An Analogue of Wilton's Formula and Values of Dedekind Zeta Functions

Home , Dedekind zeta function

arXiv:1611.08693v5 [math.NT] 5 Oct 2020 h eaiu fzt ucini h rtclsrp especi strip; critical the in us function particularly zeta is of which behaviour function the zeta Riemann for equation eeidzt ucin is us aaiadissection. Nakajima sums; Riesz function; zeta Dedekind NAAOU FWLO’ OML N VALUES AND FORMULA WILTON’S OF ANALOGUE AN OMAU AEJE AYNCARBRYADAIU HOQUE AZIZUL AND CHAKRABORTY KALYAN BANERJEE, SOUMYARUP .H ad n .E iteod[]otie napoiaef approximate an obtained [6] Littlewood E. J. and Hardy H. G. ..Ra udai ubrfil ae18 16 19 References remarks Acknowledgments case Concluding field number quadratic 5. case Imaginary field number quadratic 4.2. functions Real zeta Dedekind of 4.1. 1.1 Values Theorem the of 4. Proof setup 3. wor Background paper Earlier the functions: of 2. zeta Structure Dedekind of values 1.4. Special results of 1.3. Statement 1.2. Preliminaries 1.1. Introduction 1. 2010 e od n phrases. and words Key ahmtc ujc Classification. Subject Mathematics ela ela udai ubrfilsa n oiieintege positive any at fields number to attached quadratic values exp as zeta these well Dedekind use as for then real expressions expressi and some analogous functions find find zeta to We Dedekind Riem two region. two of critical of product the product in the functions for ro equation crucial a functional played expression approximate This functions. zeta Riemann Abstract. oPo.BueC ento his on Berndt C. Bruce Prof. To FDDKN EAFUNCTIONS ZETA DEDEKIND OF .R itnotie nepeso o h rdc ftwo of product the for expression an obtained Wilton R. J. itnsfrua eeidzt ucin pca auso values Special function; zeta Dedekind formula; Wilton’s 1. Introduction Contents respect 1 rmr:1M6 eodr:11M41. Secondary: 11M06, Primary: 80 th itdywt great with birthday lyo h critical the on ally et n the find to le n o the for ons r. fli studying in eful arbitrary n zeta ann ressions s4 ks unctional 21 21 21 7 5 5 3 2 1 f 2 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE line. Later, in the same way of thought, J. R. Wilton [19] in 1930 gave an approximate functional equation for the product of two Riemann zeta functions in the critical region. The main ingredient used to arrive on to the above mentioned approximate functional equation is the following formula:

Theorem A (Wilton’s formula). Let R(u) > 1, R(v) > 1, R(u+v) > 0 and u + v = 2, then − − 6

1 1 u 1 ∞ u 1 ζ(u)ζ(v)= ζ(u + v 1) + + 2(2π) − σ1 u v(n)n − u − u 1 v 1 − − − − Xn=1

∞ u 1 v 1 ∞ v 1 ∞ v 1 x− − sin x dx + 2(2π) − σ1 u v(n)n − v x− − sinx dx, × Z − − Z 2nπ Xn=1 2nπ z where σz(n), z being an arbitrary (complex) number, denotes the sum of d over all the divisors d of n.

Wilton’s approach to prove Theorem A involves techniques of real and complex analysis. Recently, D. Banerjee and J. Mehta [1] gave an alternative proof of Theorem A using number theoretic approach, and a similar result for Hurwitz zeta functions appeared in [18]. In this paper, we obtain a result analogous to that of Theorem A in number fields. More precisely, we find expressions for the product of two Dedekind zeta functions attached to arbitrary real (resp. imaginary) quadratic number fields. These expressions involve Meijor G-function as its error term. We then use these expressions to find some formulae for Dedekind zeta values attached to quadratic number fields at any positive integer. The ultimate goal is to find certain formulae for calculating special values of Dedekind zeta functions, which we are unable to find at this point. To the best of our knowledge these expressions are new and are of some interests.

1.1. Preliminaries. We shall begin by recalling some definitions and fix up notations to be used in this paper. Let K be a number field of degree d with signature (r1, r2) (i. e., d = r1 + 2r2) and K it’s ring of integers. Let N(a) be the norm of the ideal a . Then theO Dedekind zeta function attached to K is defined by ⊆ OK 1 1 1 − ζ (s)= = 1 , R(s) > 1, K N(a)s − N(p)s aX p Y ⊂OK ⊂OK where a and p run over the non-zero ideals and prime ideals of K respectively. If v (m) denotes the number of non-zero integral ideals inO with K OK norm m, then ζK can also be expressed as

∞ v (m) ζ (s)= K . K ms mX=1 VALUESOFDEDEKINDZETAFUNCTIONS 3

Let m n σ′ (n) := t v (t)v ( ), m,vK K K t Xt n | for any integer m, and any positive integer n. The Meijer G-function is a sum of hypergeometric functions each of which is usually an entire function. It is deﬁned by the following line integral in the complex plane (see [14])

m n Γ(bj s) Γ(1 aj + s) 1 j=1 − j=1 − m, n a1, , ap Q Q s Gp, q · · · z = q p z ds. b1, , bq 2πi · · · Γ(1 b + s) Γ(a s) Z − j j − (C) j=Qm+1 j=Qn+1 The poles of the integrand must be simple and those of Γ(b s), j = j − 1, ,m, must lie on one side of the contour C and those of Γ(1 aj +s), j = 1, · · · ,n, must lie on the other side of C. The integral then converges− for arg· · ·z < δπ, where | | 1 δ = m + n (p + q). − 2 The integral additionally converges for arg z = δπ if (q p)(Re (s)+1/2) > Re (v) + 1, where | | − q p v = b a . j − j Xj=1 Xj=1

