EULER PRODUCTS BEYOND THE BOUNDARY. 1. Introduction. (1 χ(p)p s ) 1, (1)

Transcription

1 EULER PRODUCTS BEYOND THE BOUNDARY TARO KIMURA, SHIN-YA KOYAMA, AND NOBUSHIGE KUROKAWA Abstract. We investigate the behavior of the Euer products of the Riemann zeta function and Dirichet L-functions on the critica ine. A refined version of the Riemann hypothesis, which is named the Deep Riemann Hypothesis (DRH), is examined. We aso study various anaogs for goba function fieds. We give an interpretation for the nontrivia zeros from the viewpoint of statistica mechanics.. Introduction Let χ be a primitive Dirichet character with conductor N. expressed by an Euer product The Dirichet L-function is L(s, χ) = p ( χ(p)p s ), () where p runs through a primes. The product () is absoutey convergent for Re(s) >. It is nown that L(s, χ) has a meromorphic continuation to a s C, which is entire if χ, and has a simpe poe at s = if χ =. In this paper we study the vaues L(s, χ) beyond the boundary Re(s) = of the absoute convergence region Re(s) > from the viewpoint of its reation to the vaues of the Euer product. Few resuts are nown in this context. The cassica resuts concerning the fact that the Euer product () converges to L( + it, χ) (t R, t ) can be found in textboos for either χ = ([T] Chapter 3) or χ ([M]). The ony wor we coud find beyond this is that of Godfed [G], Kuo-Murty [KM] and Conrad [C]. Godfed [G] and Kuo-Murty [KM] deat with the L-functions of eiptic curves at s =, with their resuts supporting the Birch and Swinnerton-Dyer conjecture. Conrad [C] treated more genera Euer products for Re(s) /. The (generaized) Riemann Hypothesis (GRH) for L(s, χ) asserts that L(s, χ) in Re(s) > /. When χ, it is equivaent to the foowing conjecture [C]. Conjecture. If χ, then for Re(s) > / we have L(s, χ) = im ( χ(p)p s ), n p n where the product is taen over a primes p satisfying p n. Note that the order of primes which participate in the product is important, because it is not absoutey convergent. Mathematics Subject Cassification. M6. Key words and phrases. The Riemann zeta function; Dirichet L-functions; the Riemann hypothesis; the generaized Riemann hypothesis; Euer products. Partiay supported by JSPS Research Feowships for Young Scientists (Nos , 5-43).

2 T. KIMURA, S. KOYAMA, AND N. KUROKAWA Conjecture (Deep Riemann Hypothesis (DRH)). If χ and L(s, χ) with Re(s) =, we have im n p n ( χ(p)p s ) = L(s, χ) where the product is taen over a primes p satisfying p n. { (s = and χ = ), (otherwise) We ca Conjecture the Deep Riemann Hypothesis, a deeper modification of Conjecture, iteray because we reach the boundary of the domain Re(s) > / given in Conjecture, and ogicay because Conjecture impies Conjecture. Indeed, if we denote ψ(x, χ) = χ(p) og p, m= p: p m x Conjecture is equivaent to ψ(x, χ) = O( x(og x) ), whie Conjecture is equivaent to ψ(x, χ) = o( x og x) by Conrad [C] Theorem 6.. The prototype version of this Conjecture was proposed in [C]. Conjecture to the case incuding χ =, see Aatsua [A]. For a generaization of It is an easy tas to obtain numerica support of Conjecture, since the convergence of the eft hand side is fairy fast. This ind of process, introducing a parameter to define a finite anaogue and then taing it to infinity, is often used in physics when it is difficut to anayze the infinite system directy. One can investigate how to approach infinity by anayzing the deviation from the resut in the desirabe imit. For exampe, in order to study the asymptotic behavior in an infinite voume system, it is convenient to introduce a system of some finite size Λ, and then estimate a correction by anayzing a differentia equation in terms of Λ, which is the so-caed renormaization group equation. The situation for the Riemann zeta and the Dirichet L-functions seems quite simiar: the difficuty with these functions ies essentiay invoved in treating infinity, so that convergency of the Euer product is nontrivia. In this paper we numericay examine the finite-size corrections to the zeta and L-functions appearing in the finite anaog, based on the anaogy between nontrivia zeros and eigenvaues of a certain infinite dimensiona matrix or critica phenomena observed around a phase transition point.. Function Fied Anaogs In this section, we prove an anaog of Conjecture for function fieds of one variabe over a finite fied. The theory of zeta and L-functions over such function fieds are seen, for exampe, in the textboo of Rosen [R].

