GENERALIZATIONS OF A COTANGENT SUM ASSOCIATED TO THE ESTERMANN ZETA FUNCTION

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1 GNRALIZATIONS OF A COTANGNT SUM ASSOCIATD TO TH STRMANN ZTA FUNCTION HLMUT MAIR AND MICHAL TH. RASSIAS Astract. Cotangent sums are associated to the zeros of the stermann zeta function. They have also proven to e of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure µ on R, with respect to which certain normalized cotangent sums are equidistriuted. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are otained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k. Key words: Cotangent sums, equidistriution, moments, asymptotics, stermann zeta function, Riemann Hypothesis, fractional part. 2 Mathematics Suject Classification: 11L3, 11M6. 1. Introduction Cotangent sums are associated to the zeros of the stermann zeta function. R. Balasuramanian, J. B. Conrey and D. R. Heath-Brown [3], used properties of the stermann zeta function to prove asymptotic formulas for mean-values of the product consisting of the Riemann zeta function and a Dirichlet polynomial. Period functions and families of cotangent sums appear in recent work of S. Bettin and J. B. Conrey (cf. [6]. They generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. They prove a reciprocity formula for the V. I. Vasyunin s sum [31], which appears in the Nyman-Beurling criterion for the Riemann Hypothesis. In the present paper, improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are otained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results and upper ounds for the moments of the cotangent sums under consideration. Furthermore, we otain detailed information aout the distriution of the values of these cotangent sums. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k The cotangent sum and its applications. The present paper is focused in the study of the following cotangent sum: Date: Septemer 19,

2 Definition 1.1. c := 1 m=1 where r, N, 2, 1 r and (r, = 1. m ( πmr cot, The function c (r/ is odd and periodic of period 1 and its value is an algeraic numer. Its properties of eing odd and periodic are depicted in the following graphs: Figure 1: Graph of c (r/, for 1 r, = 757, with (r, = 1. 2

3 Figure 2: Graph of c (r/, for 1 r, = 946, with (r, = Figure 3: Graph of c (r/, for 1 r, = 1471, with (r, = 1. It is interesting to mention that for hundreds of integer values of k for which we have examined the graph of c (r/ y the use of MATLAB, the resulting figure always has a shape similar to an ellipse. 3

4 Figure 4: Graph of c (r/, for 1 r, = 1619, with (r, = 1. Part of our goal is to understand this phenomenon, and we will do it to some extent. The main result in this respect is contained in Theorem 1.5, which provides information aout equidistriution and moments of these sums. Before presenting the main results of the paper regarding this cotangent sum, we shall demonstrate its significance y exhiiting its relation to other important functions in numer theory, such as the stermann and the Riemann zeta functions, and its connections to major open prolems in Mathematics, such as the Riemann Hypothesis. Definition 1.2. The stermann zeta function ( s, r, α is defined y the Dirichlet series (s, r, α = σ α (n exp (2πinr/ n s, n 1 where Re s > Re α + 1, 1, (r, = 1 and σ α (n = d n d α. It is worth mentioning that T. stermann (see [16] introduced and studied the aove function in the special case when α =. Much later, it was studied y I. Kiuchi (see [21] for α ( 1, ]. The stermann zeta function can e continued analytically to a meromorphic function, on the whole complex plane up to two simple poles s = 1 and s = 1 + α if α or a doule pole at s = 1 if α = (see [16], [18], [29]. 4

