Ivić
The last time I saw Paul Erdős was in May 1995, during the Conference on Analytic Theory held in
Urbana, Ill. in honour of H. Halberstam. It was a beautiful spring day, and we were walking (slowly, as Erdős
had some trouble with his foot) in the park of Allerton Conference Center, where the meeting took place. At
one point Erdős turned to me and said: ”You know, it is said that that those who prove the prime number
theorem will live forever. Well, Hadamard and de la Vallée–Poussin both lived to be almost a hundred. Now
Selberg and I had also to do something with the prime number theorem, so...”. The unfinished sentencehung in the air, full of understatement concerning his (and Selberg’s) rôle in the proof of the prime number
theorem. I don’t remember now my exact reply, but the essence of it was that I told him he will live anyway
through the myriad of his results.

∗ ∗ ∗Unfortunately, Erdős died on September 20, 1996. I was, as must have been many of his friends andcolleagues, utterly shocked and unprepared for the news of his death. I got to know of this sad event throughan e-mail message from Carl Pomerance late on Sep. 21. He forwarded to me a message of Miki Simonovits
from Budapest. In it was said:“Paul Erdős died Friday afternoon (20 September, 1996), in Warsaw. Early morning he felt some healthproblems, in a hotel in Warsaw. So he was carried into a hospital, where he died in the afternoon. (He was83).
As far as I know, he had a heart attack, rather serious, very early in the morning, in this Warsaw hotel,
where he stayed while visiting the Minisemester for Combinatorics, (for two weeks), gave two lectures. Vera
Sós and Andras Sarközy are leaving Budapest for Warsaw right now, Saturday morning, and Paul’s originalplans were to fly from Warsaw to Vilnius (for the Kubilius Conference) with Vera and Andras together, on
Sunday.
The doctor informed us that he had two heart attacks and the second one killed him. (Perhaps eventhe first one was serious enough to prevent him from communicating to his surrounding. This may explaine.g. that he could not reach the mathematicians. We learned about his heart attacks only after his death.)”∗ ∗ ∗This text contains my personal reminescences of Erdős. There are certainly people who knew him better
than I did, both professionaly and even more so on the personal level, but I dare say I knew him well. Overthe period of some twenty years or so, we were in contact, and the tangible result are eight papers that we
jointly published (one was also coauthored by J.-M. De Koninck, one by C. Pomerance, and one by S.W.
Graham and Pomerance). I most certainly have received more mail from him than from anybody else; it waseasy to recognize his characteristic handwriting (in big, somewhat childish letters) on letters coming from allover the world. Sometimes it was even fun to guess where the letters would be coming from: Europe, North
America, Australia, . . . . He would customarily sign them “Au revoir, E.P.”, as in Hungarian the last nameis usually put before the given name; he was born in March 1913 as Erdős Pál (the “Paul” came later).

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A. Ivić

∗ ∗ ∗Sometimes I lost track of where he was for quite some time. This was rather annoying, especially attimes when we were doing joint work, and I simply did not know how to contact him. In particular, I could
not locate his whereabouts at the beginning of this year, 1996. The Faculty of Mathematics was about- . Kurepa. Kurepa was an eminent
to organize, at the end of May, a three-day symposium in honour of D
Serbian mathematician, who died in 1993 at the age of 87. They knew each other quite well and Erdős, who
visited Belgrade many times, regularly came to Kurepa’s apartement to talk to him. When Kurepa died,Erdős in principle agreed to come to the symposium and give a conference in which he was to talk about
Kurepa’s work and his year-long relationship with him. Actually Ž. Mijajlović, the chief organizer, managed
to find money for Erdős to stay a week in a decent hotel, which was remarkable in view of the bad economic
situation in Serbia. After many efforts and lot of e-mail messages, we located Erdős in California in April,
if I remember correctly. Unfortunately it was already too late for Erdős to change his plans and come toBelgrade, moreover the trip from California to Belgrade would be rather tiring for a man of his age, though
he traveled constantly. This I regret very much, for clearly it would have been my last chance to see him
and talk to him, and besides his presence would have rendered the Kurepa symposium more attractive.

