From: Bill Dubuque Subject: Re: Rigor of Euler's sin(z) factorization ; nonstandard analysis Date: 02 Feb 1999 01:00:01 -0500 Newsgroups: sci.math Keywords: Euler and the birth of analytic number theory On such matters, the following may be of historical interest. Date: Sun, 31 Jan 1999 18:50:34 -0500 (EST) From: Bill Dubuque To: math-fun Subject: Re: primes that are 1 more than squares Jim Propp wrote: | [...] Speaking of Euler: did Euler ever give a derivation | like the preceding [ for L(1) ] ? Weil, in his historical book Number Theory, does mention some of Euler's work on the evaluation of L(1), but not along the lines you fantasize. Below are some relevant excerpts from Weil (these were OCRed so beware of errors). -Bill Dubuque At the time of Euler's apprenticeship under Johann Ber- noulli, the summation of the series zeta(n), when n is an integer >=2, was a classical problem which had exercised Leibniz and the Bernoulli brothers (cf. Corr.II.15|1737). The sum- mation of L(l) in 1674 had been perhaps the most striking early discovery of the young Leibniz (one which, as Huygens predicted, would be "for ever famous among geometers" Huy. VII.394|1674), but it depended upon the power-series expansion x x^3 x^5 arc tan x = - - --- + --- - ... , 1 3 5 which gives L(1) = pi/4; this could not be extended to zeta(n) nor to L(n) for any n > 1. Even the numerical evaluation of these series was no mean problem, in view of their slow convergence. [...] (g) Summation of zeta(2v) and related series (cf. SS.XVII,XVIII, XIX,XX). One of Euler's most sensational early discoveries, perhaps the one which established his growing reputation most firmly, was his summation of the series \sum_1^oo n^-2 and more generally of \sum_1^oo n^-2v, i.e. in modern notation zeta(2v), for all positive even integers 2v (Eu.1-14.73-86 = E 41|1735). This was a famous problem, first formulated by P. MENGOLI in 1650 (cf. G. Enestrom, Bibl.Math. (III) 12 (1912), p.138); it had resisted the efforts of all earlier analysts, including Leibniz and the Bernoullis (cf. e.g. Leibniz's letter to Joh. Bernoulli, Math.Schr.I-3.454|1697, and Joh. Bernoulli's letter to Euler, Corr.II.15|1737). Characteristically, before solving it, Euler had engaged in extensive numerical calculations in order to get good approximate values for these sums; it is largely with this intention, it seems, that he had developed the method traditionally known as "the Euler-MacLaurin sum- mation formula", and in so doing had re-discovered the "Bernoulli numbers", whose true importance for number theory was not to emerge until the next century (Eu.1-14.42- 72 = E 25|1732, Eu.I-14.108-123 = E 47|1735; cf. Part V, Chap.II, of the Institutiones Calculi Differentialis of 1755, Eu.1-10.309-336). What Euler found in 1735 is that zeta(2) = pi^2/6, and more generally, for v>=1, zeta(2v) = r_v pi^2v, where the r_v are rational numbers which eventually turned out to be closely related to the Bernoulli numbers. At first Euler obtained the value of zeta(2), and at least the next few values of zeta(2v), by a somewhat reckless application of Newton's algebraic results, on the sums of powers of the roots for an equation of finite degree, to transcendental equations of the type 1 - sin(x/a) = 0. With this procedure he was treading on thin ice, and of course he knew it. Publication in Petersburg was slow; but soon Euler had sent the news to friends and colleagues all over Europe, and it became the topic for lively discussions between him and some of his correspondents. In the meanwhile he was sparing no efforts to consolidate his methods. Less than ten years after his first discovery, he was able to include in his Introductio in Analysin Infinitorum (published in Lausanne in 1748, but ready in manuscript in 1744; cf. Corr.I.292|1744) a full account of the matter, entirely satisfactory by his stan- dards, and even, in substance, by our more demanding ones. This was based on a detailed treatment of trigonometric functions and of their expansions into infinite series and infinite products (Eu.1-8.133-212 = Chap.VIII-XI of the Introductio), to which he added a whole chapter on the "eu- lerian" products for zeta(s)) and various L-series (ibid.284-312 = Chap.XV). As to his efforts to gain information about the numbers zeta(n) for odd n > 1, it is no wonder that they remained unsuccessful, since hardly any progress has been made with this problem down to the present day. Never- theless they were not unfruitful; it was in fact while searching for such information that Euler hit upon the functional equation for zeta(s) and for the companion series oo (-1)^n L(s) = Sum ------ n = 0 (2n+1) (Eu.I-15.70-90 = E 352|1749). These remained as mere curiosities, and virtually unknown, until they