Chap. XVIII] NuMBER OF PRIMES. 433 
since the only real zeros of f{x) are the primes. The integration extends 
over a closed contour enclosing the segment of the a;-axis from 1 to n and 
narrow enough to cofitain no complex zero of /(x). 
T. Levi-Civita^^^^ gave an analytic formula, involving definite integrals 
and infinite series, for the number of primes between a and /3. 
L. Gegenbauer^" gave formulas, similar to that by von Koch,^"^ for the 
number of primes 4s ±1 or 6s ±1 which are ^n. 
A. P. Minin^^'^'' wrote ^{y)=0 or 1 according as y is composite or prime; 
*^®° e{n-l) = [n-2] + [n-b] + [n-7]+ . . .-^^P{x-\)[n-x], 
summed for all composite integers x. 
Gegenbauer^"'' proved that Sylvester's^^^ expression for the number of 
primes >n and <2n equals S)u(x)[m/a:+l/2], where x takes those integral 
values S 2n which are products of primes S y/2n. 
F. RogeP^^ gave a recursion formula for the number of primes ^m. 
T. Hayashi^^^ wrote Rf/q for the remainder obtained on dividing / by q. 
By Laurent's^^'^ result, —RFn{x)/{x'' — l)rf'~^ = or 1 according as n is 
composite or prime. Hence the sum of the jth powers of the primes between 
.andHs _j,;__FM__ 
nt.(a:"-l)n"-^-2' 
which becomes the number of primes for j = 0. If a is a primitive nth root 
of unity, Wilson's theorem shows that 
n-l 
Sa^"^ = norO (m = (n-l)! + l), 
; = 
according as n is prime or composite. Hence ^x^"~^^7(a:"— 1) = 1 or 
according as n is prime or composite. Thus 
^2a;("-i)y(a;'»-l) 
n = s 
is the number of primes between s and t. 
Hayashi^^" reproduced the second of his two preceding results and gave 
it the form 
J"^^"^ „ ^ \ cos (m—n)d — r''cos md\dd ^ 
„ '• l-2r''cosng+r^' =^" " «' 
according as n is prime or not, and gave a direct proof. 
J. V. Pexider^^^ investigated the number \l/{x) of primes ^ x. Write 
4g=G]-[^]' ^"-[f]- 
"«&Atti R. Accad. Lincei, Rendiconti, (5), 4, 1895, I, 303-9. 
"'Monatshefte Math. Phys., 7, 1896, 73. 
2"aBull. Math. Soc. Moscow, 9, 1898, No. 2; Fortschritte, 1898, 165. 
2376Monatshefte Math. Phys., 10, 1899, 370-3. 
238Archiv Math. Phys., (2), 17, 1900, 225-237. 
«9Jour. of the Phys. School in Tokio, 9, 1900; reprinted in Abhand. Gesch. Math. Wiss., 28, 
1900, 72-5. 
»^oArchiv Math. Phys., (3), 1, 1901, 246-251. 
2«Mitt. Naturforsch. Gesell. Bern, 1906, 82-91. 
