Polynomial behavior of special values of partial zeta function of real quadratic fields at s=0
aa r X i v : . [ m a t h . N T ] N ov POLYNOMIAL BEHAVIOR OF SPECIAL VALUES OF PARTIAL ZETAFUNCTION OF REAL QUADRATIC FIELDS AT S = BYUNGHEUP JUN AND JUNGYUN LEEA
BSTRACT . We compute the special values of partial zeta function at s = K n and ray class ideals b n suchthat b − n = [ δ ( n )] where the continued fraction expansion of δ ( n ) is purely periodic and each terms are polynomial in n of boundeddegree d . With an additional assumptions, we prove that the specialvalues of partial zeta function at s = L -functionsat s = χ behave as quasi-polynomial as well. We compute out explicitly the coefficients of thequasi-polynomials. Two examples satisfying the condition are pre-sented and for these families the special values of the partial zetafunctions at s = C ONTENTS
1. Introduction 22. Decomposition of L-function into partial zeta function 42.1. Partial zeta function of a ray class 42.2. Decomposition of partial Hecke L-function 52.3. Continued fractions 62.4. Quasi-polynomials 62.5. Main theorem 73. Shintani-Zagier decomposition and partial zeta values 74. Periodicity of orbit 105. Explicit computation of the coefficients 126. Two examples 176.1. Case 1: f ( n ) = n + f ( n ) = n + n + n + Date : 2011.5.23.
1. I
NTRODUCTION
This note is complementary to our previous work “the behavior ofHecke’s L-function at s = [ ] ). In [ ] , a series of real quadraticfields K n with fixed ideals b − n = [ δ ( n )] and a mod- q Dirichlet charac-ter χ , which yields a mod- q ray class character χ n : = χ ◦ N K n / Q for each K n , are considered. We gave a condition on δ ( n ) to ensure a controlledbehavior of the special value at s = L -functionof b n twisted with a ray class character χ n of modulus q in a certainway. More precisely, we called this property ‘linearity of partial Hecke L -values’ when L K n ( b n , χ n ) = A ( r ) k + B ( r ) where n = qk + r and for some constants A ( r ) , B ( r ) associated to r =
0, 1, . . . , r −
1. The coefficients A ( r ) , B ( r ) can be explicitly computed andare shown to be lying inside q Z [ χ ( ) , χ ( ) , . . . , χ ( r − )] . Roughlywritten the criterion is that the terms of the continued fractions of δ ( n ) has to be linear function in n .The proto-typical result of this linearity appeared first in Biro’s proofof Yokoi’s conjecture(cf. [ ] for the conjecture and [ ] for the proof).It was the key ingredient with a class number one criterion in solvingYokoi’s conjecture. Later this moral has been extensively applied insolving some class number one problems for some families of real qua-dratic fields when there is an appropriate class number one criterion.(cf. [ ] , [ ] , [ ] , [ ] , [ ] , [ ] ).In this paper, we deal with similar phenomenon with almost the sameassumptions as in [ ] , but we have freed the linearity assumption onthe terms of continued fraction expansion of δ ( n ) −
1. In this setting,the linearity of the values of ζ - or L -values in family is generalized to a‘polynomial’ expression . Written precisely, the special values as functionin n are taken in a packet of polynomials in a periodic way. Functionsof this type are called quasi-polynomials . So in particular the ‘linearity’in loc.cit. should read as quasi-linear function. A precise definition anda short discussion of quasi-polynomial are presented in Section 2 of thisarticle.Our main result is as follows: Theorem 2.4.
