Dt, Where Ζ(S) Is the Riemann Zeta-Function, ∞ |

ISSN2075-9827 e-ISSN2313-0210 http://www.journals.pu.if.ua/index.php/cmp

Carpathian Math. Publ. 2016, 8 (1), 16–20 Карпатськi матем. публ. 2016, Т.8, №1, С.16–20 doi:10.15330/cmp.8.1.16-20

BASIUK Y.V., TARASYUK S.I.

FOURIER COEFFICIENTS ASSOCIATED WITH THE RIEMANN ZETA-FUNCTION

We study the Riemann zeta-function ζ(s) by a Fourierseries method. The summation of log ζ(s) | | with the kernel 1/ s 6 on the critical line Re s = 1 is the main result of our investigation. Also we | | 2 obtain a new restatement of the Riemann Hypothesis. Key words and phrases: Fourier coefﬁcients, the Riemann zeta-function, Riemann Hypothesis.

Ivan Franko National University, 1 Universytetska str., 79000, Lviv, Ukraine E-mail: [email protected] (Basiuk Y.V.), [email protected] (Tarasyuk S.I.)

INTRODUCTION ∞ 1 It is known that the integral log ζ 2 + it dt, where ζ(s) is the Riemann zeta-function, ∞ | | diverges. M. Balazard, E.Saias, M.−R Yor [1] summed log ζ(s) on the critical line with the kernel | | 1 2 z 1 < 3 1/ s . Using the fact that f (z) = 1 z ζ 1 z , z 1, belongs to the Hardy space H and the | | − − | | result of Bercovici and Foias [2] on the factorization of f (z), they have proved the following theorem.

where ρ is the sequence of non-trivial zeroes of ζ(s). { j} In particular, the Riemann Hypothesis holds if and only if

Theorem ([6]). Let ρ be the sequence of non-trivial zeroes of ζ(s). Then { j}

( ρ 2 Reρ )( 2Reρ 1) | j| − j j − + ∑ 2 , > 1 ρj(ρj 1) Re ρj 2 | − |

УДК 517.53 2010 Mathematics Subject Classiﬁcation: Primary 11M06, 30D99; Secondary 11M41.

c Basiuk Y.V., Tarasyuk S.I., 2016

FOURIER COEFFICIENTS ASSOCIATED WITH THE RIEMANN ZETA-FUNCTION 17

where γ is the Euler constant. The Riemann Hypothesis holds if and only if

1 log ζ(s) | 4 | ds = 1 γ. 2π Re s= 1 s | | − Z 2 | | We make the next step studying the behaviour of the Riemann zeta-function on the critical line. The summation of log ζ(s) with the kernel 1/ s 6 on the critical line Re s = 1 is the main | | | | 2 result of our research.

1 SECTION WITH RESULTS

Our result is the following.

Theorem 1. Let ρ be the sequence of non-trivial zeroes of ζ(s). Then { j} 2 1 log ζ(s) 7 γ1 γ ρj | 6 | ds = 4γ + − + 6 ∑ log 2π Re s= 1 s | | 2 − 2 1 ρ Z 2 > 1 j | | Re ρj 2 −

( ρ 2 Reρ )(2Reρ 1 ) | j| − j j − + 4 ∑ 2 (1) > 1 ρj(ρj 1) Re ρj 2 | − | Re( ρ 2 ρ )2(2Reρ 1)(2 ρ 2 2Reρ + 1) 1 | j| − j j − | j| − j + ∑ 4 , 2 > 1 ρj(ρj 1) Re ρj 2 | − | where γ is the Euler constant,

1 log2 N γ1 = lim ∑ log m . N ∞ − m N m − 2 ! → ≤ Also we obtain a new restatement of the Riemann Hypothesis.