5 Moreover, it satisfies the functional equation: (s, r, α = 1 ( 1+α 2s Γ(1 sγ(1 + α s π 2π ( cos ( πα 2 (1 + α s, r (, α cos πs πα 2 (1 + α s, r, α, where r is such that rr 1 (mod and Γ(s stands for the Gamma function. For more details regarding the functional equation of the stermann zeta function, the reader is referred to the Appendix. R. Balasuramanian, J. B. Conrey and D. R. Heath-Brown [3], used properties of (, r, to prove an asymptotic formula for T ( I = 1 2 ( 1 2 ζ 2 + it A 2 + it dt, where A(s is a Dirichlet polynomial. Asymptotics for functions of the form of I are useful for theorems which provide a lower ound for the portion of zeros of the Riemann zeta-function ζ(s on the critical line (see [19]. M. Ishiashi (see [17] presented a nice result concerning the value of ( s, r, α at s =. Theorem 1.3. (Ishiashi Let 2, 1 r, (r, = 1, α N {}. Then (1 For even α, it holds (, r (, α = i α+1 1 m ( πmr 2 cot(α δ α,, m=1 where δ α, is the Kronecker delta function. (2 For odd α, it holds (, r, α = B α+1 2(α + 1. In the special case when r = = 1, we have (, 1, α = ( 1α+1 B α+1 2(α + 1 where y B m we denote the m-th Bernoulli numer, where B 2m+1 =, B 2m = 2 (2m! (2π 2m ν 2m. ν 1 Hence for 2, 1 r, (r, = 1, it follows that (, r, = i 2 c, where c (r/ is the cotangent sum (see Definition 1.1. This result gives a connection etween the cotangent sum c (r/ and the stermann zeta function. Period functions and families of cotangent sums appear in recent work of S. Bettin 5,

6 and J. B. Conrey [6], generalizing the Dedekind sums and sharing with it the property of satisfying a reciprocity formula. Bettin and Conrey proved the following reciprocity formula for c (r/: where c ψ (z = 2 + r c log 2πz γ πiz ( 1 r πr = i 2 ψ, 2 π ( 1 2 ζ(sζ(1 s sin πs z s ds, and γ stands for the uler-mascheroni constant. This reciprocity formula demonstrates that c (r/ can e interpreted as an imperfect quantum modular form of weight 1, in the sense of D. Zagier (see [5], [32]. The cotangent sum c (r/ can e associated to the study of the Riemann Hypothesis, also through its relation with the so-called Vasyunin sum. The Vasyunin sum is defined as follows: V := 1 m=1 { mr } cot ( πmr, where {u} = u u, u R. It can e shown (see [5], [6] that ( r V = c, where, as mentioned previously, r is such that rr 1 (mod. The Vasyunin sum is itself associated to the study of the Riemann hypothesis through the following identity (see [5], [6]: (1 1 2π(r 1/2 + ( 1 2 ( it dt ζ 2 + it log 2π γ 1 1 = 4 + t2 2 r r 2r log r π ( V 2r + V (. r Note that the only non-explicit function in the right hand side of (1 is the Vasyunin sum. The aove formula is related to the Nyman-Beurling-Baéz-Duarte-Vasyunin approach to the Riemann Hypothesis (see [2], [5]. According to this approach, the Riemann Hypothesis is true if and only if where d 2 1 N = inf D N 2π + lim d N =, N + ( ( ζ 2 + it D N 2 + it and the infimum is taken over all Dirichlet polynomials D N (s = N n=1 6 a n n s. dt t2

7 Hence, from the aove arguments it follows that from the ehavior of c (r/, we understand the ehavior of V (r/ and thus from (1 we may hope to otain crucial information related to the Nyman-Beurling-Baéz-Duarte-Vasyunin approach to the Riemann Hypothesis. Therefore, to sum up, one can see from all the aove that the cotangent sum c (r/ is strongly related to important functions of Numer Theory and its properties can e applied in the study of significant open prolems, such as Riemann s Hypothesis Main result. We now come to the main result of the paper, which states the equidistriution of certain normalized cotangent sums with respect to a positive measure, which is also constructed in the following theorem. Definition 1.4. For z R, let F (z = meas{α [, 1] : g(α z}, where meas denotes the Leesgue measure, and g(α = + l=1 1 2{lα} l C (R = {f C(R : ɛ >, a compact set K, such that f(x < ɛ, x K}. Remark. The convergence of this series has een investigated y R. de la Bretèche and G. Tenenaum (see [8]. It depends on the partial fraction expansion of the numer α. Theorem 1.5. i F is a continuous function of z. ii Let A, A 1 e fixed constants, such that 1/2 < A < A 1 < 1. Let also 1 ( 2k g(x H k = dx, π H k is a positive constant depending only on k, k N. There is a unique positive measure µ on R with the following properties: (a For α < β R we have ( µ([α, β] = (A 1 A (F (β F (α. x k dµ = (c For all f C (R, we have lim + 1 φ( { (A1 A H k/2, for even k, otherwise. r : (r,=1 A r A 1 f ( 1 c = f dµ. Remark. R. W. Bruggeman (see [9], [1] and I. Vardi (see [3] have investigated the equidistriution of Dedekind sums. In contrast with the work in this paper, they consider an additional averaging over the denominator. 7