1.2. Statement of results. Let DK, hK ,RK and wK be denote the discriminant, the class number, the regulator and the number of roots of unity in the number ﬁeld K, respectively. We further deﬁne the following auxiliary functions:

1−2z ∞ 1,z,z D Z(z) := 4(2π)2(z 1)D 2 σ (m)mz 1 cos πz G0,3 K + − K 1′ u v,vK − 3,1 − 2 − − 0, 4π m mX=1 −

1,z,z D G0,3 K , 3,1 0, 4π2m − 4π2m 2z 1 1−2z ∞ z 1 |DK | z 1/2 H(z) := (2π) − DK 2 σ1′ u v,v (m)m − t− J0(2t )dt, | | − − K Z mX=1 0 hK 1 1 T (u, v) := 1 + ζK (u + v 1), w D 2 u 1 v 1 − K| K | − − where J0 denotes the Bessel function of ﬁrst kind. The main result on the product of two Dedekind zeta functions is the following: 4 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

Theorem 1.1. Let R(u) > 1, R(v) > 1, R(u + v) > 0, u = 1, v = 1 and u + v = 2. Then for any− quadratic number− field K, 6 6 6 4RK T (u, v)+ Z(u)+ Z(v), if K is real, (1.1) ζK(u)ζK (v)= 2πT (u, v) H(u) H(v), if K is imaginary. (1.2) − − Remarks. Here, we make a few remarks before proceeding further. (I) For K = Q, Theorem 1.1 gives a formula for the product of two Riemann zeta functions, which is slightly different from Theorem A. (II) A similar line of argument as adopted in the proof should entail a version of Theorem 1.1 for higher degree number fields using other Meijer G-functions as well as for the product of finite number (> 2) of Dedekind zeta functions. Obviously, the complications will increase manifold. (III) One can extend the range of Theorem 1.1 using analytic continuation on both sides.

1.3. Special values of Dedekind zeta functions: Earlier works. The values of Dedekind zeta functions at rational integers are closely related to the algebraic characters of the attached number fields. These values are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the Dedekind zeta functions are connected. To represent these values as clearly as possible is one of the most important tasks in the present years. Over the decades, many mathematicians have been working on this project. In 1734, L. Euler established a celebrated result, which states that ζ(2m) is a rational multiple of π2m for all natural numbers m. As a generalization of this result for any totally real number field K with discriminant DK , C. L. Siegel established the following in [16]: π2dm ζK(2m) = rational , × √DK where d is the degree of K over Q. C. L. Siegel developed (using finite dimensionality of elliptic modular forms) an innovative method in [17] for computing the special values of Dedekind zeta function attached to a totally real number field at negative odd integers, and a particular interesting case is at 1. However, evaluation of special values of a Dedekind zeta function by means− of this method requires complicated computations. The residue of a Dedekind zeta function attached to an arbitrary number field K at 1 gives the analytic class number formula that involves hK , RK, wK and the absolute value of DK . The problem of expressing the special values of Dedekind zeta functions in terms of elementary functions was first studied by E. Hecke in [8]. He obtained an elementary expression involving Dedekind sum for the relative class number of totally complex quadratic extensions of a real quadratic number field. C. Meyer [11, 12] and his student H. Lange [9] evaluated L-functions for real quadratic number fields in terms of Dedekind VALUESOFDEDEKINDZETAFUNCTIONS 5 sums. The precise history of the special values of zeta and L-functions attached to a number field is quite lengthy and partly complicated due to the great popularity this subject is enjoying. D. Zagier is one of them who have made most important contributions in this subject. In [22], he obtained an expression for the values of Dedekind zeta functions attached to real quadratic number fields at negative integers using Kronecker limit formula. Siegel’s method in [17] has been exploited by D. Zagier to give an elementary expression for ζK(1 2s) in [21], where K is a real quadratic number field and s is a positive integer,− which involves only rational integers, but neither algebraic numbers nor norms of ideals. In 1976, T. Shintani [15], using an impressive linear programming method, expressed Dedekind zeta function attached to a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He also obtained a formula to express ζ ( n) K − for n = 0, 1, 2, . However, little is known about the number ζK(2m) for the number· field · · K which is not totally real. In 1986, D. Zagier [20] obtained a formula for ζK(2) for an arbitrary number field K. His method is completely geometric, where it involves the interpretation of ζK(2) as the volume of a hyperbolic manifold. Since it is the only ζK(2) which can be interpreted geometrically, this method fails to obtain the formula for ζK(2m) as well as for ζK(2m + 1) for all natural numbers m. In the same paper, D. Zagier found a nice expression for ζK(2) attached to an imaginary quadratic number field K. In this article, we find some expressions for the values of Dedekind zeta functions attached to arbitrary real (resp. imaginary) quadratic number fields at any positive integers. Zagier’s result on ζK(2) plays a crucial role in finding these expressions.

1.4. Structure of the paper. This paper comprises of 5 sections. In §2, we discuss the main ingredients that are needed to prove our results. In §3, we provide the proof of Theorem 1.1. In §4, we discuss a work of D. Zagier on the values of Dedekind zeta functions. We also discuss the expressions for the values of Dedekind zeta function for both real and imaginary quadratic number ﬁelds at any positive integers. In §5, we discuss some important remarks.