received new life at the hands of Riemann in 1859. [...] S.XVIII. "So much work has been done on the series zeta(n)", Euler writes in 1735, "that it seems hardly likely that anything new about them may still turn up ... I, too, in spite of repeated efforts, could achieve nothing more than ap- proximate values for their sums ... Now, however, quite unexpectedly, I have found an elegant formula for zeta(2), depending upon the quadrature of the circle [i.e., upon pi]" ["Tantopere iam pertractatae et investigatae sunt series reciprocae potestatum numerorum naturalium, ut vix probabile videatur de iis novi quicquam inveniri posse ... Ego etiam iam saepius ... has series diligenter sum persecutus neque tamen quicquam aliud sum assecutus, nisi ut earum summam ... proxime veram definiverim ... Deductus sum autem nuper omnino inopinato ad elegantem summae huius seriei 1 + 1/4 + 1/9 + 1/16 + etc. expressionem, quae a circuli quadratura pendet": Eu.I-14.73-74 in E 41|1735]. When he wrote these lines, he had obviously just discovered his famous result zeta(2) = pi^2/6. His paper on that subject, and on various closely related summations, was presented to the Academy on 5 December 1735; it was not printed until 1740 (Eu.I-14.73-86 = E 41), but by that time everyone who counted in the international mathematical community had been apprised of it: first of all, perhaps, Euler's friend Daniel Bernoulli in Basel (cf. Corr.II.435|12 Sept. 1736 and Corr.II.15-16|1737); also STIRLING in Edinburgh (Eu.IV A-1,no.2621|8 June 1736), EHLER and KUHN in Dantzig (PkU.345|3 April 1736), POLENI in Padova (PkU.216|13 March 1736), NAUDE in Berlin (PkU.193-199|1740), and, directly or through these, many others including of course Johann Bernoulli (cf. Corr.II.15-16|1737). As to Goldbach, he was in Petersburg at the time of the discovery and surely must have heard of it at once. In 1742, noticing (not without a shade of disappointment) that CLAIRAUT was insuffi- ciently informed of the matter, he promptly sends him full details, prefacing them with the words: ~Mr Jacq. Bernoulli ... parle de ces suites, mais ... il avoue, que malgre toutes les peines, qu'il s'etoit donnees, il n'avoit pu venir a bout, de sorte que Mrs [Jean] Bernoulli, de Moivre et Stirling, grands Maitres dans cette matiere, ont ete extremement surpris, quand je leur annoncois que j'avois trouve la somme de cette serie 1 1 1 1 1 + - + - + -- + -- + ... 4 9 16 25 et meme de celle-ci 1 1 1 1 1 + --- + --- + --- + --- + ... 2^n 3^n 4^n 5^n quand n est un nombre pair" ["Jacob Bernoulli does mention those series, but confesses that, in spite of all his efforts, he could not get through, so that Joh. Bernoulli, de Moivre and Stirling, great authorities in such matters, were highly surprised when I told them that I had found the sum of zeta(2), and even of zeta(n) for n even": Eu.IV A-5.120]. These are almost the same words which Fermat had once used in describing one of his own discoveries ("c'est une de mes in- ventions qui a quelquesfois estonne les plus grands maistres": cf. Chap.II, S.XV). From 1736 onwards, and for the next eight or ten years, Euler's result, and the degree of validity of his proof for it, become a recurrent topic in his correspondence with his friends and colleagues all over Europe. Here again Euler had been rash, applying Newton's theo- rem on the sums of the n-th powers of the roots of algebraic equations to transcendental equations of the form y = sin s, where y is given and s is the unknown. Using the power-series for the sine, and calling A, B, C, etc., the roots of the equation, he writes it as 3 5 s s s 0 = 1 - - + ------- - ----------- + ... y 1 2 3 y 1 2 3 4 5 y s s s = (1 - - )(1 - - )(1 - - ) ... A B C and concludes from this, firstly, that 1 1 1 1 - = - + - + - + ..., y A B C then, comparing terms in s^2, that 1 1 1 1 2 1 1 1 0 = -- + ... = - (- + - + ... ) - - (--- + --- + ...), AB 2 A B 2 A^2 B^2 which gives \sum 1/A^2 = 1/y^2, and similarly for higher powers. Taking in particular y = 1, he states that the roots A, B, C, etc., are q, q, - 3q, -3q, 5q, 5q, etc., where q is one fourth of the circumference of a circle of radius 1 (i.e. pi/2 in the notation he was soon to adopt). This gives: 2 1 1 1 = 2 - (1 - - + - - ...), pi 3 5 which was Leibniz's result, and further: 2 2 1 1 1 = 2 (- ) (1 + --- + --- + ...). pi 3^2 5^2 It was of course obvious to Euler that one has 1 1 1 1 1 1 1 1 + --- + --- + ... = (1 + --- + --- + ...) - (--- + --- + --- + ...) 3^2 5^2 2^2 3^2 2^2 4^2 6^2 1 1 1 3 = (1 + --- + --- + ...)(1 - ---) = - zeta(2), 2^2 3^2 2^2 4 It was of course obvious to Euler that one has 1 1 1 1 1 1 1 1 + --- + --- + ... = (1 + --- + --- + ...) - (--- + --- + --- + ...) 3^2 5^2 2^2 3^2 2^2 4^2 6^2 1 1 1 3 = (1 + --- + --- + ...)