Theorem 2. The Riemann Hypothesis holds if and only if

2 1 log ζ(s) 7 γ1 γ | 6 | ds = 4γ + − . (2) 2π Re s= 1 s | | 2 − 2 Z 2 | | Proof of Theorem 1. Observe that the conformal map z = 1 1/s transforms the domain − s : Re s > 1 onto the unit disc z : z < 1 . Consider the function 2 { | | } n o z 1 f (z) = (s 1) ζ(s) = ζ . − 1 z 1 z − − We have (s 1) ζ(s) = 1 + γ(s 1)+ γ (s 1)2 + + γ (s 1)k+1 + ..., (3) − − 1 − · · · k − where ( 1)k 1 logk+1 N γ = − lim ∑ logk m , k N, k ∞ k! N m N m − k + 1 ! ∈ → ≤ ([5, p.4]). Therefore f (z) is holomorphic in the unit disk. It was showed in [3] that the function f (z) belongs to the Hardy class Hp, 0 < p < 1. Earlier it was established in [1] and [2] that the 18 BASIUK Y.V., TARASYUK S.I.

1 function f (z) belongs to the Hardy class H 3 and σ = 0, where σ is the singular measure from the factorization (see [4])

2π iϕ 2π iϕ 1 e + z 1 e + z iϕ f (z) = B(z) exp(iC) exp iϕ dσ(ϕ) exp iϕ log f (e ) dϕ , · · −2π Z0 e z 2π Z0 e z | | − − (4) where aj aj z B(z)= ∏ | | − a 1 a z j j − j is the Blaschke product, a is the sequence of zeros of f (z) and C = Im f (0) is a real constant. { j} Consider the Fourier coefﬁcient of log f (reiθ ) : | | 2π 1 ikθ iθ ck(r, f ) = e− log f (re ) dθ, r 1. 2π Z0 | | ≤ Note that c k(r, f ) = ck(r, f ). − It follows from (3) that f (0) = 1, and (4) yields

1 c0(1, f ) = log B(0) = ∑ log − | | a j | j| and 1 2π eiϕ + reiθ log f (reiθ ) = log B(reiθ) + Re log f (eiϕ) dϕ. (5) | | | | 2π 0 eiϕ reiθ | | Z − In some neighborhood of the origin, the function F(z)= log f (z), log f (0) = 0, is holomor- 2 phic. Let F(z)= A1z + A2z + . . . be its Maclaurin expansion. According to (3)

γ γ2 A = γ; A = 1 − . 1 2 2 On the other hand,

F + F γr(eiϕ + e iϕ) (γ γ2)r2(e2iϕ + e 2iϕ) log f (reiϕ) = Re log f (reiϕ)= = − + 1 − − + ..., | | 2 2 4 where r is sufﬁciently small. The relation (5) implies, for small r,

2 γ1 γ 2 2 − r = c 2(r, B)+ r c 2(1, f ). 4 − − In [7], the expression for the Fourier coefﬁcient of the Blaschke product was obtained

Note that 2π 1 1 2iθ 1 c 2(1, f ) = + e log ζ dθ. (7) − 4 2π 1 eiθ Z0 − Return to the variable s. Taking (6) and (7) into account, we obtain

1 1 log ζ(s) 1 log ζ(s) 1 2 log ζ(s) + | 2 | ds | 4 | ds + Re s | 6 | ds 4 2π Re s= 1 s | |− 2π Re s= 1 s | | 2π Re s= 1 s | | Z 2 | | Z 2 | | Z 2 | | 2 Re( ρ 2 ρ )2(2Reρ 1)(2 ρ 2 2Reρ + 1) γ1 γ 1 | j| − j j − | j | − j = − + ∑ 4 . 4 4 > 1 ρj(ρj 1) Re ρj 2 | − | Note that

∞ 1 log ζ(s) 1 log ζ 2 + it Re s 2 ds = 2 t2 dt | 6 | 3 s | | 0 4 − 1 2 Z 1 | | Z + t Re s= 2 4 ∞ 1 ∞ 1 log ζ 2 + it log ζ 2 + it = 2 dt + dt 2 3 − Z0 1 2 Z0 1 2 4 + t 4 + t log ζ(s) 1 log ζ(s) = | 4 | ds + | 6 | ds . − Re s= 1 s | | 2 Re s= 1 s | | Z 2 | | Z 2 | | Using the results from [1] and [6], we obtain (1). The proof is completed.