2. Background setup Riesz means were introduced by M. Riesz [7] and have been studied in connection with summability of Fourier series and that of Dirichlet series (for details, we refer [3, 5]). For a given increasing sequence λn of real numbers and a given sequence α of complex numbers, the Riesz{ } sum of { n} 6 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE order κ is deﬁned by

κ(x)= κ(x) = ′(x λ )κα A Aλ − n n λXn x ≤ x κ 1 = κ (x t) − λ(t)dt Z0 − A x κ = (x t) d λ(t) Z0 − A with (x) = 0 (x) = ′ α , where the prime on the summation sign Aλ Aλ n λXn x ≤ means that when λn = x, the corresponding term is to be halved. Sometimes normalized 1 κ(x) that appears in the Perron’s formula Γ(κ+1) A κ+w 1 ′ κ 1 Γ(w)ϕ(w)x αn(x λn) = dw Γ(κ + 1) − 2πi Z(C) Γ(w + κ + 1) λXn x ≤

∞ αn is also called the Riesz sum of order κ with ϕ(w)= λw . n=1 n We consider the Dirichlet series P

∞ α ∞ a ϕ(s)= n and Φ(s)= m , λs γs nX=1 n mX=1 m where λ and γ are increasing sequences of real numbers. Here α and { n} { m} n am are complex numbers, and we assume that these series are continued to meromorphic functions over the whole complex plane, and that they satisfy suitable growth conditions. Further we consider the following integral,

κ 1 Γ(w) w+κ (C)(ϕ(u), Φ(v); x) = ϕ(u + w)Φ(v w)x dw F 2πi Z(C) Γ(w + κ + 1) − 1 ∞ a Γ(w) ∞ α = m n γw xw+κdw 2πi γv Z Γ(w + κ + 1) λu+w m mX=1 m (C) nX=1 n

1 ∞ am Γ(w) ∞ αn w+κ = (γmx) dw 2πi γv+κ Z Γ(w + κ + 1) λu+w mX=1 m (C) nX=1 n ∞ 1 am ′ u κ = v+κ αnλn− (γmx λn) . Γ(κ + 1) γm − mX=1 λnXγmx ≤ Similarly we have,

∞ κ 1 αn ′ v κ (C)(Φ(v), ϕ(u); x) = u+κ amγm− (λnx γm) . F Γ(κ + 1) λn − Xn=1 γmXλnx ≤ VALUESOFDEDEKINDZETAFUNCTIONS 7

Let ϕ(u) = ζK(u) and Φ(v) = ζK(v) with order κ = 1. Then by using the above expressions we obtain,

1 (ζ (u),ζ (v); x)+ 1 (ζ (v),ζ (u); x) F(C) K K F(C) K K ∞ v (m) v (n) ∞ v (n) v (m) = K ′ K (mx n)+ K ′ K (nx m). mv+1 nu − nu+1 mv − mX=1 nXmx nX=1 mXnx ≤ ≤ We now diﬀerentiate with respect to x and then substitute x = 1 to get

1 1 ( (ζ (u),ζ (v); 1))′ + ( (ζ (v),ζ (u); 1))′ F(C) K K F(C) K K ∞ v (m) v (n) ∞ v (n) v (m) = K ′ K + K ′ K . mv nu nu mv mX=1 nXm nX=1 mXn ≤ ≤ M. Nakajima [13] introduced a dissection (termed as ‘Nakajima dissection’) involving the splitting of a double sum

∞ ∞ ∞ = ′ + ′ . m,nX=1 mX=1 nXm nX=1 mXn ≤ ≤ This dissection technique immediately gives,

1 1 ( (ϕ(u), Φ(v); 1))′ + ( (Φ(v), ϕ(u); 1))′ F(C) F(C) ∞ v (m) ∞ v (n) = K K mv nu mX=1 nX=1 = ζK(u)ζK (v). (2.1)

3. Proof of the Theorem 1.1 The Riesz sum set up, Perron’s formula and Nakajima dissection are the main ingredients in this proof. Assume that

κ 1 Γ(w) w+κ ( B)(ζK (u),ζK (v); x)= ζK(u+w)ζK (v w)x dw F − 2πi Z( B) Γ(w + κ + 1) − − (3.1) 3 3 with max 0, R(u) 1

1 u w r1 d 1−2u−2w Γ( ) r2(1 2u 2w) (1 2u 2w) − 2− ζK (u + w) = 2− − − π− 2 − − DK 2 u+w r1 | | Γ( 2 ) Γ(1 u w)r2 − − ζK(1 u w). (3.2) × Γ(u + w)r2 − − 8 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

Now we have,

1 u w r1 1 u w u+w r1 Γ( − 2− ) Γ( − 2− )Γ(1 2 ) = − r Γ( u+w )r1 Γ( u+w )Γ(1 u+w ) 1 2 2 − 2 1 u w 1 1 u w r1 Γ( − 2− )Γ( 2 + − 2− ) = r1 π sin π (u+w) 2 2u+w r1 π r1 = sin (u + w) Γ(1 u w)r1 . (3.3) √π 2 − − Similarly, Γ(1 u w)r2 Γ(1 u w)2r2 − − = − − Γ(u + w)r2 Γ(u + w)Γ(1 u w) r2 { − − } sin π(u + w) r2 = Γ(1 u w)2r2 . (3.4) π − − Using (3.3) and (3.4) in (3.2), we obtain

1−2u−2w d(u+w) d(1 u w) π r1+r2 ζ (u + w) = 2 π− − − D 2 sin (u + w) K | K | { 2 } π cos (u + w) r2 Γ(1 u w)dζ (1 u w). × { 2 } − − K − − Thus (3.1) becomes

κ 1 d(u+w) d(1 u w) ( B)(ζK (u),ζK (v); x) = Sκ(w)f(w)2 π− − − F − 2πi Z( B) − 1−2u−2w π r +r π r D 2 sin (u + w) 1 2 cos (u + w) 2 × | K | { 2 } { 2 } Γ(1 u w)dxw+κdw. (3.5) × − − Here S (w)= Γ(w) and f(w)= ζ (1 u w)ζ (v w). κ Γ(w+κ+1) K − − K − We now consider K to be a quadratic number ﬁeld. Then (3.5) becomes