Let { K n = Q ( p f ( n )) } n ∈ N be a family of real quadraticfields where f ( n ) is a positive square free integer for each n. Suppose b n isan integral ideal relatively prime to q such that b − n = [ δ ( n )] . Assume δ ( n ) − has purely periodic continued fraction expansion δ ( n ) − = [[ a ( n ) , a ( n ) , · · · , a s − ( n )]] ARTIAL ZETA FUNTION 3 of period s independent of n and a i ( x ) are polynomials of degree smallerthan d. If N K n ( b n ( C + D δ ( n ))) modulo q is a function only depending onC , D and r for n = qk + r, then ζ q ( b n ( C + D δ ( n ))) = A ( r ) + A ( r ) n + . . . + A d ( r ) n d where n = qk + r, ≤ r < q and for some constants A ( r ) , A ( r ) , . . . , A d ( r ) depending on r. Furthermore, A i ( r ) ∈ q i + Z .A part of the result presented here was announced for the Hecke’s L -values in a family at the end of [ ] . Actually, the consideration of thepartial zeta function for ray class ideals instead of the partial Hecke’sL-function of an ideal ameliorates the previous in two folds. Firstly, wegive here an expression of Hecke’s L -function as twisted sum of the rayclass partial zeta function: L K n ( χ , s , b ) = X ( C , D ) ∈ ˜ F ′ δ χ (( C + D δ ) b ) ζ q ( s , ( C + D δ ) b ) (c.f. Proposition 2.2.). If we restrict to the case d =
1, using this iden-tification, we can recover directly the linearity of the values of Hecke’s L -functions at s = d , we obtain thehigher degree generalization of the main result presented in [ ] . Thisis previously announced in loc.cit. Secondly, in principle the partial zetafunction of a ray class ideal contains finer information than the partialHecke’s L -function of the associated ideal.Our main idea is to develop and examine appropriate cone decompo-sition similar to Shintani and Zagier for partial zeta functions(cf. [ ] , [ ] ). Once a simple cone is given, one can evaluate the Riemann sumfor the partial zeta value at s = L function. Unfortunately, the decompositions for a family of realquadratic fields with ideals are far from being uniform. But surprisinglyagain the cone decomposition for the associated ideal class behaves in auniform way. In particular, if we parameterize the orbit of the ray classideal b n acted by the totally positive unit group using a pair of integers ( C , D ) such that 0 ≤ C , D ≤ q −
1, this action has q -periodicity in thefamily parametrized by n .This paper is composed as follows. In Section 2, we recall the notionsof ray class partial zeta function and some necessary stuffs. Then westate the main results. Section 3 is devoted to an expression of the partialzeta value at 0, where we use the cone decomposition à la Shintani andZagier. In Section 4, we show that this domain decomposition is acted
BYUNGHEUP JUN AND JUNGYUN LEE by the totally positive unit group modulo units congruent to 1 modulo q ,which has certain invariance property in the family. This concludes themost important part of our main theorem that the special values behavein quasi polynomial for the family satisfying our assumption. In Section5, we compute the coefficients of the quasi polynomials in the expressionof partial zeta values to restrict possible coefficients. Finally, in Section6, we compute the quasi polynomials for two explicitly chosen families.2. D ECOMPOSITION OF L- FUNCTION INTO PARTIAL ZETA FUNCTION
Partial zeta function of a ray class.
For a number field K and afixed positive rational integer q as a conductor, the partial zeta functionof the ray class of an ideal a in K is defined as ζ q ( s , a ) : = X c ∼ q a integral N ( c ) − s where c ∼ q a means that c = ( α ) a for totally positive α ≡ ( mod q ) . For c to be integral, α should be an element of 1 + q a − and c = a if and onlyif α ∈ E + q , where E + q is the multiplicative group of totally positive unitscongruent to 1 modulo q . Thus we have(1) ζ q ( s , a ) = X α ∈ ( + q a − ) + / E + q N ( a α ) − s This should not be confused with the partial zeta function for an ordi-nary ideal class. Note that this definition works not only for an idealrelatively prime to q but for general ideal of K .From now on, we assume K to be a real quadratic field and we con-sider ray class partial zeta function of an integral ideal b relatively primeto q such that b − = [ δ ] . In this case, we have a description of thepartial zeta function of other ray class than b but in the same ideal class. Lemma 2.1.
Suppose C , D are integers such that ≤ C , D ≤ q − and (( C + D δ ) b , q ) = . Then we have ζ q ( s , ( C + D δ ) b ) = X α ∈ ( C + D δ q + b − ) + / E + q N ( q b α ) − s . Proof.
