The approximate functional equation of some Diophantine series

We prove that a family of Diophantine series satisfies an approximate functional equation. It generalizes a result by Rivoal and Roques and proves an extended version of a conjecture posed in their paper. We also characterize the convergence points.


Introduction
Consider g : R −→ R an odd 1-periodic C 1 function and f : R − Z −→ R a 1-periodic continuous function such that exists (and it is finite) for some λ = 0. This means that the only singularities of f , as a real function, are simple pole like singularities at the integers. For (w, α) ∈ (R + ) 2  g(m 2 α) m 2 f (mα) (2) where w (α) is defined by continuity for α = p/q ∈ Q with q ≤ w. Note that it makes sense because g(x) ∼ C(x − n) when x → n ∈ Z. Note also that with these assumptions, we have that g(m 2 α) f (mα)/m is uniformly bounded. Motivated by a Diophantine approximation problem raised in [1], Rivoal and Roques proved in [2] that when g(x) = sin(2π x) and f (x) = cot(π x), N (α) satisfies an approximate functional equation for N ∈ Z + . Namely, they show that has a limit when N goes to infinity for each α ∈ (0, 1], and that it is uniformly bounded in this interval. As a matter of fact, the last term could be replaced by log(1/α). They conjecture that the limiting function is not only bounded but continuous. The existence of this limiting function contrasts with the fact, also proved for this choice of g and f in [2] (see also [1] for a weaker result), that lim N →∞ N (α) exists if and only if α is a Brjuno number, that is, an irrational number such that the convergents p n /q n in its continued fraction satisfy q −1 n log q n+1 < ∞. As an aside, if we choose formally g as a constant, even replacing n 2 by n s , approximate functional equations of similar kind (very explicit ones in some cases) can be established, although the convergence conditions are tighter (see [3][4][5]). These variants are related to some formulas of Ramanujan for ζ at odd values (see [6]) and to previous works of Lerch (see [7]). For other references and a historical account on approximate functional equations, see [8, §3].
The proof of the existence of the limiting function given in [2] uses heavily the additive properties of the sine function and the partial fraction expansion of the cotangent. We show here that this approximate functional equation also holds with the general definition of N as above and that it can be deduced from a mainly combinatorial argument not depending on the choice of f and g. We also prove the continuity of the limiting function for α > 0 and its continuous extension to [0, ∞), settling in particular the conjecture posed in [2]. In the last section we apply a general result of [8] for certain approximate functional equations to characterize completely the convergence points of N when N → ∞.

Remark
The limit of T N (α) when N → ∞ can be taken separately because the series converges but, as pointed out before, the separate existence of the limits of N (α) and N α (1/α) is not assured in general (see Theorem 16). In [2], in the case g(x) = sin(2π x) and f (x) = cot(π x), the value G(0) = 0 is implicitly assigned. We prefer here to let it undefined and to link it to the continuous extension through G(0 + ) = 0. Once we have stated the convergence in (0, 1] the extension to α > 0 is rather easy. In particular G(0 + ) = 0 and G extends to a continuous function on [0, ∞).
Notation. Given a real number x, x stands for the distance from x to its nearest integer. Vinogradov's notation A B is employed here with the same meaning as A = O(B).

Some reductions
We begin with some trivial remarks that eventually lead to some reductions in the proofs of Theorem 1 and Corollary 2.
converges uniformly to a continuous bounded function on R. Proof is bounded in a neighborhood of 0 and, by the 1-periodicity, it is bounded on R.

Lemma 4 Assume that
Proof It follows easily from the definition of (α).
The first two reductions and some simple manipulations with T N are enough to conclude Corollary 2. We abbreviate lim N →∞ T N as T .

Proof of Corollary 2 By Lemma 3, it is enough to consider the case f (x) = cot(π x).
Theorem 1 assures the existence and the continuity of G in (0, 1] which are reflected into [1, ∞) thanks to Lemma 4. The continuous extension to [0, ∞) follows from the bound

Proof of the main result
The key argument to prove Theorem 1 is that the terms in α N α (1/α) are almost completely canceled by the terms in N m=1 g(m mα ) m 2 f ( mα ) such that mα is close to a positive integer. The following elementary result will be used to identify precisely these integers m. We remind the reader that if y ∈ R is not a half-integer, then y+1/2 gives the nearest integer to y. Note that for any y ∈ R, y = y + 1/2 ± y .
An important, and still elementary, remark is that although r α may not map [1, N α]∩ Z onto [1, N ] ∩ Im r α , the surjectivity only may fail for one element: if N = n/α + 1/2 with n ∈ Z + , n may not belong to [1, N α].
The surjectivity is obvious.
The proof of Theorem 1 is based on a rearrangement of the terms in N (α) − α N α (1/α) involving the functions f n and h m introduced in the previous lemmas. In the rest of this section we assume f (x) = cot(π x) and consequently λ = 1/π in the definition of T N . We can do it without loss of generality by Lemma 3.