κ 1 2(u+w) 2(1 u w) ( B)(ζK (u),ζK (v); x) = Sκ(w)f(w)2 π− − − F − 2πi Z( B) − 1−2u−2w π r +r π r D 2 sin (u + w) 1 2 cos (u + w) 2 × | K | { 2 } { 2 } Γ(1 u w)2xw+κdw. × − − Employing change of variables we have

κ 2 2(1 u+z) ( B)(ζK (u),ζK (v); x) = Sκ( z)f( z)(2π)− − F − πi Z(B) − − 1−2u+2z π r +r π r D 2 sin (u z) 1 2 cos (u z) 2 × | K | { 2 − } { 2 − } 2 z+κ Γ(1 u + z) x− dz (3.6) × − VALUESOFDEDEKINDZETAFUNCTIONS 9 where f( z) = ζ (1 u + z)ζ (v + z) − K − K ∞ v (m) ∞ v (m) = K K m1 u+z mv+z mX=1 − mX=1 v mu−1 m ∞ d m d− du−1 vK (d)vK ( d ) = | P mz mX=1 1 u v m ∞ d m d − − vK(d)vK ( d ) = | P m1 u+z mX=1 −

∞ σ1′ u v,v (m) = − − K . m1 u+z mX=1 − Therefore (3.6) implies that

2 1−2u ∞ κ 2(u 1) 2 u 1 ( B)(ζK(u),ζK (v); x) = (2π) − DK σ1′ u v,vK (m)m − F − πi | | − − mX=1 z DK π r1+r2 Sκ( z) | 2 | sin (u z) × Z(B) − 4π m { 2 − }

π r2 2 z+κ cos (u z) Γ(1 u + z) x− dz. × { 2 − } − Putting κ = 1, we have

2 1−2u ∞ 1 2(u 1) 2 u 1 ( B)(ζK(u),ζK (v); x) = (2π) − DK σ1′ u v,vK (m)m − F − πi | | − − mX=1 z DK π r1+r2 | 2 | sin (u z) × Z(B) 4π m { 2 − } π x z+1 cos (u z) r2 Γ(1 u + z)2 − dz. × { 2 − } − z(z 1) − (3.7) We treat real and imaginary cases separately. Case-I: Real quadratic number ﬁeld: In this case r1 = 2 and r2 = 0. Thus (3.7) gives

1−2u ∞ 1 1 2(u 1) 2 u 1 ( B)(ζK (u),ζK (v); x) = (2π) − DK σ1′ u v,vK (m)m − F − −2πi − − mX=1 z DK iπ(u z) iπ(u z) 2 e − + e− − 2 × Z(B) 4π m − x z+1 Γ(1 u + z)2 − dz. (3.8) × − z(z 1) − 10 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

Let z z+1 DK iπ(u z) 2 x− HB(u; x)= e − Γ(1 u + z) dz. Z 4π2m − z(z 1) (B) − We diﬀerentiate HB(u; x) with respect to x and get,

z 2 iπu DK iπ Γ(1 u + z) HB′ (u; x) = e 2 e− − dz − Z(B) 4π mx z z iπu DK iπ Γ(z)Γ(1 u + z)Γ(1 u + z) = e 2 e− − − dz. − Z(B) 4π mx Γ(z + 1) (3.9)

Also let,

z z+1 DK iπ(u z) 2 x− B(u; x)= e− − Γ(1 u + z) dz. H Z 4π2m − z(z 1) (B) − As before diﬀerentiating with respect to x gives

z 2 iπu DK iπ Γ(1 u + z) B′ (u; x) = e− 2 e − dz H − Z(B) 4π mx z z iπu DK iπ Γ(z)Γ(1 u + z)Γ(1 u + z) = e− 2 e − − dz. − Z(B) 4π mx Γ(z + 1) (3.10)

Further, let

z z+1 DK 2 x− HB(u; x) = 2 Γ(1 u + z) dz. Z 4π2m − z(z 1) (B) − As before,

z 2 DK Γ(1 u + z) HB′ (u; x) = 2 2 − dz − Z(B) 4π mx z z DK Γ(z)Γ(1 u + z)Γ(1 u + z) = 2 2 − − dz. − Z(B) 4π mx Γ(z + 1) (3.11)

The convergence of the integrals of HB′ (u; x), B′ (u; x) and HB′ (u; x) follows for H max 0, R(u) 1

iπu 0,3 1, u, u DK iπ H′ (u; x)+ ′ (u; x)+ H′ (u; x)= 2πie G e− B HB B − 3,1 0, 4π2mx −

iπu 0,3 1, u, u DK iπ 0,3 1, u, u D K 2πie− G e 4πiG . (3.12) − 3,1 0, 4π2mx − 3,1 0, 4π2mx − −

Therefore (3.8) becomes

1−2u ∞ 1 2(u 1) 2 u 1 ( ( B)(ζK(u),ζK (v); x))′ = 2(2π) − DK σ1′ u v,vK (m)m − F − − − mX=1

iπu 0,3 1, u, u DK iπ iπu 0,3 1, u, u DK iπ e G e− e− G e × − 3,1 0, 4π2mx − 3,1 0, 4π2mx − − 1, u, u D 2G0,3 K . (3.13) − 3,1 0, 4π2mx −

Now using Cauchy’s residue theorem, we have,

1 1 (C)(ζK(u),ζK (v); x) = ( B)(ζK(u),ζK (v); x)+ xζK(u)ζK (v) F F − − 2 u ζK (u + v 1)x − 4hK RK ζK(u 1)ζK (v +1)+ − 1 . − (u 2)(u 1) 2 − − wKDK

Further diﬀerentiating with respect to x gives,

1 1 ( (C)(ζK(u),ζK (v); x))′ = ( ( B)(ζK(u),ζK (v); x))′ + ζK(u)ζK (v) F F − − 1 u ζK(u + v 1)x − 4hK RK − 1 . (3.14) u 1 2 − wKDK