8 1.3. Outline of the proof and further results. In [25], M. Th. Rassias proved the following asymptotic formula: Theorem 1.6. For 2, N, we have ( 1 c = 1 π log (log 2π γ + O(1. π In that paper, a method which applies properties of fractional parts in order to approach the cotangent sum in question is descried. This method is generalized in the present paper, where some stronger results are eing proved. We initially provide a proof of an improvement of Theorem 1.6 as an asymptotic expansion. Namely, we prove the following: Theorem 1.7. Let, n N, 6N, with N = n/2 + 1.There exist asolute real constants A 1, A 2 1 and asolute real constants l, l N with l (A 1 l 2l, such that for each n N we have ( 1 c = 1 π log π (log 2π γ 1 n π + l l + Rn( where R n( (A 2 n 4n (n+1. Additionally, we investigate the cotangent sum c for a fixed aritrary positive integer value of r and for large integer values of and prove the following results. Proposition 1.8. For r, N with (r, = 1, it holds c = 1 ( 1 r c 1 r Q, where m=1 l=1 1 ( πmr rm Q = cot. Theorem 1.9. Let r, N e fixed, with (, r = 1. Let denote a positive integer with (mod r. Then, there exists a constant C 1 = C 1 (r,, with C 1 (1, =, such that c = 1 πr log πr (log 2π γ + C 1 + O(1, for large integer values of. Theorem 1.1. Let k N e fixed. Let also A, A 1 e fixed constants such that 1/2 < A < A 1 < 1. Then there exist explicit constants k > and H k >, depending only on k, such that (a r:(r,=1 A r A 1 2k Q = k (A 2k+1 1 A 2k+1 4k φ((1 + o(1, ( +. 8

9 ( (c (d r:(r,=1 A r A 1 r:(r,=1 A r A 1 2k 1 ( Q = o 4k 2 φ(, ( +. c 2k = Hk (A 1 A 2k φ((1 + o(1, ( +. r:(r,=1 A r A 1 c 2k 1 = o ( 2k 1 φ(, ( +. Using the method of moments, we deduce detailed information aout the distriution of the values of c (r/, where A r A 1 and +. Namely, we prove Theorem 1.5. Finally, we study the convergence of the series H k x 2k and prove the following theorem: Theorem The series converges only for x =. k H k x 2k, k Another interesting question which we have investigated ut have not reached a conclusion yet is whether the series H k (2k! x2k, k has a positive radius of convergence. This would lead to a simplification in the proof of our equidistriution result, since in this case we could apply results aout distriutions which are determined y their moments. References [1] L. Báez Duarte, M. Balazard and B. Landreau, Étude de l autocorrélation multiplicative de la fonction partie fractionnaire, The Ramanujan Journal, 9(25, [2] B. Bagchi, On Nyman, Beurling and Baez-Duarte s Hilert space reformulation of the Riemann hypothesis, Proc. Indian Acad. Sci. Math. 116(2(26, [3] R. Balasuramanian, J. B. Conrey and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. Reine Angew. Math. 357(1985, [4] S. Bettin, The second moment of the Riemann zeta function with unounded shifts, Int. J. Numer Theory 6(8(21, [5] S. Bettin, A generalization of Rademacher s reciprocity law, Acta Arithmetica, 159(4(213,

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