ZETA FUNCTION AND SOME OF ITS PROPERTIES Текст научной статьи по специальности «Математика»

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ZETA FUNCTION AND SOME OF ITS PROPERTIES

Nicola Fabiano

Independent researcher, Rome, Italy, e-mail: [email protected],

ORCID iD:ORCID ID: https://orcid.org/0000-0003-1645-2071

DOI: 10.5937/vojtehg68-28535; https://doi.org/10.5937/vojtehg68-28535

FIELD: Mathematics ARTICLE TYPE: Review paper Abstract:

Introduction/purpose: Some properties of the zeta function will be shown as well as its applications in calculus, in particular the "golden nugget formula" for the value of the infinite sum 1 + 2 + 3 + • • • . Some applications in physics will also be mentioned.

Methods: Complex plane integrations and properties of the Gamma function will be used from the definition of the function to its analytic extension.

Results: From the original definition of the Z(s) function valid for s > 1 a mero-morphic function is obtained on the whole complex plane with a simple pole in

s = 1.

Conclusion: The relevance of the zeta function cannot be overstated, ranging from the infinite series to the number theory, regularization in theoretical physics, the Casimir force, and many other fields.

Key words: Zeta function, analytic continuation, complex plane integration.

Definition of the Zeta Function and its generalization

Consider the complex variable s = a + it with a and t being real. For a > 1 the series

i

z <s> = £ ns (1)

n= 1

is convergent. This defines the zeta function, already known to Euler (Euler, 1738), (Euler, 1740), the properties of which were discovered by Riemann (Riemann, 1859) more than 100 years after Euler's works.

One generalization of this function is the following function

+tt 1

Z (*,«) = E Tnr^ (2)

with a > 0 due to Hurwitz (Hurwitz, 1932). For a = 1 one recovers the zeta function. Working with the Hurwitz zeta allows us to obtain more general results that can immediately be translated for the zeta function in the case

Z(s, 1).

By considering the definition of the r function

r+tt

r(s) = xs—Vxdx (3)

j 0

and by applying the substitution x ^ (n + a)x, where n is an integer, we get

f+tt

r(s) = xs-1(n + a)se—x(n+a)dx 0

(4)

that is

1

f+tt

r(s^T-r = xs-1e-x(n+a)dx . (5)

(n + a)s J o

Summing of n on both sides of (5) and using (2) leads to

r+tt +tt r+tt xS—1 e—ax

r(s)Z(s, a) = / xs—^ e—x(n+a)dx = --— dx , (6)

J0 ___n JO 1 — e

the integral converges for a > 1. This formula furnishes us with an integral expression for Z(s, a) that will be useful to analytically extend the zeta function even for a < 1. (2) could be rewritten as

1 r+tt xS—1e—ax

Z(s,a) = r(S) Jo T-e-x-dx, (7)

which bears resemblance to the r function itself (3).

The integral on the complex plane

From formula (7) we will consider the integral

Ia(s) = f

(—z)s_1e_az 1 - e-z

dz (8)

extended over the complex plane, z e C, on the contour C. When observing the integrand we see that there is a branch point at z = 0 and that there exist simple poles for z = ±2nni, where n = 1,2,3,.... Assuming there is a cut on the real positive axis, we will make use of a Hankel's type of contour for the integral (Hankel, 1869). It comes from just above the real positive axis, goes around the origin and returns back to this time below the real positive axis, so it does not contain any of the above mentioned poles, and does not pass through the branch point z = 0. We conclude that such an integral (8) provides us with an analytical function for all values of s. We can write the relation

/( ~\s-K—az c+<x j ™\s_1„-ax

(~zi)_ e_ez dz = [e™(s—10 - e—ni(s—1)] (—e_ex dx = c 0

r+TO (—x)s_1e_ax (—x)s_1e_ax

f+tt (_x)s_1e_ax f+ 2i sin[n(s — 1)1/ -dx = —2i sin(ns) -——---dx .

1 V J' Jo 1 — e_x V ' Jo 1 — e-"

(9)

Remembering the reflection property of the r function, for which

n

r(1 — s)r(s) = imcnsj • (10)

combining with the result of (9) and plugging all back in (7), we end up with an expression for Hurwitz's Z from an integral on the complex plane

«•.<•> = /^dz = — ^l.«. (11)

Now, we have already noticed that the last integral Ia(s) gives an analytical function for all values of s on the complex plane. Therefore the only possible poles of Z(s,a) could be given by the r(1 — s) function, that is at the points 1, 2, 3, . . ..

