262 



NA TURE 



\juty 12, 



Dr. J. P. Gram (MJmoires de V Academie Royale de Copen- 

 hagne, 6me. serie, vol. ii. p. 191) refers to a paper by Mertens 

 ("Ein Beitrag zur analytischen Zahlentheorie," Borckardfs 

 Jonrnil, Bd. 78), as one in which the truth of the first of the two 

 theorems is demonstrated — " fuldstoendigt Bevis af Mertens " 

 are Gram's words. 1 

 Assuming this to be the case, we shall easily find when N is 



.indefinitely great, so that S N becomes -, 



4 



Qn S n = 



(1 -4) (1 -4). . . . (1 - * 



which, according to Legend re's empirical law (Legendre, 

 " Theorie des Nombres," 3rd edition, vol. ii. p. 67, art. 397), 



2 1 0°" "N 



is equal to = — , where K = 1*104 ; and as we have written 



-Q S . = log N + (V - R), we may deduce, upon the above 

 assumptions, 



V - R = (— - 1) log N = o*8n .... log N. 



R, we know, is demonstrably less than ( 1 - - ) log N, con- 

 sequently V must be less than (0*812 + 0*215) l°g N, i.e. less 

 than 1*027 log N, and a fortiori the portion of the omnipositive 

 aggregate Q v which consists of terms whose denominators ex- 

 ceed N, when N is indefinitely great, cannot be less than 

 log N, i.e. 0*273 log N. 



Before concluding, let me add a word on Legendre's empirical 

 •formula for the value of 



(1 - \) (1 -1). . . . 1 



PsJ 



referred to in the early part of this article. 



If N is any odd number, the condition of its being a prime 

 number is that when divided by any odd prime less than its cwn 

 square root, it shall not leave a remainder zero. Now if N (an 

 unknown odd number) is divided by /, its remainder is equally 

 likely to be o, 1, 2, 3, . . . . or (/ - 1). Hence the chance 



that it is not divisible by/ is ( 1 - - J, and, if we were at liberty 



to regard the like thing happening or not for any two values of 

 p within the stated limit as independent events, the expectation 

 ■of N being a prime number would be represented by 



(!-!)(!- I) (I -|) (I - T \) 



I 



P^.i 



■which, according to the formula referred to, for infinitely large 



values of N is equal to - *-. It is rather more convenient to 



log N* 



regard N as entirely unknown instead of being given as odd, on 



which supposition the chance of its being a prime would be 



1*104 1*104 

 T- or -. 



2 log N* lo g N 



Hence for very large values of N the sum of the logarithms of 

 all the primes inferior to N might be expected to be something 

 like (i*io4)N. This does not contravene Tchebycheffs formula 

 (Serret, " Coursd'Algebre Superieure," 4me ed., vol. ii. p. 233), 

 which gives for the limits of this sum AN and BN, where 



6A 



A = 0*921292, and B = — = 1*10555; but does contravene the 



■narrower limits given by my advance upon Tchebycheffs 



1 It always seems to m: absurd to speak of a complete proof, or of a 

 theorem being rigorously demonstrated. An incomplete proof is no proof, 

 and a mathematical truth not rigorously demonstrated is not demonstrated 

 at all. I do not mean to deny that there are mathematical truths, morally 

 certain, which defy and will probably to the end of time continue to defy 

 proof, as, e.g. , that every indecomposable integer polynomial function must 

 represent an infinitude of primes. I have sometimes thought that the pro- 

 found mystery which envelops our conceptions relative to prime numbers 

 depends upon the limitation of our faculties in regard to time, which like 

 space may be in its essence poly-dimensional, and that this and such sort of 

 truths would become self-evident to a being whose mode of perception is 

 according to superficially as distinguished from our own limitation to 

 Jinearly extended time. 



method (see Am. Math. Journal, vol. iv. Part 3), according 

 to which for A, B, we may write A x , B 1( where 



Aj = 0*921423, B : = 1*076577. •* 



That the method of probabilities may sometimes be success- 

 fully applied to questions concerning prime numbers I have 

 shown reason for believing in the two tables published by me in 

 the Philosophical Magazine for 1883. ■ 



New College, June 10. J. J. Sylvester. 



SOCIETIES AND ACADEMIES. 

 London. 