Now (3.13) and (3.14) together imply,

1−2u ∞ 1 2(u 1) 2 u 1 ( (C)(ζK (u),ζK (v); x))′ = 2(2π) − DK σ1′ u v,v (m)m − F − − K mX=1

iπu 0,3 1, u, u DK iπ iπu 0,3 1, u, u DK iπ e G e− e− G e × − 3,1 0, 4π2mx − 3,1 0, 4π2mx − − 1, u, u D ζ (u + v 1) x1 u 2G0,3 K + ζ (u)ζ (v) K − − − 3,1 0, 4π2mx K K − u 1 − − 4hK RK 1 . × 2 wKDK 12 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

We now substitute x = 1,

1−2u ∞ 1 2(u 1) 2 u 1 ( (C)(ζK(u),ζK (v); 1))′ = 2(2π) − DK σ1′ u v,v (m)m − F − − K mX=1

iπu 0,3 1, u, u DK iπ iπu 0,3 1, u, u DK iπ e G e− e− G e × − 3,1 0, 4π2m − 3,1 0, 4π2m − − 1, u, u D ζ (u + v 1) 2G0,3 K + ζ (u)ζ (v) K − − 3,1 0, 4π2m K K − u 1 − − 4hK RK 1 . (3.15) × 2 wK DK

Similarly we have,

1−2v ∞ 1 2(v 1) 2 v 1 ( (C)(ζK(v),ζK (u); 1))′ = 2(2π) − DK σ1′ u v,v (m)m − F − − K mX=1

iπv 0,3 1, v, v DK iπ iπv 0,3 1, v, v DK iπ e G e− e− G e × − 3,1 0, 4π2m − 3,1 0, 4π2m − − 1, v, v D ζ (u + v 1) 2G0,3 K + ζ (u)ζ (v) K − − 3,1 0, 4π2m K K − v 1 − − 4hK RK 1 . (3.16) × 2 wKDK

Now we add (3.15) and (3.16), and further utilize (2.1) to arrive at

4hK RK 1 1 ζK(u)ζK (v) = 2ζK(u)ζK (v) 1 + ζK(u + v 1) − 2 u 1 v 1 − wKDK − − 1−2u ∞ 1, u, u D +2(2π)2(u 1)D 2 σ (m)mu 1 eiπuG0,3 K e iπ − K 1′ u v,vK − 3,1 2 − − − − 0, 4π m mX=1 −

iπu 0,3 1, u, u DK iπ 0,3 1, u, u DK 2(v 1) e− G e 2G + 2(2π) − − 3,1 0, 4π2m − 3,1 0, 4π2m − − 1−2v ∞ 0,3 1, v, v DK D 2 σ (m)mv 1 eiπvG e iπ K 1′ u v,vK − 3,1 2 − × − − − 0, 4π m mX=1 −

iπv 0,3 1, v, v DK iπ 0,3 1, v, v D K e− G e 2G . − 3,1 0, 4π2m − 3,1 0, 4π2m − −

VALUESOFDEDEKINDZETAFUNCTIONS 13

Which implies that,

4hK RK 1 1 2(u 1) ζK(u)ζK (v) = 1 + ζK(u + v 1) + 2(2π) − 2 u 1 v 1 − wKDK − − 1−2u ∞ 2 u 1 iπu 0,3 1, u, u DK iπ DK σ1′ u v,v (m)m − e G3,1 e− × − − K 0, 4π2m mX=1 −

iπu 0,3 1, u, u DK iπ 0,3 1, u, u DK 2(v 1) + e− G e + 2G + 2(2π) − 3,1 0, 4π2m 3,1 0, 4π2m − − 1−2v ∞ 2 v 1 iπv 0,3 1, v, v DK iπ DK σ1′ u v,v (m)m − e G3,1 e− × − − K 0, 4π2m mX=1 −

iπv 0,3 1, v, v DK iπ 0,3 1, v, v DK + e− G e + 2G . 3,1 0, 4π2m 3,1 0, 4π2m − −

This concludes the proof of this case. Case-II: Imaginary quadratic number ﬁeld: In this case r1 = 0 and r2 = 1. Thus (3.7) implies that,

1 1−2u ∞ 1 2(u 1) 2 u 1 ( B)(ζK (u),ζK (v); x) = (2π) − DK σ1′ u v,vK (m)m − F − πi | | − − mX=1 D z x z+1 | K| sin π(u z)Γ(1 u + z)2 − dz × Z 4π2m − − z(z 1) (B) − 1 2(u 1) 1−2u ∞ u 1 = (2π) − DK 2 σ1′ u v,v (m)m − −2π | | − − K mX=1 (I (u; x) (u; x)), (3.17) × B −IB where, z z+1 DK iπ(u z) 2 x− IB(u; x)= | | e − Γ(1 u + z) dz, Z 4π2m − z(z 1) (B) − and z z+1 DK iπ(u z) 2 x− B(u; x)= | | e− − Γ(1 u + z) dz. I Z 4π2m − z(z 1) (B) − Proceeding analogously we diﬀerentiate IB(u; x) with respect to x and apply the deﬁnition of Meijer G-function to get z 2 iπu DK iπ Γ(1 u + z) IB′ (u; x) = e | 2 | e− − dz − Z(B) 4π mx z z iπu DK iπ Γ(z)Γ(1 u + z)Γ(1 u + z) = e | 2 | e− − − dz − Z(B) 4π mx Γ(z + 1)

iπu 0,3 1, u, u DK iπ = 2πie G | | e− . (3.18) − 3,1 0, 4π2mx −

14 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

Again diﬀerentiating B(u; x) with respect to x and applying the deﬁnition of Meijer G-function weI get

z 2 iπu DK iπ Γ(1 u + z) B′ (u; x) = e− | 2 | e − dz I − Z(B) 4π mx z

iπu 0,3 1, u, u DK iπ = 2πie− G | | e . (3.19) − 3,1 0, 4π2mx −

Here we can apply the deﬁnition of Meijer G-function since the integrals in I (u; x) and (u; x) are convergent for B′ IB′ max 0, R(u) 1

m, n a1, , ap 1 πian+1 m, n+1 a1, , ap πi Gp, q · · · z = e Gp, q · · · ze− b1, , bq 2πi b1, , bq · · · · · ·

