Page:EB1911 - Volume 19.djvu/894

ℎ(𝑑) meaning the number of primitive forms for the determinant 𝑑. This is a generalisation of a theorem due to Dirichlet.

There is another formula which, in a certain sense, is the generalisation of Gauss’s sums (§ 62) in cyclotomy. Let 𝜓(𝑢, 𝑣) denote the function 𝜃₁₁(𝑢＋𝑣)÷𝜃₀₁(𝑢)𝜃₀₁(𝑣) and let 𝖣₁, 𝖣₂ be any two fundamental discriminants such that 𝖣₁𝖣₂ is also fundamental and negative: then

${\tau \theta '_{11} \over 2\pi |{\sqrt {{\mbox{D}}_{1}{\mbox{D}}_{2}}}|}\sum _{s_{1},\ s_{2}}\left({{\mbox{D}}_{1} \over s_{1}}\right)\left({{\mbox{D}}_{2} \over s_{2}}\right)\psi \left({2s_{1} \over |{\mbox{D}}_{1}|},{2s_{2} \over |{\mbox{D}}_{2}|}\right)=\sum _{a,\ b,\ c}\left[\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)+\left({{\mbox{D}}_{2} \over {\mbox{A}}}\right)\right]\sum _{m,\ n}q^{{1 \over 2}(am^{2}+bmn+cn^{2})}$

where, on the left-hand side, we are to sum for 𝑠_𝑖＝1, 2, 3 . . . |𝖣_𝑖; and on the right we are to take a complete set of representative primitive forms (𝑎, 𝑏, 𝑐) for the determinant 𝖣₁𝖣₂, and give to 𝑚, 𝑛 all positive and negative integral values such that 𝑎𝑚²＋𝑏𝑚𝑛＋𝑐𝑛² is odd. The quantity 𝜏 is 2, if 𝖣₁𝖣₂＜−4, 𝜏＝4 if 𝖣₁𝖣₂＝−4, 𝜏＝6 if 𝖣₁𝖣₂＝−3. By putting 𝖣₂＝1, we obtain, after some easy transformations,

$\sum _{s=1}^{s=\Delta }\left({-\Delta \over s}\right){\mbox{sn}}{4s{\mbox{K}} \over \Delta }={4{\sqrt {\Delta }} \over \tau \theta _{10}{}^{2}}\sum \sum q^{{1 \over 2}(am^{2}+bmn+cn^{2})}$,

which holds for any fundamental discriminant −Δ. For instance, taking 𝜔＝𝑖𝖪′/𝖪, and Δ＝3, we have 𝜃₁₀²＝2𝜅𝖪/𝜋, and Σ𝑞^{⁠1/2⁠(𝑚²＋𝑚𝑛＋𝑛²)}＝⁠2𝜅𝖪√3/𝜋⁠sn⁠4𝖪/3⁠; a verification is afforded by making 2𝖪 approach the value 𝜋, in which case 𝑞, 𝜅 vanish, while the limit of 𝑞^⁠1/2⁠/𝜅 is ⁠1/4⁠, whence the limiting value of sn⁠4𝖪/3⁠ is that of 6𝑞^⁠1/2⁠/𝜅√3, which ＝6/4√3＝√3/2, as it should be.

Several of Kronecker’s formulae connect the solution of the Pellian equation with elliptic modular functions: one example may be given here. Let 𝖣 be a positive discriminant of the form 8𝑛＋5, let (𝖳, 𝖴) be the least solution of 𝖳²−𝖣𝖴²＝1: then, if ℎ(𝖣) is the number of primitive classes for the determinant 𝖣,

(𝖳−𝖴√𝖣)^ℎ(𝖣)＝Π(2𝜅𝜅′)²

where the product on the right extends to a certain sixth part of those values of 2𝜅𝜅′ which are singular, and correspond to the field Ω(√−𝖣), or in other words are connected with the class invariant 𝑗(√−𝖣). For instance, if 𝖣＝5, the equation to find (𝜅𝜅′)² is

4⋅𝛿{(𝜅𝜅′)²−1}³＋(25＋13√5)³(𝜅𝜅′)⁴＝0

one root of which is given by (2𝜅𝜅′)²＝9−4√5＝𝖳−𝖴√5 which is right, because in this case ℎ(𝖣)＝1.