From (1) we have(2) ζ q ( s , ( C + D δ ) b ) = X β ∈ ( + q ( C + D δ ) − b − ) + / E + q N (( C + D δ ) b β ) − s .put α = β C + D δ q . This shows the equality. (cid:3) ARTIAL ZETA FUNTION 5
Let us define F : = { ( C , D ) ∈ Z | ≤ C , D ≤ q − ( C , D ) = (
0, 0 ) } and its subset(3) F δ : = { ( C , D ) ∈ Z | ≤ C , D ≤ q − (( C + D δ ) b , q ) = } .An element ( C , D ) of F sends an ideal a to another ideal ( C + D δ ) a .If ( C , D ) ∈ F δ , ( C + D δ ) sends b to another ideal relatively prime to q .The group E + of totally positive units of K acts on F by(4) ε ∗ ( C + D δ ) = C ′ + D ′ δ where ( C + D δ ) ε + q b − = C ′ + D ′ δ + q b − for ε ∈ E + . Note that thisaction is inherited to F δ .Let ˜ F ′ ⊂ F be a fundamental set for the quotient F / E + and ˜ F ′ δ ⊂ ˜ F ′ bea fundamental set for the quotient F δ / E + .2.2. Decomposition of partial Hecke L-function.
Now we considerthe partial Hecke L -function for ( b , χ ) where χ is a ray class characterof modulus q . Recall the partial Hecke L -function associated to ( b , χ ) isdefined as follows: L K ( s , χ , b ) = X c ∼ b χ ( c ) N ( c ) s where the summation runs over every integral ideal c principally equiv-alent to b .Keeping the notations introduced before, we obtain an expression ofpartial Hecke L -function of ( b , χ ) as sum of ray class partial zeta func-tions of ray class ideals principally equivalent to b twisted with valuesof χ . Since any ray class associated to an ideal class of b can be repre-sented in the form ( C + D δ ) b and E + q action on ( C , D ) preserves the rayclass, this sum is taken over ˜ F ′ . Moreover, as χ values 0 at ( C + D δ ) b for ( C , D ) outside ˜ F ′ δ , the summation actually runs over ˜ F ′ δ .Summarizing this discussion, we have Proposition 2.2.
Let q be a positive integer. For an ideal b ⊂ K relativelyprime to q and a ray class character χ modulo q, we haveL ( χ , s , b ) = X ( C , D ) ∈ ˜ F ′ δ χ (( C + D δ ) b ) ζ q ( s , ( C + D δ ) b ) . BYUNGHEUP JUN AND JUNGYUN LEE
Proof. L K ( s , χ , b ) = X a ∼ b integral χ ( a ) N ( a ) − s = X ( C , D ) ∈ ˜ F ′ δ χ (( C + D δ ) b ) X α ∈ ( C + D δ q + b − ) + / E + q N ( q b α ) − s Applying Lemma (2.1), we have done the proof. (cid:3)
Continued fractions.
To state our main result properly, we needto recall the notions of continued fractions and quasi polynomials.Let [[ a , a , . . . , a n ]] be the purely periodic continued fraction [ a , a , a , . . . , a n , a , a , . . . ] ,where [ a , a , a , . . . ] : = a + a + a + · · · .2.4. Quasi-polynomials. If f ( n ) is a function of Z such that f ( n ) : = c d ( n ) n d + c d − ( n ) n d − + . . . + c ( n ) for some periodic functions c k ( n ) then we call f ( n ) a quasi-polynomial ofdegree d . When c k ( n ) has a common period q for k =
0, 1, · · · , d , we say f ( n ) has (quasi-)period q . We call c k ( n ) the k -th coefficient (function)of f ( n ) . Proposition 2.3. f ( n ) = c d ( n ) n d + c d − ( n ) n d − + · · · + c ( n ) is quasi-polynomial of period q if and only if for n = qk + rf ( n ) = a d ( r ) k d + a d − ( r ) k d − + · · · + a ( r ) . Moreover, for j =
0, 1, · · · , d,c j ( r ) = d X i = j a i ( r ) (cid:18) ij (cid:19) ( − r ) i − j q − i . Proof.