∗ ∗ ∗Curiously enough, the last paper of mine that has just been published is “On the number of divisors ofn!”, which is joint work with Erdős , S.W. Graham and C. Pomerance. It appeared in “Analytic Number
Theory: Proceedings of a Conference in Honor of Heini Halberstam, Volume I”, pp. 337-355. The paper was
actually completed during the Halberstam Conference in Urbana in May 1995. The work was done in a roomat the Allerton Conference Center, during two evenings. What I remember most now is that the work was
rather hard, one of the main task being to convince Erdős that we have done enough for the present paper,and should stop. Erdős, however, always had (as usual) new problems, new ideas, several generalizations.
I think that he was doubtlessly the greatest problem proposer of all times. He also had the talent to judge
what to propose to whom, and to pick problems suited to the capabilities of the potential solvers.
But the story with this paper does not end here. This August in Ottawa, during the CNTA-5 Conference(this stands for the “Canadian Number Theory Association”, and this was the fifth of a series of meetings),
C. Pomerance and I got together and worked on some unfinished problems related to the abovementioned
paper. The problems were, in its original form, due to Erdős. Just as we were finishing a joint note, the sadnews of Erdős’s death came. I think that the dedication Pomerance put in our paper is a very appropriate
one:“Dedicated to the memory of our friend and teacher, Paul Erdős”∗ ∗ ∗For Erdős was exactly that: both a friend and a teacher, not only to Pomerance and myself, but
to generations of number-theorists (and very many mathematicians working in other fields) from over theworld. His “teaching” was rather subtle. He never directly gave lessons, taught theorems and methods. It
was through working with him that one got a grasp of the richness of his methods and ideas. Slowly, perhaps
even without noticing, he would take over in your mathematical way of thinking. He would painlessly
convince you, by the success of his methods, what and how to use. In fact, many times when I was working
alone on a problem, I would ask myself: “How would Erdős do it? How would he attack this problem?” It
is sometimes subconsciously that I would do this, but as time went by I realized that I often wanted to try
to do things his way, because it was the right way to do it. I’ve heard him say that God up in heaven has
the one and only copy of the BOOK. This book contains the proofs of all possible mathematical theorems

Remembering Paul Erdős

3

one could imagine, and moreover the proofs are the best and most elegant ones. When he said that a proof
came closely to being from the BOOK, that meant high praise indeed.
Perhaps it is hard to explain, without going into technicalities (which I don’t want), what were Erdős’smethods. One of the principles was that the proof should be divided in cases, according to the size of therelevant quantities. To illustrate this, let me give a fictitious example of a result proved in, what I think, isthe spirit of Erdős. In other words, this is how I believe Erdős would have proceeded. Thus consider thefollowing
Theorem. Call a natural number tolmonic if the sum of its tolmic parts does not exceed the sum of allits proper divisors. If F(x) is the number of tolmonic integers not exceeding x, thenF (x) = O



x
log x



.

Proof. If n is tolmonic,
let P (n) denote its largest prime factor. Consider the cases√a) P (n) < exp( log x),√b) exp( log x) ≤ P (n) ≤ exp logloglogx x ,c) P (n) > exp logloglogx x .

P The number of tolmonic n satisfying a) is O(x/ log x) by Brun’s sieve and estimates for ψ(x, y) =n≤x,P (n)≤y 1. For tolmonic n satisfying c) the same is also easily seen by applying a theorem of Henenbaum and Tall. Finally for those tolmonic n satisfying b), note that the defining property gives thatn is ∆-distributed. But this implies that the sum of its proper divisors lies in an interval of the form
[logA , logB x], 0 < A < B. Since the sum of tolmic parts of n does not exceed the sum of its proper divisors,
by a result of Erdős from 1937 the number of such n is ≪ x/ log x, unless P (n) ≤ logC x, but this is coveredby case a) already.