3 EULER PRODUCTS BEYOND THE BOUNDARY 3 Let F q be the finite fied of q eements. We fix a conductor f(t ) F q [T ] and introduce a Dirichet character χ : (F q [T ]/(f)) C, which is extended to F q [T ] by χ(h) = for h such that (h, f) (). We define the Dirichet L-function by the Euer product: L Fq(T )(s, χ) = h ( χ(h)n(h) s ), where h = h(t ) F q [T ] runs through monic irreducibe poynomias, and N(h) = q deg h. In the ceebrated wor of Kornbum [K], it is proved that the above Euer product is absoutey convergent in Re(s) >, and is a poynomia in q s of degree ess than deg(f) if χ [W]. We prove the foowing theorem. Theorem (DRH over function fieds). Let q, f and χ be as above. Put K = F q (T ) and assume χ. Then the foowing () and () are true. () For Re(s) > /, we have im n deg h n () For t R with L K ( + it, χ), it hods that im n deg h n ( χ(h)n(h) s ) = L K (s, χ). ( χ(h)n(h) it ) = L K ( + it, χ ) { (χ =, t π og p Z). (otherwise) Proof of Theorem. We prove () first. We estimate the product E n = ( χ(h)n(h) it) deg h n by deaing with its ogarithm og E n = χ(h) q ( +it) deg h. deg h n = We divide the sum into three parts as og E n = A(n) + B(n) + C(n) with A(n) = = deg h n/ B(n) = n/ deg h n C(n) = =3 n/<deg h n χ(h) q ( +it) deg h, χ(h) q ( +it) deg h, χ(h) q ( +it) deg h. By the above mentioned Kornbum s theorem, we put r L K (s, χ) = ( λ j q s ) j=

4 4 T. KIMURA, S. KOYAMA, AND N. KUROKAWA with λ j = q or [D][Gr][W]. Then by taing the ogarithmic derivatives of r ( χ(h)n(h) s ) = ( λ j q s ) (Re(s) > ) h and comparing the coefficients of q s, we have (deg h) j= (deg h)χ(h) deg h = r λ j ( ). j= By this identity, the first partia sum A(n) is cacuated as A(n) = n = By the Deigne s theorem we have q λ j q ( +it) r n j= = λ j q +it ( (deg h) λ j q +it (deg h)χ(h) deg h ). and the assumption L K( + it, χ) tes that. Then by the Tayor expansion for og( x), it hods that +it ( ) r im A(n) = og λ j n j= q +it ( ) = og L K + it, χ. Next for estimating B(n), we use the generaized Mertens theorem [R] that N(h) og n (n ). When χ = and t Hence deg h<n π og q Z, we compute that B(n) = = n/ deg h n deg h n (+it) deg h q q (+it) deg h = ( (og n + C + O(n ) ) = ( og + O(n ) ). im B(n) = og. n In a other cases it hods that B(n) as n. deg h<n/ q (+it) deg h ( og n )) + C + O(n ) Finay, C(n) as n by a simiar argument to Lemma 3. in [C]. For proving (), we use the decomposition into A(n) and B(n) + C(n), in pace of that into A(n), B(n) and C(n) above. In this case both B(n) and C(n) are concerning absoutey convergent series ie C(n) in the proof of (). Thus B(n) + C(n) as n. Conjecture and Theorem are generaized to automorphic L-functions by Lownes [L].