Substitute k with q − i ( n − r ) in the first expression of f ( n ) . Rear-ranging this as a form of polynomial in n , we obtain the second expres-sion. c j ( r ) is easily obtain from the rearrangement. (cid:3) ARTIAL ZETA FUNTION 7
Main theorem.
Our main theorem is as follows:
Theorem 2.4.
Let { K n = Q ( p f ( n )) } n ∈ N be a family of real quadraticfields where f ( n ) is a positive square free integer for each n. Suppose b n isan integral ideal relatively prime to q such that b − n = [ δ ( n )] . Assume δ ( n ) − has purely periodic continued fraction expansion δ ( n ) − = [[ a ( n ) , a ( n ) , · · · , a s − ( n )]] of period s independent of s and a i ( x ) are polynomials of degree smallerthan d. If N K n ( b n ( C + D δ ( n ))) modulo q is a function only depending onC , D and r for n = qk + r, then ζ q ( b n ( C + D δ ( n ))) = A ( r ) + A ( r ) n + . . . + A d ( r ) n d where n = qk + r , 0 ≤ r < q and for some constants A ( r ) , A ( r ) , . . . , A d ( r ) depending on r. Furthermore, A i ( r ) ∈ q i + Z . Corollary 2.5.
Let χ be a Dirichlet character modulo q for a positiveinteger q and χ n be a ray class character modulo q defined by χ ◦ N K n .With the same assumption of Theorem 2.4, we have L K n ( χ n , b n ) is quasi-polynomial with degree d and period q and the values of i-th coefficientfunctions are in q i + Z [ χ ( ) , χ ( ) · · · χ ( q )] .Note in [ ] , the linearity is the quasi-linearity in the second form inProposition 2.3 written as a polynomial in k , while we take the first formas shape of polynomial in n in this article.3. S HINTANI -Z AGIER DECOMPOSITION AND PARTIAL ZETA VALUES
A real quadratic field K is diagonally embedded into its Minkowskispace K R ≃ R by ι = ( τ , τ ) , where τ , τ are two real embeddings of K . The multiplicative action of E q + on K + induces an action on ( R ) + byextending in coordinate-wise way: ε ◦ ( x , y ) = ( τ ( ε ) x , τ ( ε ) y ) .A fundamental domain D R of ( R ) + / E + q is given as(5) D R : = { x ι ( ) + y ι ( ε − λ ) | x > y ≥ } ⊂ ( R ) + where ε λ is the totally positive generator of E q and λ = [ E + : E + q ] .We fix an integral ideal b such that b − = [ δ ] . Moreover we assumethat δ > < δ ′ <
1. If we take the convex hull of ι ( b − ) ∩ ( R ) + in ( R ) + , the lattice points on the boundary are { P i } i ∈ Z for P i ∈ ι ( b − ) . P i are uniquely determined by imposing that P = ι ( ) , P − = ι ( δ ) and x ( P i ) < x ( P i − ) where x ( P k ) is the first coordinate of P k . Since P λ m = ι ( ε − λ ) BYUNGHEUP JUN AND JUNGYUN LEE for some positive integer m (See Proposition 2.4 (5) in [ ] ), D R isfurther decomposed into ( λ · m ) -disjoint union of