∗ ∗ ∗Erdős visited many times Sarajevo, of course before the war started. In the summer of 1992 he sent me
the following message in one of his letters:“Why war in Bosnia? Everybody loses.”
There is no doubt that Erdős was right. He was a very compassionate man, and this showed (among
other things) in the understanding of the hardships that the war created in Serbia (refugees, sanctions,
negative image of Serbs in Western media, . . .). In the summer of 1993, which was definitely the worst year,
I had the pleasure of seeing him twice in Hungary. The first time was in June, and I came to Budapest one
week before the Lillafüred conference on analytic number theory. Imre Kátai kindly provided the financing
for my stay, but nevertheless I was tight for money. Erdős took me out for lunch in a nice reataurant near
the Mathematical Institute in the Reáltanoda street. While discussing some problems involving the largest
prime factor of n, Erdős suddenly pulled out of his jacket pocket a fistful of banknotes of money of several
different countries. “Take it”, he kindly said in a soft way, “I know you are hard up for foreign currency”. I
was quite embarassed, and told him that, having in mind the political and economic situation in Belgrade, I
certainly could not repay the debt in the foreseeable future. Erdős insisted, but I managed to refuse, however
the gesture I will never forget.

∗ ∗ ∗Erdős had the unique talent of being able to say something concrete about almost any imaginableproblem. Most mathematicians usuallly cannot say anything about a problem, and in a few instances they
can say quite a lot. Erdős was exceptional, in the sense that he had the very rare gift of being able to “attack”

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A. Ivić

almost any problem. Thus he was a living encyclopedia in some sense, and I’ve heard often people say to eachother: “If you don’t know, ask Erdős”. His memory also was, even in his last years, fascinating. He knew
literally thousands of results, with rather precise bibliographical data. He was literally a mathematician “quiouvre la voie” (“who opens the path”), as G. Tenenbaum put in a dedication to Erdős in own of his papers.
Once I was working with him on a problem involving squarefree numbers. We needed a certain resultin the course of our investigations, and I did not know where to look for it. Erdős said:“There is a relevant
paper by Mirsky in the “Quarterly J. Mathematics (Oxford)” in 1947 or 1948 at the latest”. I went to
the library and there it was, in all its semi-forgotten splendour: L. Mirsky, Note on an asymptotic formulaconnected with r-free integers, Quart. J. Math. Oxford 18 (1947), 178-182. Of course, this kind of situation
occurred often, and I knew well when Erdős started the sentence with “There is a relevant paper...” thatthe paper in question was relevant indeed. And the additional beauty is that he very likely knew the author
and probably had a joint paper or two with him.

∗ ∗ ∗Erdős was truly prolific, and in the matter of the number of published papers he is probably without
peer in the history in mathematics (some great mathematicians, like Euler, probably published more pages
than Erdős did, but math was certainly different in Euler’s days). Several years ago I asked him how many
does he have, and he said:“1200 ± ε ”. By now the number has increased to about 1500 I believe, and tensof joint papers will appear posthumously. I pity the person(s) who will work on publishing his bibliography!
The number of his collaborators must be well over 400, and Erdős told me that alone with A. Sárközy hehas more than 50 joint papers. I suspect that it is Erdős himself who came up with the notion of the ”Erdősnumber”. This is defined as follows. Erdős had the number 0, those who had a joint paper with Erdős had
Erdős number 1, those who had a joint paper with someone having Erdős number 1 had Erdős number 2 etc.
It is, of course, a joke that every living mathematician who had a joint paper had a finite Erdős number,but certainly a very large number of mathematicians have a finite Erdős number. And it is truly fascinatingwhen you start looking at some of the fields he worked on. He had papers from approximation theory, grouptheory, set theory, polynomials, numerical analysis - you name it, he sure has got it. For example, I was
surprised that in the ‘50’s Erdős wrote a joint paper with Jovan Karamata, the great Serbian mathematician.
The paper was on summability methods, a field one usually does not associate Erdős with. When I asked
him about this particular paper, since Karamata’s work always interested me, he smiled and said that heknew Karamata quite well. He explained, in rather remarkable detail, the content of the paper, and then hewent on and told me how he regularly visited Karamata in Geneva (where he was forced to emigrate afterWorld War II by the Yugoslav communist goverment), told me stories about him, anecdotes . . . As I alreadysaid, there is hardly a contemporary of his that Erdős did not know, and very likely he had a paper or two
written with him.