Royal Society, May 3. — " Electro- Chemical Effects on 

 Magnetizing Iron." Part II. By Thomas Andrews, F.R. S.E. 

 Communicated by Prof. G. G. Stokes, P..R.S. 



The present paper contains the results of a further study of the 

 electro-chemical effects observed between a magnetized and an 

 unmagnetized bar when in circuit in certain electrolytes, recorded 

 in Part I. of this research. The method of experimentation 

 was generally similar to that pursued and described in Part I., 



1 Viz. A] = -— °- 2 A, and Bi = S959 - 5 A, the values of which are incorrectly 

 50999 50999 



stated in the memoir. Strange to say, Dr. Gram, in his prize essay, pre- 

 viously quoted, on the number of prime numbers under a given limit, has 

 omitted all reference to this paper in his bibliographical summary of the sub- 

 ject, which is only to be accounted for by its having escaped his notice ; a 

 narrowing of the asymptotic limits assigned to the sum of the logarithms of 

 the prime numbers series being the most notable fact in the history of the 

 subject since the publication of 'tchebycheffs me noir. Subjectively, this 

 paper has a peculiar claim upon the regard of its author, for it was his medi- 

 tation upon the two simultaneous difference-equations which occur in it that 

 formed the starting-point, or incunabulum, of that new and boundless world 

 of thought to which he has given the name of Universal Algebra. But, apart 

 from this, that the superior limit given by TchebychefF as 1*1055 should be 

 brought down by a more stringent solution of his own inequalities t ) only 

 1*076577 — in other words, that the excess above the probable mean value 

 (unity) should be reduced to little more than jfrds of its original amount — is in 

 itself a surprising fact. Perhaps the numerous (or innumerable) misprints 

 and arithmetical miscalculations which disfigure the paper may help to 

 account for the singular neglect which it has experienced. It will be noticed 

 that the mean of the limits of TchebychefF is 1 01342, the mean of the new 

 limits being 0*99900. The excess in th^ one case ab ^ve and the defect in 

 the other below the probable true mean are respectively 001342 and 

 o'ooioo. 



- A principle precisely similar to that employed above if applied to deter- 

 mining the number of reduced proper fractions whose denominators do not 

 exceed a given number n, leads to a correct result. The expectation of two 

 numbers being prime to each other will be the product ofthe expectations 

 of their not being each divisible by any the same prime number. But the 



probability of one of them being divisible by i is -, and therefore of two of 



z 



them being not each divisible by i is -. Hence the probability of their 



having no common factor is 



(1 - i)(i - \) 1 - A) (rii) • ■ • • ad inf., i.e. is %. 



If. then, we take two sets of numbers, each limited to n, the probable number 

 of relatively prime combinations of each of one set with each of the other 



should be -- , and the number of reduced proper fractions whose denomin- 



ators do not exceed n should be the half of this or — - . I believe M. Cesaro 



has claimed the prior publication of this mode of reasoning, to which he is 

 heartily welcome. The number of these fractions is the same thing as the 

 sum of the totients of all numbers not exceeding n. In the Philosophical 

 Magazine for 1883 (vol. xv. p. 251), a table of the.se sums of totients has been 

 published by me for all values of n not exceeding 500, and in the same year 

 (vol. xvi p. 231) the table was extended to values of n not exceeding 1000 

 In every case without any exception the estimated value of this totient sum 

 is found to be intermediate between 



V* and 3(" + e£. 



7T-' 77- 



Calling the totient sum to n, T(«), I stated the exact equation 



n" + n 



T(«) + t(;)+t(*)+t(2) + 



from which it is capable of proof, without making any assumption as to the 



form of T«, that its asymptotic value is „ -* The functional equation itself 



is merely an integration (so to say) of the well known theorem that any 

 number is equal to the sum ofthe totients of its several divisors. The intro- 

 duction to these tables will be found very suggestive, and besides contains an 

 interesting bibliography of the subject of Karey series {suites de Fairy), 

 comprising, among other writers upon it, the names of Cauchy, Glaisher, and 

 Sir G. Airy, the last-named as author of a paper on toothed wheels, pub- 

 lished, I believe, in the " Selected Papers " of the Institute of Mechanical 

 Engineers. The last word on the subject, as far as I am aware, forms one of 

 the interludes, or rather the postscript, to my "Constructive Theory of 

 Partitions," published in the American Journal 0/ Mathematics. 