On the other hand, from definition (2), we already know that Z(s, a) converges for a > 1. We, therefore, conclude that the only possible pole for Z(s, a) is to be found at the point s = 1, and it is a simple pole, because the r function has only simple poles for negative integers, of the form

r<*> = nfe+n)+i-1)nn?n±il +O(x + n) , (12)

where ^(x) = d/dx[ln(r(x))]. For s = 1, /a(1) is written as

la (1)=/i-z dz = 2ni Res --z = 2ni, (13)

1 — e—z z=o \1 — e—z '

because for a > 0 the integrand is zero at infinity, and its residue is +1. Hence, from (11) we obtain that

lim Z(s,a\ = -1 . (14)

s-m r(1 - s) ( )

Equation (12) tells us that r(1 - s) has a single pole for s = 1 with the residue of -1. It, therefore, follows that Z(s,a) has s = 1 as the only singularity, which is a single pole with the residue of +1.

Functional equation

Formula (11) provides an expression for the Hurwitz zeta function Z(s, a) which is valid for all values of s e C\{1} by means of an expression containing a complex integral (8) Ia(s) (Hurwitz, 1932).

Consider now the real positive point (2N + 1)n, N being an integer, and define the Hankel's contour CN in analogy to the previous one encountered. The path runs from the point (2N+1)n towards 0 just above the positive real axis, goes around the origin z = 0 in a counterclockwise direction without intersecting it, and returns to the original point just below the real positive axis, and does not contain any of the simple points ±2nni of the integrand. Consider then a circle CN centered in the origin with the radius (2N + 1)n, ending at the beginning of CN. The full path CN+CN is closed, and the origin lies outside of it, therefore we obtain

1

(—z)

s— 1 g—«z

N

1- e-

dz = J](R+ + R—)

(15)

Cn

n=1

where R+ and R- are the residues of the integrand at the points +2nni and -2nni respectively, for n = 1,..., N. This is true because all the poles are inside the above defined contour, and the minus sign is necessary as this contour runs clockwise. The residues at points ±2nin are given by

R± = (2nn)s—1eT 2n(s—^e^m«

hence

(16)

R+ + R— = (2nn)s—1 2sin (|s + 2nna)

2 sin (2s) cos(2nan) (2n)1—s n

1s

+

2 cos (2s) sin(2nan) (2n)1—s n

1s

(17)

For a large N, the first part of the contour, Cn, becomes the Hankel's contour already encountered in (8), limNCn = C. The integral on the circle CN does not contribute in the limit N ^ In fact, parametrizing the variable on the circle path z = (2N + 1)nei(9, —n < 6 < and for a > 0 one has

(—z)s—1e—az 1 - e—z

dz

< f+[(2N + 1)n]se—(2N+1)nd6

J—n

2n[(2N + 1)n]se—(2N+1)n, (18)

and this expression clearly goes to zero for N ^

In the limit N ^ applied to (15), we therefore obtain

lim--

(—z)s—1e—az 1 - e—z

dz = —— 2ni

(—z)s—1e—az 1 - e—z

dz =

2 sin (2s) cos(2nan) 2 cos (§s) sin(2nan)

1s

£

n=1

n

1s

+

(2n

1s

E

n=1

n

1s

(19)

and with the aid of (11) we can eventually write the converging series

m ^ r(1 - s)

Z (s,a) = W37

2 sin (V c°s(1nan) + 2 cos (V sin(fan) V2 n1-s V2 n1-s

n=1 n=1

(20)

By writing a = 1 in (20) we obtain the functional equation linking the values of Z(s) and Z(1 - s) discovered by Riemann

Z(s) = (2n)ssin (2s) Z(1 - s) , (21)

and by making use of the r function reflection property (10), we obtain the alternative expression for the Z function reflection property

21-sr(s)Z(s) cos (2s) = nsZ(1 - s) . (22)

Some special values of Z

In the previous section, we have seen that Z(s) is a meromorphic function on the complex plane, with a simple pole for s = 1. We will now discuss some notable values of this function.

Z(2): the Basel problem

Historically, the first who posed the problem of the values of ^ +=11/n2 was Mengoli in 1650 (Mengoli, 1650). Euler solved this problem in the years after 1730, and he named it after his hometown (Euler, 1740).

Consider the series

3 5 7

x x x

sin(x) = x - "3! + ^! - 7 + ••• (23)

and divide this expression by x, obtaining the infinite polynomial

sin(x) x2 x4 x6

—— = 1 - ^ + ^ - ^ + ••• . (24)

x 3! 5! 7!