smaller cones: D R = λ m G i = { x P i − + y P i | x > y ≥ } .Accordingly the fundamental set of the quotient ( ι ( C + D δ q + b − ) T ( R ) + ) / E + q inside D R , which we denote by D is given by a disjoint union: D : = λ m G i = (cid:16) ι ( C + D δ q + b − ) \ { x P i − + y P i | x > y ≥ } (cid:17) .Since { P i − , P i } is a Z -basis of ι ( b − ) , there is a unique ( x iC + D δ , y iC + D δ ) ∈ (
0, 1 ] × [
0, 1 ) such that(6) x iC + D δ P i − + y iC + D δ P i ∈ ι ( C + D δ q + b − ) ,for each i , C , D ∈ Z . Thus ι (cid:0) C + D δ q + b − (cid:1) \ { x P i − + y P i | x > y ≥ } = { ( x iC + D δ + n ) P i − + ( y iC + D δ + n ) P i | n , n ∈ Z ≥ } .(7)In [ ] , Yamamoto found a recursive relation satisfied by ( x iC + D δ , y iC + D δ ) : x i + C + D δ = 〈 b i x iC + D δ + y iC + D δ 〉 , y i + C + D δ = − x iC + D δ ,(8)where 〈·〉 is as defined as 〈 x 〉 = x − [ x ] (resp. 1) for x Z (resp. for x ∈ Z )(See (2.1.3) of loc.sit. ).Let A i : = x ( P i ) and δ i = A i − A i for all i ∈ Z . Then from Eq.(7), we obtainthe following: ζ q ( s , ( C + D δ ) b ) = λ m X i = X n , n ≥ N (( x iC + D δ + n ) δ i + ( y iC + D δ + n )) − s A − si .From Shintani and Yamamoto’s evaluation of zeta function at s = [ ] ), we find an expression of the partial zeta value interms of the values of 1st and 2nd Bernoulli polynomials:(9) ζ q ( ( C + D δ ) b ) = λ m X i = − B ( x iC + D δ ) B ( x i − C + D δ ) + b i B ( x iC + D δ ) Moreover we can express x mi + jC + D δ , y mi + jC + D δ using epsilon action ∗ definedin (4). ARTIAL ZETA FUNTION 9
Lemma 3.1.
Let ε be the totally positive fundamental unit of K. Then fori > we havex mi + jC + D δ = x j ε i ∗ ( C + D δ ) and y mi + jC + D δ = y j ε i ∗ ( C + D δ ) , for j =
0, 1, 2, · · · , m − .Proof. See Lemma 2.7 in [ ] . (cid:3) From above lemma and equation (9), we obtain the following Lemma
Lemma 3.2. ζ q ( ( C + D δ ) b )= m X i = λ − X j = − B ( x i ε j ∗ ( C + D δ ) ) B ( y i ε j ∗ ( C + D δ ) ) + b i B ( x i ε j ∗ ( C + D δ ) ) .Let F δ be a set defind in Eq.(3). Lemma 3.3. If b is an integral ideal such that b − = [ δ ] with ( b , q ) = .Then for C + D δ ∈ F δ , we find thatOr b ( C + D δ ) = { ε j ∗ ( C + D δ ) | j =
0, 1, · · · λ − } , where ε is a totally positive fundamental unit of E + and λ = [ E + : E + q ] Proof.
By definition of ε ∗ in (4) we find that ε j ∗ ( C + D δ ) = C + D δ ifand only if(10) ( ε j − )( C + D δ ) ∈ q b − .Since ( q , b ( C + D δ )) =
1, (10) is equal to ε j ∈ E + q . (cid:3) Combining Lemma 3.2 and Lemma 3.3, we obtain
Proposition 3.4. ζ q ( ( C + D δ ) b ) = X ( A , B ) ∈ Or b ( C + D δ ) m X i = − B ( x iA + B δ ) B ( x i − A + B δ )+ b i B ( x iA + B δ ) .
