When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the

Abstract. Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance: The number of n ≤ x coprime to a satisfying

1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1

Abstract. Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the type Q(ζq, q √ a), we prove the following prime analogue of a conjecture of Erdős & Pomerance (1985) concerning the exponent function fa(p) (defined to be the minimal exponent e for which a e ≡ 1 modulo p): Fa(x; A, B) (‡) lim x→ ∞ π(x)

For an arbitrary positive integers a> 1, for every n ∈ N we define ω(n) to be the number of distinct prime factors of n, and we define the function fa(n) as:

Abstract. We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes. 1.

Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1

Dedicated to T N Shorey on his sixtieth birthday Abstract. We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n. In particular, we show that P(τ(n)) ≥ (logn) 33/31+o(1) for infinitely many n, and P(τ(p)τ(p 2)τ(p 3 log log plogloglog p))> (1+o(1)) loglogloglog p for every prime p with τ(p) ̸ = 0. Keywords. 1.

We give a relatively easy proof of the Erdős-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.