5 EULER PRODUCTS BEYOND THE BOUNDARY 5 The foowing theorems are for the case of the trivia character. Theorem. Let X be a projective smooth curve over F q. Then im n N(x) q n ( N(x) / ) exp ( n = q / ) = ( q ) (X, ζ. ) Notice that n = q / = og q q n q d q (u) u og u, where q n f(u)d q (u) = n f(q )(q q ) = is Jacson s q-integra [KC][J]. Thus, it is considered as a modified q-ogarithmic integra. The situation is extended to the case of the Riemann zeta function studied by Aatsua [A], where a modified ogarithmic integra appears. Proof of Theorem. Let g be the genus of the curve X. exist α j C with α j = q for j =,, 3,..., g such that By Deigne s theorem [D] there ζ(x, s) = g j= ( α jq s )( α j q s ) ( q s )( q s. ) Note that α j q, because α j + α j Z. Thus we have On the other hand we compute ζ ( X, ) g j= = ( α jq / )( α j q / ) ( q / )( q /. ) og ( N(x) / ) N(x) q n = og = deg(x) n deg(x) n = =,n deg(x) n ) ( q deg(x) q deg(x) q deg(x) + q deg(x) + n <deg(x) n =3 q n <deg(x) n deg(x). When n, the second term tends to og by the generaized Mertens theorem [R], and the third term goes to, because we have x X N(x) α < for any α >. The first term

6 6 T. KIMURA, S. KOYAMA, AND N. KUROKAWA is cacuated as foows.,n deg(x) n q deg(x) = = = = = n = deg(x) q / deg(x) n X(F q ) = q / n q + g j= (α j + α j ) = n = n = q / q / q / g j= + (α j + α j ) q / + o() = g j= + og ( α jq / )( α j q / ) q / + o(), where we used the fact that α j = q (α j q) for convergence of the Tayor expansion of the ogarithms. Therefore it hods that og ( N(x) / ) N(x) q n n = = q / + og + og g j= ( α jq / )( α j q / ) q / + o(). Hence N(x) q n ( N(x) / ) exp ( n = q / ) ( q ) ζ (X, ). Theorem is the deeper anaogue for smooth curves of the foowing Theorem 3 for proper smooth schemes, which in its turn is a function fied anaogue of Mertens theorem [R]. In the situation of Theorem, it hods that ( N(x) ) (Res s= ζ(x, s)) e γ og t as t. N(x) t Theorem 3. Let X be a proper smooth scheme over F p. Then we have ( N(x) dim(x) ) ( Res s=dim(x) ζ(x, s) ) e γ og t as t. N(x) t

7 EULER PRODUCTS BEYOND THE BOUNDARY 7 Proof of Theorem 3. og ( N(x) dim(x) ) N(x) q n = og ( q dim(x) deg(x) ) = deg(x) n deg(x) n = =,n deg(x) n q dim(x) deg(x) q dim(x) deg(x) + = q dim(x) deg(x). n <deg(x) n The second term goes to as n, because we have x X N(x) α < for any α > dim(x). The first term is cacuated as foows. By putting = deg(x), we compute q dim(x) deg(x) = n deg(x) q dim(x). () deg(x) n = deg(x) By the resuts of Grothendiec [Gr] and Deigne [D], there exist α i, β j C with α i, β j < q dim(x) such that deg(x) deg(x) = X(F q ) = q dim(x) + j β j i α i. Hence as n. Since () = n = n + = = og n + γ + og ζ(x, s) = j ( βj q dim(x) ) i ( αi q dim(x) i( α iq dim(x) ) j ( β jq dim(x) ) + o(), i ( α iq s ) ( q dim(x) s ) j ( β jq s ), we see that s = dim(x) is the argest poe of ζ(x, s), which is simpe with Res s=dim(x) ζ(x, s) = og q i( α iq dim(x) ) j ( β jq dim(x) ). Taing a terms into account, we concude that ) N(x) q n ( N(x) dim(x) ) ne γ (og q)res s=dim(x) ζ(x, s) = (og q n ) e γ Res s=dim(x) ζ(x, s). We conjecture that Theorem 3 woud hod for genera schemes:

8 8 T. KIMURA, S. KOYAMA, AND N. KUROKAWA Conjecture 3. Let X be a proper smooth scheme over Z. Then ( N(x) dim(x) ) ( Res s=dim(x) ζ(x, s) ) e γ og t as t. N(x) t 3. Numerica Cacuations In this section we show some numerica data supporting the Deep Riemann Hypothesis (Conjecture ). If this conjecture is true, the partia Euer product L x (s, χ) = p x( χ(p)p s ), converges to L(s, χ) or L(s, χ) as x even on the critica ine Re(s) = /. We formay put L x (s, χ) = L(s, χ) for x =. First we give Tabe, which shows the accuracy of Conjecture at s = /. We find that the ratio of L(, χ) and L x(, χ) is amost equa to for x = 7, when χ is quadratic. d L E ( L)/E ( ( d p Tabe. L := L (, ( )) d, E := p 7 ) p ). In what foows we put χ 7a and χ 7b to be the character χ moduo 7 with χ and χ =, respectivey. Namey, if we define the character χ moduo 7 by giving the vaue at the primitive root 3 Z/7Z, we define χ 7a (3) = exp(π /3) and χ 7b (3) =. We aso denote by χ 3 the nontrivia character moduo 3, which satisfies χ 3 =. Denote by p n the n-th prime number. Figures,, 3, 4, 5 and 6 show the datum for the vaues L x ( + it, χ ), L x ( it, χ ), L x ( + it, χ)

9 EULER PRODUCTS BEYOND THE BOUNDARY Figure. Rea part (eft) and imaginary part (right) of L x (/ + it, χ 7a ) Figure. Rea part (eft) and imaginary part (right) of L x (/ + it, χ 7b ) Figure 3. Rea part (eft) and imaginary part (right) of L x (3/4 + it, χ 7a ) Figure 4. Rea part (eft) and imaginary part (right) of L x (3/4 + it, χ 7b ) for x = p (green), x = p (bue), x = p (yeow) and (red). Figures, 3 and 5 are for χ 7a, and Figures, 4 and 6 for χ 7b. As t, we apparenty see that L x (/ + it, χ) L(/, χ) for χ, that L x (/ + it, χ) L(/, χ) for χ =, and that L x (3/4 + it, χ) L(3/4, χ), L x ( + it, χ) L(, χ) for both cases χ = and χ. This supports the DRH (Conjecture ).

10 T. KIMURA, S. KOYAMA, AND N. KUROKAWA Figure 5. Rea part (eft) and imaginary part (right) of L x ( + it, χ 7a ) Figure 6. Rea part (eft) and imaginary part (right) of L x ( + it, χ 7b ) s α (χ 7a ) α (χ 7b ) / / Tabe. Exponents of δl x (s, χ) x α for χ 7a and χ 7b. We introduce the foowing error function in order to estimate the speed of convergence for L x (s, χ): δl x (s, χ) = Lx(s,χ) L(s,χ) L(s,χ) (s = / and χ = ). (otherwise) Lx(s,χ) L(s,χ) L(s,χ) Figure 7 shows the vaues of δl x (s, χ). When we approximate the error function as δl x (s, χ) x α, the exponents are determined so that they fit the numerica resuts (Tabe ). We see the speed of convergence becomes faster as s gets arger, if s is rea..... e-5 e-6 e+6 e e-5 e-6 e+6 e+7 Figure 7. δl x (s, χ) for s = / (red), s = 3/4 (green) and s = (bue) with χ 7a (eft) and χ 7b (right)