4. P
ERIODICITY OF ORBIT
In this section, we adapt the discussion of the previous section to afamily of real quadratic fields. With the assumptions of this paper, inthe considered family, the Shintani-Zagier cone decomposition varies.This variation of decomposition is far from being periodic in q but thefundamental unit acts on F δ with period q .We restate the assumption for the family ( K n , b n ) . Let { K n = Q ( p f ( n )) } n ∈ N be a family of real quadratic fields where f ( n ) is a positive square freeinteger for each n . Suppose b n is an integral ideal relatively prime to q such that b − n = [ δ ( n )] for a δ ( n ) satisfying δ ( n ) >
2, 0 < δ ( n ) ′ < δ ( n ) − δ ( n ) − = [[ a ( n ) , · · · , a s − ( n )]] .Then it is known that the minus continued fraction expansion of δ ( n ) is δ ( n ) = (( b ( n ) , b ( n ) , · · · , b m ( n ) − ( n ))) where b i ( n ) = ¨ a j ( n ) + i = S j ( n ) S j ( n ) depending on n . S j ( n ) is defined from a i ( n ) asfollows: S j ( n ) = ¨ j = S j − ( n ) + a j − ( n ) for j ≥ m ( n ) of the minus continued fraction of δ ( n ) is(11) m ( n ) = ( S s ( n ) for even sS s ( n ) for odd s (cf. page 177-178 in [ ] , Lemma 3.1 in [ ] ).Setting µ ( s ) = s (resp. odd s ), we have m ( n ) = S µ ( s ) s ( n ) .Let ε n > K n and ε λ ( n ) n > E q ( K n ) + . Then λ ( n ) = [ E + ( K n ) : E + q ( K n )] equalsthe cardinality of the orbit of ( C , D ) ∈ F δ ( n ) as we have discussed in theprevious section. ARTIAL ZETA FUNTION 11
For each ( C , D ) ∈ F δ ( n ) pair, we have the sequence { x iC + D δ ( n ) , y iC + D δ ( n ) } i ≥− defined as in Eq.(6) of the previous section: x C + D δ ( n ) = D Dq E ,(12) y C + D δ ( n ) = Cq ,(13) x i + C + D δ ( n ) = D b i ( n ) x iC + D δ ( n ) − x i − C + D δ ( n ) E ,(14) y i + C + D δ ( n ) = − x iC + D δ ( n ) .(15)For i ≥
0, we define 0 ≤ C i ( n ) , D i ( n ) ≤ q − ε in ∗ ( C + D δ ( n )) = C i ( n ) + D i ( n ) ,with C ( n ) = C , D ( n ) = D .Using Lemma 3.1 , we have x m ( n ) iC + D δ ( n ) = x ε in ∗ ( C + D δ ( n )) = x C i ( n )+ D i ( n ) δ ( n ) = (cid:28) D i ( n ) q (cid:29) and y m ( n ) iC + D δ ( n ) = − x m ( n ) i − C + D δ ( n ) = y ε in ∗ ( C + D δ ( n )) = y C i ( n )+ D i ( n ) δ ( n ) = C i ( n ) q . Lemma 4.1.
For i ≥ , m ( n ) i = S µ ( s ) si ( n ) . Proof.
We can rewrite S µ ( s ) si ( n ) as following: i X j = s µ ( s ) X k = a ( j − ) µ ( s ) s + k − ( n ) = i X j = s µ ( s ) X k = a k − ( n ) = iS µ ( s ) s ( n ) = im ( n ) . (cid:3) Finally we obtain (cid:28) D i ( n ) q (cid:29) = x S µ ( s ) si ( n ) C + D δ ( n ) and C i ( n ) q = − x S µ ( s ) si ( n ) − C + D δ ( n ) .Note that for j > x S j ( n ) C + D δ ( n ) ( n ) and x S j ( n ) − C + D δ ( n ) are determined only by r for n = qk + r (See Property 3 in p.16 and Proposition 3.5 of [ ] ).Thus we have q -invariant property of C i ( n ) , D i ( n ) in n = qk + r : Proposition 4.2.
For n , n ′ such that n ′ = n + qk, assume ( C , D ) ∈ F δ ( n ) ∩ F δ ( n ′ ) . Then ( C i ( n ) , D i ( n )) = ( C i ( n ′ ) , D i ( n ′ )) ∈ F δ ( n ) ∩ F δ ( n ′ ) . Consequently, we see λ ( n ) = λ ( n ′ ) and E + ( n ) / E + q ( n ) ≃ E + ( n ′ ) / E + q ( n ′ ) for n ′ = n + qk . Furthermore, from the original assumption that N (( C + D δ ( n )) ( mod q ) is determined by r the residue of n , we have F δ ( n ) = F δ ( n ′ ) . This identification preserves the E + ( n ) / E + q ( n ) -action. Lemma 4.3.
If N (( C + D δ ( n )) b n ) ( mod q ) is determined by r the residueof n mod q for n = qk + r then F δ ( n ) is invariant as k varies.Proof. Use the fact (( C + D δ ( n )) b n , q ) = ( N (( C + D δ ( n ) b n ) , q ) =
1. This is again equivalent to ( C , D ) ∈ F δ ( n ) . From the q -invarianceof ( N (( C + D δ ( n )) b n ) , q ) = n , we obtain that ( C , D ) ∈ F δ ( n ) iff ( C , D ) ∈ F δ ( n ′ ) for n ′ = n + qk . (cid:3) In particular, we obtain the q -invariance in n of an orbit in F δ ( n ) as inthe following: Proposition 4.4.
