Approximating the divisor functions

We exploit the properties of a sequence of functions that approximate the divisor functions and combine them with an analytical formula of a delta-like sequence to give a new proof of a theorem of Gronwall on the asymptotic of the divisor functions.


Introduction and statement of the main results
Given a complex number α, for any m ∈ N = {1, 2, . ..} the sum of positive divisors function σ α (m) is defined as the sum of the αth powers of the positive divisors of m [1].It is usually expressed as where k|m means that k ∈ N is a divisor of m.The functions so defined are called divisor functions.They play an important role in number theory, namely within the study of divisibility properties of integers and the distribution of prime numbers.Such functions appear in connection with the Riemann zeta function Indeed, given any α ∈ R, it is plain that In particular, for α > 1 this yields In 1913, Grönwall [4] showed in a quite elementary way that if α > 1, then where [1, 2, . . ., m] is the least common multiple of 1, 2, . . ., m.
The proof of this theorem is arguably more involved than Grönwall's one.However, a novelty of our approach lies in the fact that, unlike Grönwall, we do not use the multiplicativity of σ α , i.e., σ α (mn) = σ α (m)σ α (n) for any coprime m, n [1], but we exploit the properties of a sequence of C ∞ functions whose pointwise limit is σ α (see §2).
Our main tool is the following theorem, which, to the best of our knowledge, is new in the literature.
which is continuous in a neighborhood of the integers, we have where ⌊η⌋ denotes the integer part of η ∈ R and χ Z is the characteristic function of Z.
Our hope is also that this alternative approach might provide new insights and lead to some new result in the future.
Acknowledgements.The approximation of the divisor functions with the sequences of C ∞ functions presented in this paper is an original idea of the first Author of this paper and the core of his Master thesis at Florida International University.

Notation
For brevity, we write lim , and the symbol {•} n∈N for a sequence is abbreviated as {•} n .The power (cos(y)) β is written as cos β (y), and the same we do for the sine function.
Given a real number x ≥ 1, in sums like x k=1 we mean that k ∈ N with k ≤ ⌊x⌋, where ⌊x⌋ denotes the integer part of x ∈ R, i.e., the greatest integer such that x − ⌊x⌋ ≥ 0.
The characteristic function of a set A ⊆ R is denoted by χ A .

Approximation of the divisor function
In this section a sequence of functions that approximate σ α is defined.To this end, we consider σ α for real α > 0, and we extend such a function to [0, ∞) by letting For any given real numbers α > 0 and M > 1, let us consider the sequence where The main property of the functions (2.1) is given by the following proposition.
Proposition 2.1.For any given real numbers α > 0, M > 1 and x ∈ [0, M ], the sequence n → C α,n (M ; x) is non-increasing, and In this case, the sequence n → cos 2n (πx/k) is decreasing and lim n cos 2n (πx/k) = 0. On the other hand, 3 Delta-like sequences and proof of Theorem 1.2 In the literature (see e.g.[5], [2]) a sequence of non-negative functions If, in addition, the functions g n are smooth, they are also called mollifiers, or approximations of the identity.
A well-known set of delta-like sequences is made of non-negative functions g n ∈ L 1 (R m ) satisfying the properties (a) and (b) below.
(a) For every ǫ > 0, there exists δ > 0 such that Here and in what follows, we have let |x| = x 2 1 + ... + x 2 m .We prove the following where each g n is non-negative, satisfies the previous property (a), and Proof.From (b') it follows that for every fixed 0 < ǫ < 1 there exists N > 0 such that for some δ > 0. Thus, by using (a) we see that if n ≥ N , then We also have that which yields (3.1).
4 Proof of Theorem 1.1 For x, n ∈ N, with x > 1, let us take M = 2x for the approximants of σ α (x) defined in (2.1), namely Recall that the function f n (t) := t α cos 2n (πx/t) extends to a continuous function in [0, ∞), with f n (0) = f n (2x) = 0. Further, it satisfies the hypotheses of Theorem 1.2 in [0, 2x].Thus, By the change of variables t → xt one has where Thus, Theorem 2.1 yields Since it is plain that f n (xt) ≥ 0 for all t ∈ [0, 2], we can take a sufficiently large integer m, set m ′ = 4(m!), and write after the change of variable xt → t, Now, let us take x = m! so that x/k = m!/k ∈ N and ∈ Z for all k ∈ {1, . . ., m}.Thus, the previous inequality becomes Again, we apply Theorem 1.It is plain that in order to prove the same limit for G α ([1, 2, . . ., m]) it suffices to take x = [1, 2, . . ., m] and proceed in a completely analogous way.Theorem 1.1 is completely proved.