Cg{n), and certain series of arithmetical functions 267 

 so that 



It follows that 



V /i (q) c„ jn) ^ j_j / -CT^ \ ^ /-S7^-OT 



= n--^— n {i-w-(s-i)i 



Thus ,._.,„) = |i;s^ (6.1). 



This formula is equivalent to Raman ujan's formula (9-6), and 

 reduces to (1*4) when s = 2. 



The case s = 1. 

 7. The case in which s = 1, which is not discussed by Ram- 

 anujan, is of particular interest. 

 We observe first that 



n-(-i) (/),_, (n) ^{s) = [1 -p-(-^)} {1 -pr^^-^)} ...^(s) 

 tends to zero, when s-^1, unless n is a prime p or a power of a 

 prime jo, in which case its limit is log p. Thus 

 lim »i-(^-'' (/>,_! (u) ^{s) = A (n), 



A (n) having its usual significance in the theory of primes*. 

 If then we suppose that the series (6-1) is convergent for s = 1, and 

 that its sum is continuous, we obtain 



A(,) = 2^iM;# (M). 



It is not difficult to prove that this formula is incorrect. 



In order to prove this, and to obtain the correct formula, I 

 consider the function 



supposing first that a, the real part of s, is greater than unity. It 

 is plain that 



where t7_= 1 — 



ot*-i(ot- 1)' 

 so that 



* A (n) = log or if « = ro'", and A («) = otherwise. 

 VOL. XX. PART II. 18 



