Inequalities for finite trigonometric sums. An interplay: with some series related to Harmonic numbers

An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of H. Chen (2010).


Introduction
Many identities that evaluate trigonometric sums in closed form can be found in the literature. For example, in a solution to a problem in SIAM Review [8, p.157], M. Fisher shows that General results giving closed forms for the power sums secants p−1 k=1 sec 2n ( kπ 2p ) and p k=1 sec 2n ( kπ 2p+1 ), for many values of the positive integer n, can be found in [4] and [6]. Also, in [13] the author proves that p k=1 sec 2kπ 2p + 1 = p if p is even, −p − 1 if p is odd.
However, while there are many cases where closed forms for finite trigonometric sums can be obtained it seems that there are no such formulae for the sums we are interested in.
In this paper we study the trigonometric sums I p and J p defined for positive integers p by the formulae: with empty sums interpreted as 0.
To the author's knowledge there is no known closed form for I p , and the same can be said about the sum J p . Therefore, we will look for asymptotic expansions for these sums and will give some tight inequalities that bound I p and J p . This investigation complements the work of H. Chen in [5,Chapter 7.] where it was asked, as an open problem, whether the inequality 2p π (ln p + γ − ln(π/2)) holds true for p ≥ 3, (here γ is the so called Euler-Mascheroni constant.) In fact, it will be proved that for every positive integer p and every nonnegative integer n, we have where the b 2k 's are Bernoulli numbers (see Corollary 3.4. The corresponding inequalities for J p are also proved (see Corollary 3.9.) Harmonic numbers play an important role in this investigation. Recall that the nth harmonic number H n is defined by H n = n k=1 1/k (with the convention H 0 = 0). In this work, a link between our trigonometric sums I p and J p and the sum of several series related to harmonic numbers is uncovered. Indeed, the well-known fact that H n = ln n + γ + 1 2n + O 1 n 2 proves the convergence of the numerical series, for every positive integer p.
An interplay between the considered trigonometric sums and the sum of these series will allow us to prove sharp inequalities for I p and J p , and to find the expression of the sums C p , D p and E p in terms of I p and J p .
The main tool will be the following formulation [14,Corollary 8.2] of the Euler-Maclaurin summation formula: Theorem 1.1. Consider a positive integer m, and a function f that has a continuous where the b 2k 's are Bernoulli numbers, B 2m−1 is the Bernoulli polynomial of degree 2m−1, and the notation δg for a function g : [0, 1] → C means g(1) − g(0).
For more information on the Euler-Maclaurin formula, Bernoulli polynomials and Bernoulli numbers the reader may refer to [1,7,10,14,15] and the references therein. This paper is organized as follows. In section 2 we find the asymptotic expansions of C p and D p for large p. In section 3, the trigonometric sums I p and J p are studied.

The sum of certain series related to harmonic numbers
In the next lemma, the asymptotic expansion of (H n ) n∈N is presented. It can be found implicitly in [9, Chapter 9] we present a proof for the convenience of the reader.
Proof. Note that for j ≥ 1 we have Adding these equalities as j varies from 1 to n − 1 we conclude that Thus, letting n tend to ∞, and using the Monotone Convergence Theorem, we conclude dt.
It follows that So, let us consider the function f n : Note that f n (0) = 0, f n (1) = 1/n, and that f n is infinitely continuously derivable with In particular, f Applying Theorem 1.1 to f n , and using the above data, we get The important estimate is the lower bound, i.e. R n,m > 0. In fact, considering separately the cases m odd and m even, we obtain, for every nonnegative integer m : This yields the following more precise estimate for the error term: which is valid for every positive integer m. Now, consider the two sequences (c n ) n≥1 and (d n ) n≥1 defined by For a positive integer p, we know according to Lemma 2.1 that c pn = O 1 n 2 , it follows that the series ∞ n=1 c pn is convergent. Similarly, since d pn = c pn + 1 2pn and the series ∞ n=1 (−1) n−1 /n is convergent, we conclude that ∞ n=1 (−1) n−1 d pn is also convergent. In what follows we aim to find asymptotic expansions, (for large p,) of the following sums: Proposition 2.2. If p and m are positive integers and C p is defined by (2.1), then where ζ is the well-known Riemann zeta function.
Proof. Indeed, we conclude from Lemma 2.1 that with 0 < r pn,m ≤ |b 2m |. It follows that and the desired conclusion follows with ε p,m =r p,m /ζ(2m).
In the next proposition we have the analogous result corresponding to D p .
Proof. Indeed, let us define a n,m by the formula with empty sum equal to 0. We have shown in the proof of Lemma 2.1 that where g n,m is the positive decreasing function on [0, 1/2] defined by Now, for every t ∈ [0, 1/2] the sequence (g np,m (t)) n≥1 is positive and decreasing to 0. So, using the alternating series criterion [3, Theorem 7.8, and Corollary 7.9] we see that, for every N ≥ 1 and t ∈ [0, 1/2], This proves the uniform convergence on [0, 1/2] of the series Now using the properties of alternating series, we see that for t ∈ (0, 1/2) we have On the other hand we have Now, the important estimate for ρ p,m is the lower bound, i.e. ρ p,m > 0. In fact, considering separately the cases m odd and m even, we obtain, for every nonnegative integer m : This yields the following more precise estimate for the error term: and the desired conclusion follows.
The case of E p which is the sum of another alternating series (2.3) is discussed in the next lemma where it is shown that E p can be easily expressed in terms of D p . Lemma 2.4. For a positive integer p, we have where D p is the sum defined by (2.2).
Proof. Indeed