74. Frequency of Primes.—The distribution of primes in a finite interval (𝑎, 𝑎＋𝑏) is very irregular, if we change 𝑎 and keep 𝑏 constant. Thus if we put 𝑛!＝𝜇, the numbers 𝜇＋2, 𝜇＋3, . . . (𝜇＋𝑛−1) are all composite, so that we can form a run of consecutive composite numbers as extensive as we please; on the other hand, there is possibly no limit to the number of cases in which 𝑝 and 𝑝＋2 are both primes. Legendre was the first to find an approximate formula for 𝖥(𝑥), the number of primes not exceeding 𝑥. He found by induction

𝖥(𝑥)＝𝑥 ÷ (log_𝑒𝑥−1·08366)

which answers fairly well when 𝑥 lies between 100 and 1,000,000, but becomes more and more inaccurate as 𝑥 increases. Gauss found, by theoretical considerations (which, however, he does not explain), the approximate formula

${\mbox{F}}(x)={\mbox{L}}(x)=\int _{2}^{x}{dx \over \log x}$

(where, as in all that follows, log 𝑥 is taken to the base 𝑒). This value is ultimately too large, but when 𝑥 exceeds a million it is nearer the truth than the value given by Legendre’s formula.

By a singularly profound and original analysis, Riemann succeeded in finding a formula, of the same type as Gauss’s, but more exact for very large values of 𝑥. In its complete form it is very complicated; but, by omitting terms which ultimately vanish (for sufficiently large values of 𝑥) in comparison with those retained, the formula reduces to

${\mbox{F}}(x)={\mbox{A}}+\sum _{m}(-1)^{\mu }{1 \over m}{\mbox{L}}(x^{1/m})\quad (m=1,2,3,5,6,7,11,...)$

where the summation extends to all positive integral values of 𝑚 which have no square factor, and 𝜇 is the number of different prime factors of 𝑚, with the convention that when 𝑚＝1, (−1)^𝜇＝1. The symbol 𝖠 denotes a constant, namely

${\mbox{A}}=\sum {(-1)^{\mu } \over m}\times \left\{{\tfrac {1}{2}}-\int _{2}^{\infty }{dx \over x(x^{2}-1)\log x}\right\}$

and 𝖫 is used in the sense given above.

P. L. Tchébichev obtained some remarkable results on the frequency of primes by an ingenious application of Stirling’s theorem. One of these is that there will certainly be (𝑘＋1) primes between 𝑎 and 𝑏, provided that

𝑎＜⁠5𝑏/6⁠ − 2√𝑏 − ⁠16/25⁠ 𝖱 log 6 (log 𝑏)² − ⁠5/24𝖱⁠ (4𝑘＋25) − ⁠+25/6𝖱⁠

where 𝖱＝⁠1/2⁠ log 2 ＋ ⁠1/3⁠ log 3 ＋ ⁠1/5⁠ log 5 − ⁠1/30⁠ log 30＝0·921292 . . . . From this may be inferred the truth of Bertrand’s conjecture that there is always at least one prime between 𝑎 and (2𝑎 − 2) if 2𝑎＞7. Tchébichev’s results were generalized and made more precise by Sylvester.

The actual calculation of the number of primes in a given interval may be effected by a formula constructed and used by D. F. E. Meissel. The following table gives the values of 𝖥(𝑛) for various values of 𝑛, according to Meissel’s determinations:—

𝑛		𝖥(𝑛)
20,000		2,262
100,000		9,592
500,000		41,538
1,000,000		78,498

Riemann’s analysis mainly depends upon the properties of the function

$\zeta (s)=\sum _{n}n^{-s}\qquad \qquad (n=1,\ 2,\ 3,\ .\ .\ .\ )$,

considered as a function of the complex variable 𝑠. The above definition is only valid when the real part of 𝑠 exceeds 1; but it can be generalized by writing

$2\sin \pi z\Gamma (z)\zeta (z)=i\int _{\infty }^{\infty }{\frac {(-x)^{z-1}dx}{e^{x}-1}}$

where the integral is taken from 𝑥＝＋∞ along the axis of real quantities to 𝑥＝𝜖, where 𝜖 is a very small positive quantity, then round a circle of radius 𝜖 and centre at the origin, and finally from 𝑥＝𝜖 to 𝑥＝＋∞ along the axis of real quantities. This function ζ(𝑧) is of great importance, and has been recently studied by von Mangoldt Landau and others.