Proof of Theorem 2. If the Riemann Hypothesis is true, then the series at the right hand side of (1) are absent, and we have (2) 2 1 log ζ(s) 7 γ1 γ | 6 | ds = 4γ + − . 2π Re s= 1 s | | 2 − 2 Z 2 | | Now assume that relation (2) holds. If the Riemann Hypothesis is not true, then in (1)

ρ ( ρ 2 Reρ )(2Reρ 1) j | j| − j j − > 6 ∑ log + 4 ∑ 2 0. > 1 1 ρj > 1 ρj(ρj 1) Re ρj 2 − Re ρj 2 | − |

Examine carefully the series Re( ρ 2 ρ )2(2Reρ 1)(2 ρ 2 2Reρ + 1) | j| − j j − | j| − j ∑ 4 . > 1 ρj(ρj 1) Re ρj 2 | − | 20 BASIUK Y.V., TARASYUK S.I.

We are interested in when all terms of this series are positive. The following conditions appear Re( ρ 2 ρ )2 > 0. | j| − j If 0 < Re ρ < 1 and Im ρ > 1 + 1 , then Re( ρ 2 ρ )2 > 0. j | j | 2 √2 | j| − j It is known (see [8]) that the ﬁrst 10 22 + 1 non-trivial zeros of the Riemann zeta-function lie on the critical line. In particular, Im ρ1 = 14, 1347 . . . . These facts imply Re( ρ 2 ρ )2 > 0 for all non-trivial zeros ρ that lie inside the critical | j| − j j strip 0 < Re s < 1. Hence, if the Riemann Hypothesis is not true, then

REFERENCES

[1] Balazard M., Saias E., Yor M. Notes sur la fonction ζ de Riemann, 2. Adv. Math. 1999, 143 (2), 284–287. doi:10.1006/aima.1998.1797 (in French) [2] Bercovici H., Foias C. A real variable restatement of Riemann’s hypothesis. Israel J. Math. 1984, 48 (1), 57–68. doi:10.1007/BF02760524 [3] Ero˘glu K.I., Ostrovskii I.V. On an application of the Hardy classes to the Riemann zeta-function. Turkish J. Math. 2001, 25, 545–551. [4] Hoffman K. Banach spaces of analytic functions. Dover, New York, 1988. [5] Ivic A. The Riemann zeta-function. Wiley, New York, 1985. [6] Kondratyuk A., Yatsulka P. Summation of the Riemann zeta-function logarithm on the critical line. Proc. of the fourth intern. conf. on analytic number theory and spatial fessellations ”Voronoy’s impact on modern science”, Kyiv, Ukraine, September 22–28, 2008, 59–63. [7] McLane G.R., Rubel L.A. On the growth of Blaschke products. Canad. J. Math. 1969, 21, 595–601. doi: 10.4153/CJM-1969-067-3 [8] Odlyzko A. The 1022-nd zero of the Riemann zeta function. Contemp. Math. 2001, 290, 139–144.

Received 17.11.2015

Revised 17.02.2016

Басюк Ю.В., Тарасюк С.I. Коефiцiєнти Фур’є, асоцiйованi з дзета-функцiєю Рiмана // Карпатськi матем. публ. — 2016. — Т.8, №1. — C. 16–20. Ми вивчаємо дзета-функцiю Рiмана ζ(s), використовуючи метод коефiцiєнтiв Фур’є. Пiд- сумовування log ζ(s) з ядром 1/ s 6 на критичнiй прямiй Re s = 1 є головним результатом | | | | 2 нашого дослiдження. Також отримали твердження, рiвносильне гiпотезi Рiмана. Ключовi слова i фрази: коефiцiєнти Фур’є, дзета-функцiя Рiмана, гiпотеза Рiмана.