πian+1 m, n+1 a1, , ap πi e− Gp, q · · · ze (3.20) − b1, , bq · · · whenever n p 1. Subtracting≤ (3.19)− from (3.18) and then applying (3.20) we obtain,

iπu 0,3 1, u, u DK iπ I′ (u; x) ′ (u; x) = 2πi e G | | e− B −IB − 3,1 0, 4π2mx −

iπu 0,3 1, u, u DK iπ e− G | | e − 3,1 0, 4π2mx − 1, u, u D = 4π2G0,2 | K | . (3.21) 3,1 0, 4π2mx −

We further recall the following important relation (see, relation (9) [2, pp.209]) to rewrite (3.21) in suitable form:

m, n a1, , ap 1 n, m 1 b1, , 1 bq Gp, q · · · = Gq, p − · · · − z . (3.22) b1, , bq z 1 a1, , 1 ap · · · − · · · −

Applying relation (3.22) in (3.21), we obtain

VALUESOFDEDEKINDZETAFUNCTIONS 15 we have 1 1 Γ(1 u s) Xs G2,0 X = − − ds 1,30, 1 u, 1 u −2πi Z Γ(u + s) s − − (B) X 1 Γ(1 u s) s 1 = − − t − ds dt −2πi Z0 Z(B) Γ(u + s) X 1,0 = G − t dt − Z 0,2 u, u 0 − − X u 1/2 = t− J0(2t )dt. (3.24) − Z0 We now diﬀerentiate (3.17) with respect to x to get

1−2u ∞ 1 2u 1 2 ( ( B)(ζK (u),ζK (v); x))′ = (2π) − DK σ1′ u v,vK (m) F − | | − − mX=1 X u 1 u 1/2 m − t− J0(2t )dt. × Z0 Using Cauchy’s residue theorem, we get

1−2u ∞ 1 2u 1 2 ( (C)(ζK(u),ζK (v); x))′ = (2π) − DK σ1′ u v,v (m) F | | − − K mX=1 X u 1 u 1/2 m − t− J0(2t )dt + ζK(u)ζK (v) × Z0 1 u ζK(u + v 1)x − 2πhK − 1 . − u 1 w D 2 − K| K | Further substituting x = 1,

Adding (3.25) and (3.26), and then applying (2.1) we obtain,

2πhK 1 1 1−2u 2u 1 2 ζK(u)ζK (v) = 1 + ζK(u + v 1) (2π) − DK w D 2 u 1 v 1 − − | | K | K | − − 4π2m ∞ u 1 |DK | u 1/2 2v 1 1−2v σ1′ u v,v (m)m − t− J0(2t )dt (2π) − DK 2 × − − K Z − | | mX=1 0 4π2m ∞ v 1 |DK | v 1/2 σ1′ u v,v (m)m − t− J0(2t )dt. × − − K Z mX=1 0

4. Values of Dedekind zeta functions The special values of Dedekind zeta functions attached to an arbitrary quadratic number ﬁeld is not completely known. In this section, we will discuss about some formulae for Dedekind zeta functions attached to both real and imaginary quadratic number ﬁelds at any positive integers. Zagier’s result on ζK(2) is used to get these formulae. To state his result, we need to introduce the following real valued function x 1 4 A(x)= 2 log 2 dt. Z0 1+ t 1+ t

The precise result of D. Zagier [20] for ζK(2) is the following:

Theorem B. Let K be an arbitrary number ﬁeld with discriminant DK , and let r1 and r2 be its numbers of real and complex places, respectively. Then π2r1+2r2 ζK(2) = cνA(xν,1) A(xν,r ). D × · · · 2 | K | Xν p ﬁnite

Here cν ’s are rational numbers and xν,j’s are real algebraic numbers.

This formula does not work for ﬁnding ζK(2m) with m 2. However, he conjectured that in this case too, an analogous result hol≥ds. We need a more general real-valued function to state the conjecture which appeared in VALUESOFDEDEKINDZETAFUNCTIONS 17

[20]. For each natural number m, let Am be the real-valued function defined by 2m 1 2m 1 2 − ∞ t − Am(x)= dt. (2m 1)! Z x sinh2 t + x 1 cosh2 t − 0 − It is noted that the function Am(x) agrees with the function A(x) when m = 1. Conjecture 4.1 (Zagier). Let K be an arbitrary number field with discriminant DK , and let r1 and r2 be its numbers of real and complex places, π2m(r1+r2) respectively. Then ζK(2m) equals times a rational linear combi- √ DK | | nation of the products of r2 values of Am(x) at algebraic arguments. In [20], he proved this conjecture for certain abelian number fields. More precisely, he proved the following: Theorem C. If K is an abelian number field, then Conjecture 4.1 is true. πt In fact, the argument x can be chosen of the form x = cot N , where N is the smallest natural number such that K Q(e2πi/N ). ⊂ For m = 1, and for an imaginary quadratic number field K, Theorem C reduces to the following simplest form: 2 π DK πn ζK(2) = A cot , (4.1) 6 DK × n DK | | 0