This function is well defined for all x e R and has its zeroes at the points ±nn, for n = 1,2,3,.... Hence, this polynomial can be written as an infinite product of the factors

sin(x)

1 — X) (1 + x

n/ V n

=1

n2

1 — (1 + X) (1 — X) Î1 + -

4n2

2n/

2n/

9n2 /

3n/

n

n=1

3n

1

n2n2

(25)

That has to be compared with Z(2), namely

z (2) = Z n1 = H + 9+ ■)■

n=1 x

(26)

Consider the coefficient of the polynomial (25) in x2. It is obtained by picking exactly once each x2 term in the factors, multiplyng it with all the constant terms and then adding together all those terms. An analogy that helps would be with the characteristic polynomial of a matrix, where the coefficient of the lowest term of the polynomial (in our case, x2) is given by the trace of the matrix that is the sum of all eigenvalues. This gives for the x2 term in (25)

x 1 n2 + 4n2 + 9n2 +

x2 / 1 1

=--0 1 + T + 77 +

n2

49

= — Z (2)

n2

Comparing this result with the x2 term of (24), one obtains

(27)

x2 x2

— X- = — ^ Z (2) 3! n2

(28)

giving eventually the required value for Z

n

Z (2) = — w 1.6449 6

(29)

Notice that, even though this brilliant procedure leads to the correct value, it is wrong. In fact, one could multiply (25) by an arbitrary positive function, say exp(x), and retain the same result, because the roots of the polynomial will not change. Yet the series expansion of this new function would be quite different from (24). The culprit is that it is not valid to treat an infinite product or sum expecting it to behave like a finite one.

x

2

x

2

1

1

1

Z (-1)

The infinite sum

1

1 + 2 + 3 + 4 + 5 + ••• = ^ n_ (30)

n-1

n=1

is equivalent to Z(-1). By writing s = -1 in (21), we get

Z<-1> = 2n^ (-2) Z(2) (31)

remembering the properties of the r function and eq. (29) for the value of Z(2) we have

Z (-1) = -12 = 1 + 2 + 3 + 4 + 5 + ••• , (32)

so we end up with an astonishing result: the sum of all positive integers is not only finite, but also negative! Of course this does not mean that the usual addition rules turned out to be magically spoiled. In fact, consider the finite sum

1 + 2 + 3 + ... + N = £> = N(W2+ , (33)

n=1

and this expression becomes infinite when N ^ +rc>.

What it really means is that the definition of Z given in (1) is no longer valid whenever a < 1, like in our case of s = -1. The Z(s) that obeys to (21) is the analytical continuation on the whole complex plane of the one defined in (1), and they do coincide only when a > 1.

One could also think of equation (30) as an equivalence to a so-called regularized Z function defined in (1). This technique is used very often in theoretical physics and lays on a rigorous basis, too vast to be described here. Loosely speaking, it is like considering the behavior of a function near a pole, then discarding the divergent part while retaining the finite part. The latter assigns the value of the function on the pole.

For instance, by considering the behavior of the r function close to negative integers described in (12), near the origin we have

1 2 + n2

lim r(x) = - - y + — x + O(x2) (34)

xio x 12

so that a regularized r function should assume the value

Reg[r(0)] = —y « -0.5772 . (35)

It is worth noticing that the value of Z(—1) appears in physics when computing the Casimir force in one dimension, that arises as a fluctuation of the vacuum energy when quantizing the electromagnetic field, or when computing the ground state energy of the bosonic string theory, a model which was an attempt to unify gravitational with other fundamental forces.

Z (0)

The infinite sum

-

1 + 1 + 1 + 1 + 1 + ■■■ = £ n-o (36)

n=1

is equivalent to Z(0). By writing s = 0 in (21), we get

Z(0) = lim 1r(1)sin (2s) z(1 — s) , (37)

n V 2 /

and, remembering that Z has a simple pole with residue +1 in s = 1, we have

lim sin(|s)c(1 — .) = — 2 (38)

Thus, by putting all the values in (37) we obtain

Z (0) = — 1 = 1 + 1 + 1 + 1 + 1 + ••• , (39)

obtaining once again a negative value for an infinite sum of positive terms. The same considerations given for the case of Z(—1) apply here. Other useful information on complex functions could be found for instance in (Denjoy, 1926), (Wolff, 1926), (Dosenovic, 2018), (Todorcevic, 2019).

References

Denjoy, A. 1926. Sur l'itération des fonctions analytiques. C.R. Acad. Sci. Paris, 182, pp.255-257 (in French).