If N (( C + D δ ( n )) b n ) ( mod q ) is a function dependingonly on C , D and r for n = qk + r then Or b ( C + D δ ( n )) is invariant as kvaries.
5. E
XPLICIT COMPUTATION OF THE COEFFICIENTS
Recall some calculations from Section 3.2 in [ ] . For this we definea sequence { ν iC D ( r ) } i ≥− for 0 ≤ r ≤ q −
1. As we have constrained that a i ( x ) ∈ Z [ x ] , 〈 a i ( n ) 〉 q is independent of k for n = qk + r but determinedonly by r . Thus we can define γ i ( r ) : = 〈 a i ( n ) 〉 q = 〈 a i ( r ) 〉 q .Let Γ j ( r ) : = ¨ Γ j − ( r ) + γ j − ( r ) for j ≥
10 for j = i ≥ c i ( r ) = ¨ γ j ( r ) + i = Γ j ( r ) { ν iC D ( r ) } i ≥− satisfying ν − C D ( r ) = q − Cq , ν C D ( r ) = 〈 Dq 〉 and ν i + C D ( r ) = 〈 c i ( r ) ν iC D ( r ) − ν i − C D ( r ) 〉 .When C , D are fixed and clear from the context, we denote x iC + D δ ( n ) and ν iC D ( r ) by x i ( n ) and ν i ( r ) , respectively. ARTIAL ZETA FUNTION 13
Proposition 5.1.
For j ≥ and integer n such that a j + ( n ) ≥ q, { x i ( n ) } S j ( n ) ≤ i ≤ S j + ( n ) has period q. Explicitly we havex S j ( n )+ q + i ( n ) = x S j ( n )+ i ( n ) for ≤ i ≤ a j + ( n ) − q . Proof.
See Proposition 3.4 in [ ] (cid:3) Proposition 5.2.
For integers j ≥ , if n = qk + r thenx S j ( n )+ i ( n ) = ν Γ j ( r )+ i ( r ) for ≤ i ≤ γ j + ( r ) . Proof.
See Proposition 3.5 in [ ] (cid:3) For 0 ≤ i ≤ s −
1, let a i ( x ) = P dj = α i j x j ∈ Z [ x ] . Then for 0 ≤ r ≤ q −
1, there are an unique γ i ( r ) ∈ [ q ] ∩ Z such that a i ( r ) = q τ i ( r ) + γ i ( r ) for τ i ( r ) ∈ Z . Lemma 5.3.
If n = qk + r, then we havea i ( n ) = q d X m = A im ( r ) k m + q τ i ( r ) + γ i ( r ) , where A im ( r ) = P dj = m α i j (cid:18) jm (cid:19) q m − r j − m .We recall that { x iC + D δ ( n ) } S j ( n ) ≤ i ≤ S j + ( n ) is and arithmetic progressionmod Z with difference 〈 x S j ( n )+ C + D δ ( n ) − x S j ( n ) C + D δ ( n ) 〉 [ See Proposition 3.2 in [ ]] .From proposition 5.2, we find that if n = qk + r then(16) 〈 x S j ( n )+ C + D δ ( n ) − x S j ( n ) C + D δ ( n ) 〉 = 〈 ν Γ j ( r )+ C D ( r ) − ν Γ j ( r ) C D ( r ) 〉 .Let d jC D ( r ) : = 〈 ν Γ j ( r )+ C D ( r ) − ν Γ j ( r ) C D ( r ) 〉 .Now we will express the value ζ ( ( C + D δ ( n )) b ( n )) using q k r where n = qk + r . We note that form Proposition 3.4 ζ q ( ( C + D δ ( n )) b ( n ))= X ( A , B ) ∈ Or b ( C + D δ ( n )) m ( n ) X i = − B ( x iA + B δ ( n ) ) B ( x i − A + B δ ( n ) ) + b i ( n ) B ( x iA + B δ ( n ) ) .(17)For simplification, we define F ( x , y ) : = − B ( x ) B ( y ) + B ( x ) = ( x − )( − y ) + x − x +
16 .