∗ ∗ ∗The language of Erdős (his English always had an unmistakable Hungarian accent, and he was fluent
in French and German besides of course Hungarian, his mother tongue) was full of amusing idiosyncrasies.
For example, ”noise” meant ”music”, ”poison” meant ”alcoholic drink”, ”boss” meant ”wife”, ”ǫ’s” meant
children. Besides, Uncle Joe and Uncle Sam stood of course for the Soviet Union and the USA, respectively.
Naturally, when Erdős asked for an ǫ of wine, this meant a small, or symbolic quantity. He did not drinkor smoke, and in fact disliked smoking. When I saw the effigy of A. Rényi in the hall of the Mathematical
Institute of Budapest, I remarked to Erdős that Rényi died relatively young (49), and asked him for thecause of his death. He replied curtly “He smoked himself to death”. Obviously the very thought of thisupset him, since he was close to Rényi. When he calmed down he explained then that Rényi died from lungcancer, which was most likely caused by excesive smoking. Perhaps his closest friend and collaborator was
Paul Turán, who also died of cancer in 1976. Erdős mentioned with kindness many mathematicians, but I
think that it was Turán who occupied a special place in his heart.

Remembering Paul Erdős

5

∗ ∗ ∗
When a mathematician publishes the unbelievable number of 1500 papers, people are bound to ask
questions. How was that humanly possible? Are the papers all really that good? Does he even remember
what was in them? I will try to answer these questions, to the best of my knowledge. First, Erdős started very
early publishing papers, when he was not yet even twenty, and continued to do so (and practically nothingelse, since he never had a regular job to speak of) for well over sixty years. Besides, he was doubtlessly in
the genius class (he discovered negative integers when he was merely three years old!). His working habitswere formidable, he had no working hours, he worked (sometimes literally) day and night. I have the feelingthat, when he was in his prime, he could have proved anything if he wanted to. If you compare an ordinary,
successfull mathematician to a computer, then we should compare Erdős to a transputer, because he had arare talent of working parallel on several problems at the same time. (He always reminded me in that aspectof the great Brazilian composer Heitor Villa-Lobos, who used to compose music while listening to the radioplay). I’ve witnessed this: people would come to him, he would talk for several minutes (quite seriously) fora person about one problem, then with the next person about the second problem etc.
As for the quality of his papers, no, they are not all of the same level. Some are good, some are very
good, and some are great. Erdős had this basic philosophy: if a result is new and correct, publish it. Now
there are many very fine mathematicians who do not agree with this philosophy. They say no, a result has
to be polished, great if possible to be published. They say one’s papers should be in increasing quality,
nothing to be changed, nothing to be added. This is fine with me, if it can be achieved. But I, having been
“Erdősised” to the core, accept and adhere to Erdős’s philosophy. Although Erdős never said this explicitly(at least not to me, and frankly I did not ask) I think I can pretty well get the idea behind his attitude
about the publishing of papers. It is the very work itself that is, in a higher sense, the purpose of thiswhole game of mathematics. It is fulfilling in itself and brings meaning, and Erdős sacrificed his personal
life (had no family, never married) to the research in mathematics. Now publishing was the visible, tangible
aspect of the game, but the countless efforts, tries, conjectures, false leads, failures, heated discussions withhis colleagues remain unseen (once, in Oberwolfach, I had a “heated” discussion with him that lasted morethan an hour. People later told me they heard quite loud shouting coming from the room - God didn’t giveme a quiet voice - but after the verbal storm Erdős and I came out of the room the best of friends like
we were before the discussion). And, of course, practice makes perfect even in writing papers and Erdőssurely had lots of it. After a while writing papers becomes, in a sense, almost a habit (“addiction” would
be perhaps too strong to say). So Erdős publishes, he throws seeds around. This is because often his papers
lead to sharpenings, generalizations, new problems. And his seeds grow to new papers, new talents are being
attracted to the game, and the whole process flows continuosly for decades. “Prove and conjecture” was his
credo. Certainly Erdős could have, with additional effort, improved many of his own results. I am quite surethat he did not want to do that, in many cases, on puprose, since it would take him valuable time for which
he had other use.
As for the question how much he knew of his own theorems, I’ve already explained that he had an
exceptional memory. Of course, he must have forgotten quite a few of his results, but in general I think hekept in his head much more of his own mathematics than people usually thought that he did.