11 EULER PRODUCTS BEYOND THE BOUNDARY 4. Finite Size Scaing In this section, we show another specia feature that L x (s, χ) has. Since L x (s, χ) is a finite Euer product, it obviousy has no zeros on the critica ine. Nevertheess, L x (s, χ) gives a certain sequence of compex numbers, which seemingy grows up to the nontrivia zeros of L(s, χ), as x. In other words, the finite partia Euer product L x (s, χ) aready nows the nontrivia zeros of L(s, χ). In Figures 8, 9 and, the bue curves show the vaues ρ x (t) = π Im d ( ) dt og L x + it, χ (3) with x = p for χ 3, χ 7a, χ 7b, respectivey. The red curves are L ( + it, χ). This function (3) is an anaog of the eigenvaue density function in random matrix theory. The Riemann zeta function on the critica ine s = / + it can be seen as a characteristic poynomia of a certain infinite dimensiona matrix [KS, BH]: With the Riemann-Siege theta function ( it ϑ(t) = Im og Γ + ) t 4 og π, the function Z(t) = e iϑ(t) ζ ( + it) turns out to be rea. This is because the competed ζ- function s(s ) ( s ξ(s) = π s/ Γ ζ(s) ) is rea on Re(s) = / due to the functiona equation ξ(s) = ξ( s). Dirichet L-functions aso have simiar representations. The rea function Z(t) changes its signature at nontrivia zeros of the Riemann zeta function. Thus Z(t) is expressed as a reguarized product reg (t tj ) j= where t j satisfies ζ ( + it j) =. This means the argument of Z(t) jumps by π at the zeros. Therefore when we define the density function of the nontrivia zeros on the critica ine as ρ(t) = δ(t t j ) j= = π Im j= t t j = π Im d dt og reg (t tj ), the function (3) shoud converge to this density function in the imit of x, up to the factor coming from ϑ(t). Here we simpy write the deta function as δ(x) = π Im x, which is originay represented as δ(x) = im ɛ + ± π Im x iɛ. Apparenty the ocation of the zeros of L( + it, χ) agrees to that of the peas of ρ x(t) in Figures 8, 9 and. j= This suggests that a finite set of first few primes aready nows the nontrivia zeros of L(s, χ), and that the Euer product woud be meaningfu beyond the boundary. We aso observe that the bue curve osciates near t = if and ony if χ =.

12 T. KIMURA, S. KOYAMA, AND N. KUROKAWA Figure 8. ρ x (t) for χ Figure 9. ρ x (t) for χ 7a Figure. ρ x (t) for χ 7b Figure. Peas in ρ(t) with the smaest zero for χ 3 (eft), χ 7a (center) and χ 7b (right) Figure shows how the peas of ρ(t) with the smaest zero in Figures 8, 9 and get coser to the zeros of L(s, χ) for x = p (green), x = p (bue), x = p (yeow). We see these peas getting higher and narrower, and approaching the Dirac deta function. This ind of scaing behavior is often found in critica phenomena associated with some phase transitions. Especiay, in this case, the situation is simiar to percoation theory [SA]. Figures, 3 and 4 indicate the vaues R x (t) = π Im og L x ( ) + it, χ

13 EULER PRODUCTS BEYOND THE BOUNDARY 3 for χ 3, χ 7a, χ 7b, respectivey, for x = p (green), x = p (bue), x = p (yeow) and (red). This aso seems to refect the property of DRH. The green, bue and yeow curves appear to converge to the red one more smoothy ony when χ (Figure 3). In the other two cases, the curves osciate many times near the origin. The eaps in the red curves correspond to the zeros of L(s, χ). We normaize that the jumps at zeros are equa to one. This refects the conjecture that the mutipicity of such zeros shoud be a one. In other words, if we express their derivatives by the Dirac deta function, the coefficients are one Figure. R x (t) for χ Figure 3. R x (t) for χ 7a Figure 4. R x (t) for χ 7b We define another function N x (t) from R x (t) by subtracting the contribution of the L- function versions of the Riemann-Siege theta function. This counts the number of the nontrivia zeros on the critica ine in the imit of x. Figures 5, 6 and 7 show the vaues of N x (t) for χ 3, χ 7a, χ 7b, respectivey. The panes of Figures 8, 9, show N x (t) around the smaest nontrivia zeros of the L-functions with x = p (green), x = p 5 (ight bue), x = p (bue), x = p 5 (purpe), x = p (yeow) and x = (red). As the case of R x (t), we see a sharp step structure as the cut-off parameter x getting arger. These figures aso te us that the vaues Im og L( + it) are amost stabe for nontrivia zeros + it of the L-function, no matter how many prime numbers we tae into account. This suggests that the nontrivia zeros are anaogs of the critica points in statistica mechanics, which are stabe to the finite-size correction. To examine the anaogy to critica phenomena in statistica mechanics, we sha chec the scaing property around the critica point. Being the smaest zero +it, we define the scaing variabe z = t t t x λ.