Theorem D. Let K = Q(√DK) be an imaginary quadratic number field of discriminant DK. Then 2 π λν ζK(2) = ηνA , (4.2) 6 D D × D | K | | K | Xν | K | p finite p 18 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE for some finite collection of numbers ην Z and λν Q with λν + ∈ ∈ ν η Q D ν Q. | K| ∈ p 4.1. Real quadratic number field case. This section deals with an expression for the values of Dedekind zeta functions attached to real quadratic number fields. We note that an expression for the values of Dedekind zeta functions at even positive integers can be obtained from two prominent results. One of them was given by Euler which entails the expression of zeta function at even positive integers as ( 1)n+1B (2π)2n ζ(2n)= − 2n , (4.3) 2 (2n)! where B2n is the 2n-th Bernoulli number. Let χ be any Dirichlet character with modulus f such that χ( 1) = 1. Then we can derive the following relation from a result of H. -W.− Leopoldt (See, relation (12) in [10]):

τ(χ) 2π 2n B L(2n,χ) = ( 1)n+1 2n,χ , (4.4) − 2 f (2n)! where τ(χ) is the Gauss sum associated to χ and B2n,χ is the 2n-th generalized Bernoulli number associated to χ. Let K = Q(√D) be a quadratic number ﬁeld with discriminant DK. The Kronecker symbol (DK /.) is a real primitive Dirichlet character with modulus Dk . We denote this Dirichlet character by χDK and it’s associated Dirichlet |L-series| would be

∞ s L(s,χDK )= (DK /m)m− . mX=1 This is convergent in the half-plane σ> 0. We have

ζK(s)= ζ(s)L(s,χDK ). (4.5) We now apply (4.3) and (4.4) in (4.5) to obtain τ(χ )(2π)4nB B DK 2n 2n,χ¯DK ζK(2n)= 2 2n , (4.6) 4((2n)!) DK when K is real quadratic number ﬁeld.

We remark that L(s,χDK ) at odd positive integers is still not known. Thus the expression for Dedekind zeta function of real quadratic number ﬁeld at odd positive integers can not be obtained from (4.5). Therefore we are interested to get an expression for this case too. We use Theorem 1.1 to get the same. Let σ(1)(m)= σ (m) and C = hK RK . We deﬁne ′ 2n 2,vK 1/2 − − wK DK

0,3 1,x,x DK iπ P (x) := G e− , 3,1 0, 4π2m −

VALUESOFDEDEKINDZETAFUNCTIONS 19

1,x,x D Q(x) := G0,3 K eiπ , 3,1 0, 4π2m − and 1,x,x D R(x) := G0,3 K . 3,1 0, 4π2m −

Substite u = 2 and v = 2n + 1 in (1.1) and then apply (4.6) to obtain the following expression for Dedekind zeta values of real quadratic number ﬁelds at odd integers. More precisely we have:

Theorem 4.1. For a real quadratic number field K with discriminant DK , we have 4n+4 C 1 τ(χDK )(2π) B2n+2B2n+2,χD ζ (2n +1) = 1+ K K 2 2n+2 ζK(2) 2n ((2n + 2)!) DK 2(2π)2 3 ∞ 2(2π)4n −1−4n + D− 2 mσ(1)(m) P (2) + Q(2) + 2R(2) + D 2 ζ (2) K ζ (2) K K mX=1 K ∞ m2nσ(1)(m) P (2n +1)+ Q(2n + 1) + 2R(2n + 1) . × mX=1 4.2. Imaginary quadratic number field case. In this section, we discuss an expression for Dedekind zeta values attached to imaginary quadratic number fields. This expression for odd positive values is derived from the following two important results. Proposition 4.1 (Ramanujan’s Identity). For αβ = π2, we have

n 1 n 1 α− ζ(2n +1)+ S (2α) = ( β)− ζ(2n +1)+ S (2β) 2 2n+1 − 2 2n+1 n+1 2n k B2kB2n+2 2k n+1 k k 2 ( 1) − α − β , − − (2k)!(2n + 2 2k)! Xk=0 −

∞ 1 where Sn(r)= kn(eπrk 1) . kP=1 − In [4], M. Chamberland and P. Lopatto obtained an explicit formula for Riemann zeta odd positive values from Proposition 4.1. More precisely, they obtained the following: Proposition 4.2 (Chamberland–Lopatto). For n = 4m 1, − 4m 2 2m 1 F2m 14 − G2m 14 − ζ(4m 1) = − S4m 1(π)+ − S4m 1(2π) − − Dm − Dm − 2m 1 F2m 14 − − S4m 1(4π), (4.7) − Dm − 20 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE and for n = 4m + 1, 16m(G 16m) + 2E ζ(4m +1) = 2m m S (π) − ( 1 + 16m)E 4m+1 − m 2G K 16m(2 16m ( 4)m + 1) 2m m · − − S (2π) − ( 1 + 16m)E 4m+1 − m G 16m + 4G 16mK + 2E 2m 2m m m S (4π), (4.8) − ( 1 + 16m)E 4m+1 − m where m+1 k B2kB2m+2 2k F = ( 1) − , m − (2k)!(2m + 2 2k)! Xk=0 − m+1 k B2kB2m+2 2k G = ( 4) − , m − (2k)!(2m + 2 2k)! Xk=0 − 2m 1 2m 1 Dm = 4 − [(4 − + 1)F2m 1 G2m 1]/2, − − − 1 (1 42m) K = 2 − , m 1 + ( 4)m 24m+1 − − 42m E = G 24m+1K H 24mK G , m 2 2m − m m − m 2m m m+k B4mB4m+2 4k H = ( 4) − . m − (4k)!(4m + 2 4k)! Xk=0 − Let χ be any Dirichlet character with modulus f such that χ( 1) = 1. In that case H. -W. Leopoldt [10] obtained the odd L-values, − − τ(χ) 2π 2n+1 B L(2n + 1,χ) = ( 1)n+1 2n+1,χ¯ , (4.9) − 2i f (2n + 1)! where τ(χ) is the Gauss sum associated to χ and B2n+1,χ is (2n + 1)-th generalized Bernoulli number associated to χ. One can ﬁnd the formulae for ζK (4n 1) and ζK(4n + 1) by applying (4.7), (4.8) and (4.9) in (4.5). −

