826 
Mathematics. — “On an arithmetical function connected with the 
decomposition of the positive integers into prime factors.” 1. 
By J. G. VAN DER Corput. (Communicated by Prof. J. C. 
KLUYVER). 
(Communicated in the meeting of May 27, 1916). 
Let # be any arbitrary integer > 1 and resolve w into prime factors; 
let ¢, represent the smallest exponent of these factors and let a, 
indicate how many times ¢, occurs in the series of this exponents. 
Moreover we take e, =O and v, represents the greatest divisor of 
u, for which ¢, >> m, m being any arbitrary positive integer. The 
object of this paper is to deduce a formula obtaining two general 
arithmetical functions /’ and f, satisfying four relations, n representing 
any positive integer, viz. 
1. for Bs mand also Toren 1, 1441 7 
F(a) == 05 
29d, if e, Sm, 
7. =U; 
ae. for. éy = Ul, a, =n, . 
; VO el CAT 
4th, F (u) = O (vy), 
u having a constant value < ——_———_. 
m (m + 1) 
The integers 7 and » are called the parameters of the function 
F and f the function corresponding to £. 
This article, now, is intended to demonstrate the formula 
1 1 \ 
SF OIS pa +0 En for == 1% 
a8 log « (log x)? 
“u=l i 2 nt, CS 
Ef ee 
log x log « 
for any arbitrary integral positive value of 7 and this proof will 
be given in $ 2 for n—1, in $ 3 for the other case. The modulus 
of the congruences, for which this modulus has not been mentioned, 
is in this paper the arbitrary positive integer #, x represents a 
number > 1, / an integer, prime to 4, a has a constant value, viz, 
—— oui > AGP 
adn ie 
ui 
ug = | 