Lemma 10
For N , N * and α as in Lemma 7, we have the decomposition Proof Let us set a m = g(m mα ) cot(π mα )/m 2 and write a m = b m + c m with b m = a m if m ∈ Im r α and b m = 0 otherwise. With this notation, according to Lemma 5, we have N * (α) = N * m=1 (b m + c m ). Lemmas 8 and 7 show that where in the last sum, abusing of the notation, we assume that the undefined values of r −1 α (m) when m / ∈ Im r α are ignored because b m = 0. Using the identity for m ∈ Im r α and recalling that ψ is even and 1-periodic, the last sum is N α (α) because 1 − ψ(n/α) cot(π n/α ) is uniformly bounded. It remains to prove that For m ≤ α −1 /2, clearly m / ∈ Im r α and hence h m (α) = 1. Using the estimate cot x − 1/x = O(x), we obtain that the part of the sum corresponding to this range is O(α log(2α −1 )). On the other hand, in the complementary range m > α −1 /2, we have g(m 2 α)/m 3 α 2 and it only remains to show that First note that if mα ≤ α/4, then h m (α) = 0 (see the proof of Lemma 11). Now we have This inequality follows by writing m = q α −1 /2 + r with 0 ≤ r < α −1 /2 and by noting that mα presents gaps of size α when r varies. The last double sum amounts O(α log(2α −1 )) which proves (5).
Proposition 13 For N ≥ 1 and α ∈ (0, 1], we have Proof The trivial bound proves that N (α) − T N (α) and N * (α) − T N * (α) differ in O(N −1 ) which is absorbed by the error term, then it is enough to prove that N * (α) − and (N α) −1 . Thanks to (4) and (6), it equals which is α log(2α −1 ) according to Lemma 12. On the other hand, arranging the last two terms as S and following the proof of Lemma 11, we have S

Convergence of 8 N
If we slightly strengthen the regularity requirements for g, it is possible to characterize the convergence points of N when N → ∞, extending [2,Th. 2]. Namely, we impose in this section g ∈ C 1,γ for some 0 < γ < 1. This means that g satisfies locally the Hölder condition g (y) − g (x) = O |y − x| γ . In fact it is enough to require it for The characterization of the convergence points is a quite direct application of a general result in [8] on approximate functional equations (see [8,Prop. 2] and [8, §6.2]). The point to be checked is a simple analytic fact.
Lemma 14 Let g be a bounded function in C 1,γ for some 0 < γ < 1 with g(0) = 0. Then Proof We can restrict the sum to m < x −1/2 because the rest of the terms give a bounded contribution to the function by the trivial estimate. On the other hand, we know that g (ξ ) − g (0) = O(|ξ | γ ) and the mean value theorem shows that The first sum is g (0) log x −1/2 + O(1) and the second is O(1).
The previous result shows that the case g (0) = 0 is special and that in fact, the sum in Theorem 1 could be omitted. The series appearing in the next result is a model of what happens in this case under g ∈ C 1,γ . The purpose of introducing it here is to give a common treatment to g (0) = 0 and g (0) = 0 in the proof of Theorem 16 (in fact, the absolute convergence in the latter case can be obtained as a quite direct consequence of it). A secondary reason is to show that an extra factor n 2 x γ forces the convergence everywhere of the series in [1, §4] (recall that n nx = n 2 x ). Proof Of course, nx = 0 only occurs if x = a/q ∈ Q with q | n. On the other hand, Then subdividing the range of n into blocks of length q, the series for x = a/q is bounded by ∞ k=1 (qk) −2 q log q, which converges and is uniformly bounded.
For x / ∈ Q, let S be the part of the series corresponding to q ≤ n < Q with p/q and P/Q consecutive convergents in the continued fraction of x. It is enough to prove S = O(q −σ ) for some σ > 0 because the denominators of the convergents grow exponentially [10,Th.12]. Replacing n nx by 1, the terms in S with q n contribute O q −1 log(2q) as shown in the proof of [4,Prop.3.1] (where they are labeled S 2 and S 3 ) taking k = 1, s = 2. If q | n, [4,Lem. 2.3] proves that n nx ≤ 1 and n nx ≥ 1 imply, respectively, n √ q Q and n √ q Q because q Q nx n. Summing up, Writing n = kq, we obtain that each sum is O(q −1 ). Now, we are ready to decide about the convergence of N when N → ∞. To keep the coherence with the previous parts, we state the result for positive numbers, but the periodicity allows to extend it to R.
Theorem 16 Let f and g as in the introduction but imposing also g ∈ C 1,γ for some 0 < γ < 1. For x ∈ R + and g (0) = 0, the series Proof By the 1-periodicity of the series, we can restrict ourselves to x ∈ (0, 1]. And we can also suppose that f (x) = cot(π x) (see Lemma 3). In this interval, the result for x irrational is an application of Proposition 2 in [8], which adresses approximated functional equations involving the Gauss map (denoted by α in [8]) T (x) = 1 x mod 1.
We denote by T k the iterates of T for k ≥ 1. Now set where H (x) is the function in Lemma 14 multiplied by λ. This infinite series corresponds to S G+H (x) in [8] (see [8, (16)]). Note that S(x) converges for every irrational x because G + H is bounded and xT (x) . . . T k−1 (x) ≤ F k+2 where F k stands for the k-th Fibonacci number (see [11, (11) and (