DosenoviC, T., Kopellaar, H. and Radenovic, S. 2018. On some known fixed point results in the complex domain: Survey. Vojnotehnicki glasnik/Military Technical Courier, 66(3), pp.563-579. Available at: https://doi.org/10.5937/vojtehg66-17 103.

Euler, L. 1738. De summatione innumerabilium progressionum. Euler Archive -All Works, 20 (in Latin) [online]. Availabe at: https://scholarlycommons.pacific.e du/euler-works/20 [Accessed: 15 September 2020].

Euler, L. 1740. De summis serierum reciprocarum. Euler Archive - All Works, 41 (in Latin) [online]. Availabe at: https://scholarlycommons.pacific.edu/euler-wor ks/41 [Accessed: 15 September 2020].

Hankel, H. 1869. Die Cylinderfunctionen erster und zweiter Art. Math. Ann., 1, pp.467-501 (in German).

Hurwitz, A. 1932. Einige Eigenschaften der Dirichlet'schen Funktionen F (s) = XXD/n) • 1/(ns), die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. In: Abteilung für Mathematik und Physik der Eidgenössischen Technischen Hochschule (eds) Mathematische Werke. Basel: Springer (in German). Available at: https://doi.org/10.1007/978-3-0348-4161-0_3.

Mengoli, P. 1650. Novœ Quadraturœ Arithmeticœ. Bononiœ: exTypographia lacobi Montij (in Latin).

Riemann, B. 1859. Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie (in German) [online]. Available at: http://www.emis.mi.sanu.ac.rs/EMIS/classics/Riemann/Zeta.pdf [Accessed: 15 September 2020].

TodorceviC, V. 2019. Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics. Springer Nature Switzerland AG. ISBN: 978-3-030-22591-9.

Wolff, J. 1926. Sur l'itération des fonctions bornées. C. R. Acad. Sci., 182, pp.200-201.

ДЗЕТА-ФУНКЦИЯ И ЕЕ ОСОБЕННОСТИ

Никола Фабиано

независимый исследователь, Рим, Италия

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:

27.25.17 Метрическая теория функций, 27.33.00 Интегральные уравнения, 27.39.29 Приближенные методы функционального анализа ВИД СТАТЬИ: обзорная статья

Резюме:

Введение/цель: В данной статье представлены некоторые особенности дзета-функции, а также ее применение в математическом анализе, особое внимание уделено формуле «golden nugget» при вычислении бесконечной суммы 1+2 + 3 + •••. В статье также приводятся примеры ее применения в физике.

Методы: Интеграция в комплексной плоскости и свойства гамма-функции используются на всех этапах: от определения функции до ее аналитического расширения. Результаты: Из первоначального определения функции Z (s), относящейся к s> 1, выводится мероморфная функция на всей комплексной плоскости с простым полюсом в s = 1. Выводы: Дзета-функция безусловно играет важнейшую роль во многих областях, начиная от бесконечных рядов и заканчивая теорией чисел, регуляризации в теоретической физике, силе Казимира и пр.

Ключевые слова: дзета-функция, аналитическое продолжение, интеграция в комплексной плоскости.

ЗЕТА-ФУНКЦША И НЕКЕ ^ЕНЕ ОСОБИНЕ

Никола Фабиано

независни истраживач, Рим, Итали]а

ОБЛАСТ: математика ВРСТА ЧЛАНКА: прегледни рад

Сажетак:

Увод/цил>: У раду су приказане неке особине зета-функци}е, као и ъена примена у математичко} анализи, нарочито формула „golden nugget" за вредност бесконачног збира 1+2+3+.... Тако^е, поменуте су и неке ъене примене у фи-зици.

Методе: Интеграци]е комплексне равни и особине гама-функци]е биЬе искоришЯене од дефиници]е функци]е до к>е-не аналитичке екстензц'е.

Резултати: Од оригиналне дефиници]е функци]е Z(s) валид-не за s>1, доби}а се мероморфна функци]а на целоj комплексно] равни са простим полом на s = 1.

Закъучак: Изузетан значаj зета-функци}е jе несумъив, од бесконачних низова до теорбе бро}ева, регуларизацще у теори]ско] физици, Казимирове силе и многих других области.

Къучне речи: зета-функци}а, аналитичка континуаци}а, ин-теграци}а комплексне равни.

Paper received on / Дата получения работы / Датум приема чланка: 24.09.2020. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 09.10.2020.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 11.10.2020.

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© 2019 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons" (http://creativecommons.org/licenses/by/3.0/rs/).

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