Reference has already been made to the fact that if 𝑙, 𝑚 are coprimes the linear form 𝑙𝑥＋𝑚 includes an infinite number of primes. Now let (𝑎, 𝑏, 𝑐) be any primitive quadratic form with a total generic character 𝖢; and let 𝑙𝑥＋𝑚 be a primitive linear form chosen so that all its values have the character 𝖢. Then it has been proved by Weber and Meyer that (𝑎, 𝑏, 𝑐) is capable of representing an infinity of primes all of the linear form 𝑙𝑥＋𝑚.

75. Arithmetical Functions.—This term is applied to symbols such as φ(𝑛), Φ(𝑛), &c., which are associated with 𝑛 by an intrinsic arithmetical definition. The function Φ(𝑛) was written ʃ𝑛 by Euler, who investigated its properties, and by proving the formula $\prod _{1}^{\infty }(1-q^{s})=\sum _{-\infty }^{+\infty }q^{{1 \over 2}3s^{2}+s}$ deduced the result that

$\int n=\int (n-1)+\int (n-2)-\int (n-5)-...=\sum (-1)^{s-1}\int \left(n-{3s^{2}\pm s \over 2}\right)$

where on the right hand we are to take all positive values of 𝑠 such that 𝑛－⁠1/2⁠(3𝑠²±𝑠) is not negative, and to interpret 𝑓0 as 𝑛, if this term occurs. J. Liouville makes frequent use of this function in his papers, but denotes it by ζ(𝑛).

If the quantity 𝑥 is positive and not integral, the symbol E(𝑥) or [𝑥] is used to denote the integer (including zero) which is obtained by omitting the fractional part of 𝑥; thus E(√2) =1, E(0·7)=0, and so on. For some purposes it is convenient to extend the definition by putting E(−𝑥) = −E(𝑥), and agreeing that when 𝑥 is a positive integer, E(𝑥) =𝑥−⁠1/2⁠; it is then possible to find a Fourier sine-series representing 𝑥−E(𝑥) for all real values of 𝑥. The function E(𝑥) has many curious and important properties, which have been investigated by Gauss, Hermite, Hacks, Pringsheim, Stern and others. What is perhaps the simplest roof of the law of quadratic reciprocity depends upon the fact that if 𝑝, 𝑞 are two odd primes, and we put 𝑝=2ℎ+1, 𝑞=2𝑘+1

$\sum _{r=1}^{r=h}{\mbox{E}}{\Big (}{\frac {rp}{q}}{\Big )}+\sum _{s=k}^{s=1}{\mbox{E}}{\Big (}{\frac {sp}{q}}{\Big )}=hk={\tfrac {1}{4}}(p-1)(q-1)$

the truth of which is obvious, if we rule a rectangle 𝑝″×𝑞″ into unit squares, and draw its diagonal. This formula is Gauss’s, but the geometrical proof is due to Eisenstein. Another useful formula is

$\sum _{r=1}^{r=m-1}{\mbox{E}}{\Big (}x+{\frac {r}{m}}{\Big )}={\mbox{E}}(mx)-{\mbox{E}}(x)$, which is due to Hermite.

Various other arithmetical functions have been devised for particular purposes; two that deserve mention (both due to Kronecker) are δ_ℎ𝑘, which means 0 or 1 according as ℎ, 𝑘 are unequal or equal, and sgn 𝑥, which means 𝑥÷|𝑥|.

76. Transcendental Numbers.—It has been proved by Cantor that the aggregate of all algebraic numbers is countable. Hence immediately follows the proposition (first proved by Liouville) that there are numbers, both real and complex, which cannot be defined by any combination of a finite number of equations with rational integral coefficients. Such numbers are said to be transcendental. Hermite first completely proved the transcendent character of 𝑒; and Lindemann, by a similar method, proved the transcendence of π. Thus it is now finally established that the quadrature of the circle is impossible, not only by rule and compass, but even with the help of any number of algebraic curves of any order when the coefficients in their equations are rational (see Hermite, C.R. lxxvii., 1873, and Lindemann, Math. Ann. xx., 1882). Another number which is almost certainly transcendent is Euler's constant C. It may be convenient to give here the following numerical values:—