∗ ∗ ∗Let me get back to the fictitious Theorem about the mysterious “tolmonic numbers” (this I made up
since it sounded vaguely familiar to me, but I sure can’t remember where I got it from!). Anyway, what
could have happened is the following. After having read the proof, L.J. Picolas observed that the problem
could be made to depend on exponential sums. He proved that F (x), the counting function of tolmonic
numbers, actually satisfies F (x) = O(x log−A x) for any fixed A > 0 by using simple estimates from thetheory of exponential sums. Then by using the more sophisticated theory of exponent pairs Baker and Cook
showed that F (x) = O(x0.95 ) holds, and H. Nuxley announced further improvements. This pleased Erdős,
who came up with the following flood of new questions:6

A. Ivić

√a) Erdős can prove F (x) = Ω(logC x) for some C > 0. Is it true that F (x) = Ω( x)? Is it true that1
F (x) = O(x 2 +ε )?
b) Let 1 = a1 < a2 < · · · denote the sequence of tolmonic numbers. Is it true that an+1 < an withprobability 1/2?
1/2c) Is it true that aP
n+1 − an ≪ an ? (Erdős can prove it with exponent 11/12).
d) Is it true that n≤x (an+1 − an )2 ≪ x1+ε ?
√e) If ω(n) is the number of distinct prime factors of n, is it true that ω(n) = O( log log n) if n is
tolmonic?
f) Is it true . . . - but I better stop here, since by now you must have gotten the picture.

∗ ∗ ∗As I said, some of his papers were truly great. I’ve asked him precisely three times what he thought
were his greatest achievements in Number Theory. Each time he replied without hesitation: the following
three things.
1. The proof that ω(n), Ω(n) (the number of distinct and the number of all prime factors of n) obey theGaussian normal distribution law. This is joint work with M. Kac, and is one of the crowning achievementsof Probabilistic Number Theory (On the Gaussian law of errors in the theory of additive functions, Proc.
Nat. Acad. Sci. U.S.A. 25 (1939), 206-207).
2. The characterization of the logarithm as an additive function and the work on the distribution of
additive functions (On the distribution of additive functions, Annals of Mathematics 47 (1946), 1-20).
3. The elementary proof of the Prime Number Theorem (joint work with A. Selberg). This is: On a
new method in elementary number theory which leads to an elementary proof of the prime number theorem,
Proc. Nat. Acad. Sci. 35 (1949), 374-384.
Now the last paper is by Erdős alone, since it is well known that the elementary proof of the Prime
Number Theorem developed into a situation. Since I was not a witness in 1948 of what really happened
between Erdős and Selberg, since I haven’t heard Selberg’s version and since Erdős told me (twice: in 1981
and 1983) to keep confidential about his version I will not dwell on this topic here. After all, this is a matterfor the history of mathematics.

∗ ∗ ∗What I just said was corroborated in public by Erdős himself. In the summer of 1993 the János Bolyay
Society organized a big conference in Keszthely (on the Balaton lake) in Combinatorics. The conference
was, quite appropriately, in honour of Erdős’s 80th birthday. Erdős himself was supposed to give a talk on
his work. Well, since that was a conference on Combinatorics (I had a talk on combinatorial number theory,
so this is how I attended) I thought he was going to talk primarily on his work in Combinatorics. After all,he is one of the founders of Graph Theory and much else besides, and what he had done in Combinatorics
could hardly be squeezed into an one hour talk. However, all of us who listened to him (well over 200 people)
were in for a big surprise. Actually there were three surprises. Erdős usually talked in the old-fashioned
way, with blackboard, writing his formulas and thoughts with no notes. This time, he used an overhead
projector and transparencies (after the talk, I snatched a couple of these - with his permission - as a souvenir,
since he honoured me by quoting me at one place). The second surprise is that never, but I mean never
have I heard him talk so clearly, lucidly, full of youthful vigour. At certain points during his lecture I waswondering whether this was really happening, whether an 80 year old man could give such a well organizedand prepared talk. The third, and perhaps biggest, surprise is that practically 80 percent or so of his time
was spent on his results from Number Theory, and not Combinatorics. Why did he do it? I have no idea,
and I did not dare ask (actually in Keszthely it was hard to get physically to him, there was literally acrowd of people around him all the time). Was it that he valued his results from Number Theory more thanRemembering Paul Erdős