(1 - ---) = - zeta(2), 2^2 3^2 2^2 4 so that the above conclusions give zeta(2) = pi^2/6. The same procedure, pushed further, gives to Euler (somewhat la- boriously, "multo labore": Eu.I-14.85) the values of zeta(n) for n = 4,6,8,10,12, and those of L(n) for n = 3,5,7. Applied to other values of y than y = 1, it gives the value of still other series (Dirichlet series in modern terminology). All this is rapidly sketched in Euler's article of 1735; one can almost feel in it the feverish excitement of the writer. Euler's method had been open to grave objections, as he perceived himself (cf. Eu.I-14.79 in E 41|1735, and Eu.I-14.139-140 in E 61|1743) even before they were pointed out by his correspondents: Johann Bernoulli in 1737 (Corr.II.16-17), as well as Daniel Bernoulli (cf. Corr.II.477|1741), Daniel's cousin Nicolas Bernoulli (cf. Corr.II.683|1742), Cramer in Geneva (cf. Corr.II.477,683) and perhaps others. Apart from the question of the con- vergence, or possible divergence, of the series and products involved (a question raised, somewhat clumsily, by Nicolas Bernoulli: Corr.II.683-684 and 691|1742), one could well ask whether the transcendental equation 1 3 1 = s - ----- s + ... 1.2.3 had not, besides its "visible" real roots, other "imaginary" roots which would vitiate the whole calculation (Corr.II.16; cf. Eu.IV A-5.121); nor was this objection limited to the possible existence of roots of the form a + b sqrt(-1), even if this could have been meaningfully asked before Euler de- fined the sine for such values of the argument (cf. infra, S.XIX); for the so-called fundamental theorem of algebra, which at least had the status of a plausible conjecture for algebraic equations (cf. above, S.VII) could not even be rea- sonably formulated for equations "of infinite degree" such as the one Euler was trying to handle. More specifically, it was objected to him that if an ellipse was substituted for the circle in the geometric definition of the sine, his con- clusions would become obviously false, while the visible real roots of his equation would remain the same (Corr. 11.477). At first Euler held his doubts in abeyance by observing that his method agreed with Leibniz's result in the case of L(1) (so that, as he says, if the equation 1 = sin x had "im- aginary roots" besides the visible real ones, at least the sum of their reciprocals would be 0), and also by comparing his new values for zeta(2), zeta(4), etc., with the numerical values he had computed previously (cf. above, S.XVII). Thus, he says, "I did not hesitate to publish these sums as being perfectly true" ("non dubitavi istas summas tanquam verissimas producere": Eu.I-14.140 in E 61|1743). Nevertheless, for the next ten years, Euler never relaxed his efforts to put his conclusions on a sound basis. At first this led him only to an analytical proof for zeta(2) = pi^2/6 (or rather for the equivalent result 1 + 1/3^2 + 1/5^2 + ... = pi^2/8) which did not seem capable of being extended even to zeta(4) (cf. his letter of 1737 to Johann Bernoulli, Bibl.Math.(III) 5 (1904), pp.258-259, and P. Stackel's article, Eu.I-14.156-176). But now his attention had been drawn by his heuristic proof of 1735, as well as by his earlier work on what is now called the gamma function (cf. Corr.I.3- 6|1729, etc.) to infinite products in their relation to infinite series, and this, too, led him to a momentous discovery. In a paper presented to the Academy in 1737 (Eu.I-14.216- 244 = E 72), after discussing some rather queer series sug- gested to him by Goldbach, he goes on to consider some infinite products "no less admirable", he says, "since in them the factors proceed according to the prime numbers, whose progression is no less irregular [than the terms in Goldbach's series]" ("... neque minus erunt admirabilia ... cum ... in his ... termini progrediantur secundum numeros primos, quorum progressio non minus est abstrusa": Eu.I-14.227). What he means by this is the famous "eulerian" product for zeta(s): zeta(s) = Prod (1 - p^-s)^-1 p extended to all primes p, or in his notation 1 1 1 2^n 3^n 5^n 7^n ... 1 + --- + --- + --- + ... = -------------------------------- 2^n 3^n 4^n (2^n-1)(3^n-1)(5^n-1)(7^n-1) ... (Eu.1-14.230), and the similar product pi 1 1 1 1 3 5 7 11 13 ... -- = 1 - - + - - - + - - ... = --------------- 4 3 5 7 9 4 4 8 12 12 ... or in modern notation p L(1) = Prod ---- p!=2 p+-1 where the sign is determined by p+-1 = 0 (mod 4) (Eu.I- 14.233; cf. Corr.I.577-578|1752). From the former result he deduces that the sum \sum 1/p is infinite "as the logarithm of the sum of the harmonic series \sum 1/n"; as he knew that the sum \sum_1^N 1/n is of the order of magnitude of log N (cf. e.g. Eu.I-14.87-l00 = E 43|1734) this meant that the sum \sum_{p