14 4 T. KIMURA, S. KOYAMA, AND N. KUROKAWA Figure 5. N x (t) for χ Figure 6. N x (t) for χ 7a Figure 7. N x (t) for χ 7b Figure 8. N x (t) (eft) and Ñx(z) (right) for χ Figure 9. N x (t) (eft) and Ñx(z) (right) for χ 7a Correspondingy we introduce a scaed function Ñx(z), defined as N x (t) = Ñx(z = t t t x λ ). Right panes of Figures 8, 9 and show the vaues of Ñx(z). By choosing a proper exponent λ, a the curves are amost approximated by ony one curve. This means that the dependence on the cut-off parameter x appears ony in the form of the scaing variabe z. This scaing behavior supports the simiarity to the critica phenomena.

15 EULER PRODUCTS BEYOND THE BOUNDARY Figure. N x (t) (eft) and Ñx(z) (right) for χ 7b character t λ χ χ 7a χ 7b Tabe 3. Numericay evauated exponents around the smaest zeros + it for χ 3, χ 7a and χ 7b Tabe 3 shows the numerica vaues of the smaest zeros of the L-functions and the corresponding exponents for χ 3, χ 7a and χ 7b. These exponents are numericay determined by fitting the curves of Ñ x (z) by changing the parameter x =, 5,, 5,. In the case of the ordinary critica phenomena, there is ony one critica point. On the other hand, there are infinitey many zeros on the critica ine of the L-function, which are anaogs of the critica point. Thus, even if we focus on ony the smaest zero, as discussed in this study, there shoud be correction to its scaing bahavior from such other zeros: we have to tae care of the scaing property for others simutaneousy. References [A] H. Aatsua: The Euer product for the Riemann zeta-function on the critica ine. (preprint, ) [BH] E. Brézin and S. Hiami: Characteristic poynomias of random matrices, Comm. Math. Phys. 4 () 35. [C] K. Conrad: Partia Euer products on the critica ine, Canad. J. Math. 57 (5) [D] P. Deigne: La conjecture de Wei II, IHES Pub. Math. 5 (98) [G] D. Godfed: Sur es produits parties euériens attachés aux courbes eiptiques, C. R. Acad. Sci. Paris [Gr] Sér. I Math. 94 (98) A. Grothendiec: Cohomoogie -adique et fonctions L, Seminaire de Geometrie Agebrique du Bois-Marie , SGA 5, Springer Lecture Notes in Math. 589 (977). [J] F. H. Jacson: On q-definite integras, Q. J. Pure App. Math. 4 (9) [K] H. Kornbum: Über die Primfuntionen in einer arithmetischen Progression, Math. Zeit., 5, (99). [KC] V. Kac and P. Cheung: Quantum cacuus, Universitext, Springer-Verag,. [KM] W. Kuo and R. Murty: On a conjecture of Birch and Swinnerton-Dyer, Canad. J. Math. 57 (5) [KS] J. P. Keating and N. C. Snaith: Random matrix theory and ζ(/ + it), Comm. Math. Phys. 4 () [L] C. Lownes: Deep Riemann hypothesis for GL(n). (preprint, ) [M] F. Mertens: Ein Beitrag zur anaytischen Zahentheorie, J. Reine Angew. Math. 78 (874) [R] M. Rosen: A generaization of Mertens theorem, J. Ramanujan Math. Soc. 4 (999) 9. [R] M. Rosen: Number theory in function fieds, Springer New Yor,. [SA] D. Stauffer and A. Aharony: Introduction to percoation theory, CRC press, 994. [T] E. C. Titchmarsh: The theory of the Riemann zeta function, Oxford University Press, 987. [W] A. Wei: Sur es courbes agébriques et es variétés qui s en déduisent, Actuaités Sci. Ind. 4 = Pub. Inst. Math. Univ. Strasbourg 7 (945), Hermann, 948.

16 6 T. KIMURA, S. KOYAMA, AND N. KUROKAWA [W] A. Wei: Basic number theory, Springer Berin Heideberg, 974. Mathematica Physics Laboratory, RIKEN Nishina Center, - Hirosawa, Wao, Saitama, 35-98, Japan. E-mai address: taro.imura@rien.jp Department of Biomedica Engineering, Toyo University, Kujirai, Kawagoe, Saitama, , Japan. E-mai address: oyama@toyo.jp Department of Mathematics, Toyo Institute of Technoogy, -- Oh-oayama, Meguro-u, Toyo 5-855, Japan. E-mai address: uroawa@math.titech.ac.jp