7

those from other fields, Combinatorics included? Perhaps that was the answer, since he clearly said that heconsidered as his greatest achievements
1. The proof that ω(n), Ω(n) (the number of distinct and the number of all prime factors of n) obey
the Gaussian normal distribution law. This is joint work with M. Kac (On the Gaussian law of errors in thetheory of additive functions, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 206-207).
2. The characterization of the logarithm as an additive function and the work on the distribution of
additive functions (On the distribution of additive functions, Annals of Mathematics 47 (1946), 1-20).
3. The elementary proof of the Prime Number Theorem (joint work with A. Selberg). The paper by
Erdős is: On a new method in elementary number theory which leads to an elementary proof of the prime
number theorem, Proc. Nat. Acad. Sci. 35 (1949), 374-384.
I apologize for repeating this, but I consider this quite an important point, and this is why I did it.

∗ ∗ ∗I’ve often asked Erdős to send me reprints of the ∞ of his published work. He did that many times,
but nevertheless the number of his works that I have is far smaller than I would have liked. Erdős told methat his mother (both of his parents were teachers of mathematics) kept track of his published work until
she died at the age of over 90. Actually almost until her death his mother accompanied him on his constanttravels, in general organizing life for him. Erdős in fact owned an apartment in Budapest, but said he did
not want to live in it when he was in Budapest, because his mother died there, and that brought on painful
memories to him. So he seems to have let his friends use the apartment; in 1993 he told me Paul Turán’s
son lived there with his family. He stayed in a guest room of the Hungarian Academy of Sciences, whose
prominent member he was for many years. It seems that the only salary that he received on a regular basis
was from the Academy of Sciences in Budapest (the sum which, I believe, every academician receives) andfrom Haifa (Erdős was Jewish, but not in the strict sense, and he was certainly more cosmopolitan thananyone else I knew). The only catch is that, both in Budapest and in Haifa, money was available only when
he was physycally there. Otherwise, he would lecture at some University and get (very moderately) paid, or
he would be invited as a guest to a conference (this happened a dozen times a year, I believe), or he would
simply spend some time as a guest in some mathematician’s home.

∗ ∗ ∗Of course Erdős was the prototype of the itinerant mathematician, who basically never stayed at thesame place. He went on carrying very little personal belongings, which never really seemed to fill up abattered old suitcase. He liked silk shirts, usually wore sandals, and had dark non-descript worn suit.
His rooms were always filled with letters, manuscripts and a motley collection of coloured pills, which heswallowed with unmistakable regularity. “These are vitamins” he used to explain curtly, although I suspect
there were surely other things that he must have been taking. When I complained several times that, in theinterest of his health, he should be careful about what he was taking, he sneered and said that at his age
nothing could harm him much. “Two worst things in life are old age and stupidity!” he used to say, and by
this remark he managed always to end up a conversation.

∗ ∗ ∗When we were together at a meeting and wanted to work, we had the following deal. I would come up
to his room between 6 and 6.30 in the morning and we would work about 8, when usually breakfast wasavailable. Since I am an early riser this, in principle, suited me, but it used to be tiring at certain times.
The work was done as much as we could, before either one of us tired; then we would start talking about

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A. Ivić

other things. First it was mathematics, then we would slip on to other subjects. Erdős was a remarkably
well-informed man. He used to carry a big old transistor radio, which was strong enough to capture radiotransmissions from Budapest from most parts of the globe, I presume. The reason for this is that he wanted
to hear “as soon as possible” when/if the downfall of communism came, for which I had a full understanding.
I have many memories from those early hours spent with Erdős. Once in Amalfi (Italy) in 1989, when I
stayed up late the previous night, I completely forgot about the appointment with Erdős the next morning.
But he didn’t, and exactly at 6.30 the telephone rang. “Where are you?” simply said the well-known voice, alittle irritated. I apologized and rushed to his room getting dressed in the hotel corridor. The chambermaid’s“Mamma mia!” explains well what I must have looked like when I was passing her.