Inequalities for trigonometric sums
As we mentioned in the introduction, we are interested in the sum of cosecants I p defined by (1.1) and the sum of cotangents J p defined by (1.2). Many other trigonometric sums can be expressed in terms of I p and J p . The next lemma lists some of these identities.
Proof. First, note that the change of summation variable k ← p − k proves that K p = K p . So, using the trigonometric identity tan θ + cot θ = 2 csc(2θ) we obtain (i) as follows: Similarly, (ii) follows from the change of summation variable k ← p − k in L p : Thus, using (i) and the trigonometric identity cot(θ/2) − cot θ = csc θ we obtain This concludes the proof of (iii).
Proposition 3.2. For p ≥ 2, let I p be the sum of cosecants defined by the (1.1). Then where D p and E p are defined by formulae (2.2) and (2.3) respectively.
Proof. Indeed, our starting point will be the "simple fractions" expansion [2, Chapter 5, §2] of the cosecant function: which is valid for α ∈ C \ Z. Using this formula with α = k/p for k = 1, 2, . . . , p − 1 and adding, we conclude that and this result can be expressed in terms of the Harmonic numbers as follows Using the well-known result ( [16], [7,Formula 9.542]): and considering separately the cases m even and m odd we obtain the following corollary.
Corollary 3.4. For every positive integer p and every nonnegative integer n, the sum of cosecants I p defined by (1.1) satisfies the following inequalities: As an example, for n = 0 we obtain the following inequality, valid for every p ≥ 1: This answers positively the open problem proposed in [5,Section 7.4].
Remark 3.5. The asymptotic expansion of I p was proposed as an exercise in [10, Exercise 13, p. 460], and it was attributed to P. Waldvogel, but the result there is less precise than Corollary 3.4 because here we have inequalities valid in the whole range of p.
Now we turn our attention to the other trigonometric sum J p . The first step to we find an analogous result to Proposition 3.2 for the trigonometric sum J p , is the next lemma, where an asymptotic expansion for J p is proved but it has a harmonic number as an undesired term, later it will be removed.
Lemma 3.6. For every positive integers p, there is a real number θ p ∈ (0, 1) such that Proof. Indeed, let ϕ be the function defined by According to the partial fraction expansion formula for the cotangent function [2, Chapter 5, §2] we know that Thus, ϕ is defined and analytic on the interval (−1, 2). Let us show that ϕ is concave on this interval. Indeed, it is straight forward to check that, for −1 < x < 2 we have So, we can use Theorem 1.1 with m = 1 applied to the function x → ϕ x+k p for 1 ≤ k < p to get Adding these inequalities and noting that ϕ(0) = 2, ϕ (0) = 1, ϕ(1) = 1 and ϕ (1) = −π 2 /3, we get  Proof. Recall that c n = H n − ln n − γ − 1 2n satisfies c n = O(1/n 2 ). Thus, both series c pn and C p = ∞ n=1 (−1) n−1 c pn are convergent. Further, we note that C p = D p − ln 2 2p where D p is defined by (2.2). According to Proposition 3.2 we have Now, noting that we conclude that C p − C p = 2C 2p , or equivalently On the other hand, for a positive integer p let us define F p by It is easy to check, using Lemma 3.1 (iii), that We conclude from (3.2) and (3.4) that C p − 2C 2p = F p − 2F 2p , or equivalently Hence, ∀ m ≥ 1, and letting m tend to +∞ we obtain C p = F p , which is equivalent to the announced result.
Using the values of the ζ(2k)'s [7, Formula 9.542]), and considering separately the cases m even and m odd we obtain the next corollary. Corollary 3.9. For every positive integer p and every nonnegative integer n, the sum of cotangents J p defined by (1.2) satisfies the following inequalities: As an example, for n = 0 we obtain the following double inequality, which is valid for p ≥ 1 : 0 < 1 π −p 2 ln p + (ln(2π) − γ)p 2 − p − J p < π 36