On the other hand, L(s,χdK ) at such integers is still not known. Thus an expression for Dedekind zeta values associated to imaginary quadratic number field at even positive integers can not be obtained from (4.5). Therefore it is interesting to find an expression for this case too. We use Theorem 1.1 to find such an expression. We put u = 2 and v = 2n in (1.2) and apply the above resulting expressions for ζ (4n 1) and ζ (4n + 1) to get the expression for ζ (2n). K − K K Remark 4.1. One can get an expression for ζK(4n 1) by substituting u = v = 2m in (1.2). This expression is simpler than that− of the above one, and it does not require Zagier’s result for ζK(2). One can find a relationship 2 between (ζK (2m + 1)) and ζK(4m + 1) by substituting u = v = 2m + 1 (1.2). Some other formulas can also be deduced from Theorem 1.1. VALUESOFDEDEKINDZETAFUNCTIONS 21

5. Concluding remarks D. Zagier in [20] asked the following questions related to Theorem D:

(Q1) Whether there is a formula of the type Eq. 4.2 for ζK (4), ζK(6), etc attached an arbitrary imaginary quadratic number field K? (Q2) Whether Theorem B can be sharpened similar to Theorem D for any arbitrary number field? He also remarked that the answer to this question is possible if one can prove Theorem B and Theorem D using methods available in analytic number theory. Here, we have obtained the expressions for ζK(m) attached to an arbitrary quadratic number field K at any positive integer (m 2). We believe that these expressions can be useful to provide satisfactory≥ answer to (Q1). It is possible to obtain analogous expression for higher degree number fields which may be helpful to answer (Q2). We conclude by making the following comments: (1) It could be possible to determine special values of Dedekind zeta functions for any quadratic number fields from the explicit formulae. But, this would involve the evaluation of Meijer G-function and other complicated terms. (2) One can obtain approximate functional equation for the product of two Dedekind zeta functions in the critical region by following the arguments of J. R. Wilton [19].

Acknowledgments S. Banerjee is partially supported by Infosys grant and A. Hoque is supported by the SERB-NPDF (PDF/2017/001958), Govt. of India. The authors are grateful to Professors Masumi Nakajima and Bruce C. Berndt for their careful reading, helpful comments and suggestions. The authors are thankful to the anonymous referee(s) for his/her/their valuable comments and suggestions which have helped improving the presentation immensely and drawing some useful remarks.

References [1] D. Banerjee and J. Mehta, Linearlized product of two Riemann zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 123–126. [2] H. Bateman, Higher Transcendental Functions, Vol.1, McGraw-Hill Book Company , New York, (1953). [3] K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functions II, World Scientiﬁc, Singapore, 2007. [4] M. Chamberland and P. Lopatto Formulas for odd zeta values and powers of π, J. integer seq., 14, (2011), 1–5. [5] K. Chandrasekharan and S. Minakshisundaram, Typical means, Oxford Univ. Press, Oxford, 1952. [6] G. H. Hardy, J. E. Littlewood, The zeros of Riemann’s zeta-function on the critical line, Math. Z. 10 (1921), no. 3-4, 283––317. 22 SOUMYARUP BANERJEE, KALYAN CHAKRABORTY AND AZIZUL HOQUE

[7] G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New Yorks, 1964. [8] E. Hecke, Bestimmung der Klassenzahl einer neuen Reihe von algebraischen Zahlkörpern, Nachr. Akad. Wiss. Göttingen Math. -Phys. Kl. II (1921), 1-23. [9] H. Lange, Uber¨ eine Gattung elemetar-arithmetischer Klasseninvarianten reell- quadratischer Zahlkörper, J. Reine Angew. Math. 233 (1968), 123–175. [10] H.-W. Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg. 22 (1958), 131–140. [11] C. Meyer, Uber¨ einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math. 198 (1957), 143–203. [12] C. Meyer, Uber¨ die Bildung von elementar-arithmetischen Klasseninvarianten in reell-quadratischen Zahlkörpern, “Algebraische Zahlentheorie”, Mathematisches Forschungsintitut Oberwolfach, Berichte, Heft 2 (1966), 165–215. [13] M. Nakajima, A new expression for the product of the two Dirichlet series, I, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), 19–22. [14] R. Narain, The G-functions as unsymmetrical Fourier kernels-I, Proc. Amer. Math. Soc. 13 (1962), 950-959. [15] T. Shintani, On the evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 393–417. [16] C. L. Siegel, Uber¨ die analytische Theorie der quadratischen Formen. III, Ann. of Math. (2) 38 (1937), no. 1, 212–291. [17] C. L. Siegel, Berechung von Zetafunktionen an Ganzzahligen Stellen, Nachr. Akad. Wiss. Gött. 10 (1969), 87–102 (Ges. Abh. 4, 82–97). [18] N. L. Wang and S. Banerjee, On the product of Hurwitz zeta-functions, Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 5, 31–36. [19] J. R. Wilton, An approximate functional equation for the product of two ζ-functions, Proc. London Math. Soc., S2-31 (1930), 11–17. [20] D. Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math. 83 (1986), 285–301. [21] D. Zagier, On the values at negative integers of the zeta function of a real quadratic field, Enseignement Math. 22 (1976), 55–95. [22] D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 253–284.

Soumyarup Banerjee @Department of Mathematics, IIT Gandhinagar, Palaj, Gandhinagar-382355, Gujarat, India. Email address: [email protected]

Kalyan Chakraborty @Kerala School of Mathematics, Kunnamangalam- 673571, Kozhikode, Kerala, India. Email address: [email protected]

Azizul Hoque @Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India. Email address: [email protected]