∗ ∗ ∗Only once I lost temper with him. This happened in the summer of 1982, when he came to Belgrade byplane, on a very hot day. As we were driving from the airport to the city, he started asking questions. “Whydid we have to wait so much for the luggage? Why do you have these containers overflowing with garbage?
Why are the roads so badly paved? Why are we waiting so long for the street lights to turn green?” Thisavalanche of questions went for quite some time, and although the remarks were basically all in place I lost
temper at one point and yelled at him: “There is no why in communism!” An uneasy silence crept in forsome time, but the incident was quickly behind us.

∗ ∗ ∗That same summer Erdős really scared me. Let me tell you the story. As it was terribly hot in myapartment, Erdős suggested we go to the nearby park where it was somewhat less unpleasant (“I must havefresh air!” he used to say). We took along my two year old daughter Natalija, since we couldn’t leave heralone. Anyway, by the time we got to the park and had our notebooks ready for some problems on the
distribution of P (n), I remembered that there was a little chore I had to do. “Fine”, Erdős said,“leave thechild here with me for a few minutes, and we’ll continue when you get back”. With some hesitation I agreed,
and rushed off. However, when I came back puffing and sweating some 10-15 minutes later, the park was
empty. No Erdős, no Natalija. I felt sick in the stomach and looked around once, twice, three times. No
one in sight. Panic was starting to take over, and dark thoughts of death, kidnapping, traffic accidents cameto my head. I ran from one side of the park to the other – nothing. Finally, when I was already thinkingof calling the police, from across the street I saw them. They were walking slowly, Natalija holding Erdős’shand, and they seemed to be talking (in what language I never knew!) and smiling at each other. In her
other hand there was some candy Erdős apaprently bought for her, and they both seemed to be perfectly
happy in each other’s company. Fear gave place to relief, and I ran towards them and hugged them both.
“You shouldn’t have worried” Erdős said with a sly smile, “I have my ways with children.” And he did; manytimes he gave me chocolates for my daughters (Emilija was born two years later, in 1984), always asked how
were they doing in school etc. It is amazing that always he remembered their exact age, like he kept thedata in a computer file in his brain.

∗ ∗ ∗Besides arithmetic functions I had, from the late seventies, an interest in the theory of ζ(s). In fact, I
became in some sense a “split personality”, because my work is equally divided between what I call “Erdős
theory” and zeta-function theory. One of the reasons for this is that, no matter how exciting, “Erdős
theory” usually seriously interests only a limited number of people, whereas ζ(s) is one of the central themesof Analytic Number Theory. Erdős knew well of this. Not only did he approve (“Your work in the theory of
ζ(s) is certainly more important than in arithmetic functions” he told me bluntly long ago) but he showedRemembering Paul Erdős

9

a remarkable interest and knowledge of zeta-function theory. I told him, of course, that I actually did not
believe the notorious Riemann Hypothesis, RH, that all complex zeros of ζ(s) have real parts equal to 1/2.
He told me that P. Turán “probably more disbelieved the RH than he believed in it”, although he did notknow Turán’s arguments. When we met, he often asked me about the newest results in the theory of ζ(s),and I was only happy to tell him as much as I knew. “What about the RH - still nothing?” (meaning still
unsettled) he used to ask, with a slightly impish grin on his face. As for his own views about the RH, he did
not divulge them, probably he believed the RH, being an optimist by nature. And besides approving of myworking in the theory of ζ(s) he gave me, during a bad personal situation of mine, a piece of advice I will
never forget. “Aleksandar”, he wrote,“there is no justice in Heaven or Earth, but there is certainly
justice in MATHEMATICS! Thus try to be as good as you can in Mathematics.” And with thisquotation of his I will finish these notes. When I close my eyes I will always remember him, walking in somedeep thoughts, with glasses and graying hair, a slim, stooped figure in a worn suit and sandals, always readyto pour out formulas, ideas, problems... The world of Mathematics will be never the same without him.

e-mail: aleks@ivic.matf.bg.ac.yu

eivica@ubbg.etf.bg.ac.yu

Belgrade, October 1996