Because b i ( n ) = i = S j ( n ) for some j , we can find that m ( n ) X i = − (cid:0) B ( x iC + D δ ( n ) ) B ( y iC + D δ ( n ) ) + b i ( n ) B ( x iC + D δ ( n ) ) (cid:1) = s µ ( s ) X l = (cid:0) − B ( x S l ( n ) C + D δ ( n ) ) B ( x S l ( n ) − C + D δ ( n ) ) + a l ( n ) + B ( x S l ( n ) C + D δ ( n ) ) (cid:1) + s µ ( s ) − X l = S l + ( n ) − X i = S l ( n )+ F ( x iC + D δ ( n ) , x i − C + D δ ( n ) ) (18)From the fact that { x iC + D δ ( n ) } S j ( n ) ≤ i ≤ S j + ( n ) is and arithmetic progressionmod Z with difference d jC D ( r ) for n = qk + r , we obtain the following:If 1 ≤ γ ≤ q and a l + ( n ) ≥ γ then S l ( n )+ γ X i = S l ( n )+ F ( x i ( n ) , x i − ( n ))= (cid:16) ( γ d l ( r ) + ( − d l ( r ))[ ν Γ l ( r ) ( r ) + d l ( r ) γ ] + B ( x S l ( n )+ γ ( n )) − B ( x S l ( n ) ( n ))) − γ (cid:17) (19)Moreover if γ = q then from the periodicity of x i ( n ) [ See Proposition5.1 ] , we have the periodicity of the values of the 2nd Bernoulli polyno-mial: B ( x S l ( n )+ q ( n )) = B ( x S l ( n ) ( n )) .Thus(20) S l ( n )+ q X i = S l ( n )+ F ( x i ( n ) , x i − ( n )) = (cid:16) ( qd l ( r ) +( − d l ( r ))[ ν Γ l ( r ) ( r )+ d l ( r ) q ] ) − q (cid:17) .We note that Or b ( C + D δ ( n )) is a set depending only C , D and r for n = qk + r [ See Proposition 4.4 ] under the conditions of the followingproposition. Thus for n = qk + r we can define Or b ( C + D δ ( n )) = : Or b
C D ( r ) . Proposition 5.4. If b − n = [ δ ( n )] and δ ( n ) − = [[ a ( n ) , a ( n ) , · · · a s − ( n )]] for a i ( x ) = P dj = α i j x j ∈ Z [ x ] and if N (( C + D δ ( n )) b n ) ( mod q ) is afunction only depending on C , D and r then we have for n = qk + r ζ q (( C + D δ ( n )) b n ) = X ( A , B ) ∈ Or b CD ( r ) B AB ( r ) + B AB ( r ) k + · · · B dAB ( r ) k d , ARTIAL ZETA FUNTION 15 where for m ≥ B mAB ( r ) = q s µ ( s ) X l = A l , m ( r ) B ( ν Γ l ( r ) AB ( r ))+ s µ ( s ) − X l = A l + m ( r ) (cid:16) ( qd lAB ( r ) + ( − d lAB ( r ))[ ν Γ l ( r ) AB ( r ) + d lAB ( r ) q ] ) − q (cid:17) andB AB ( r ) = s µ ( s ) X l = − B ( ν Γ l ( r ) AB ( r )) B ( ν Γ l ( r ) − AB ( r )) + q τ l ( r ) + γ l ( r ) + B ( ν Γ l ( r ) AB ( r ))+ s µ ( s ) − X l = h τ l + ( r ) (cid:16) ( qd lAB ( r ) + ( − d lAB ( r ))[ ν Γ l ( r ) AB ( r ) + d lAB ( r ) q ] ) − q (cid:17) + B ( ν Γ l + ( r ) − AB ( r )) − B ( ν Γ l ( r ) AB ( r )) + (cid:16) ( γ l + ( r ) − ) d lAB ( r ) + ( − d lAB ( r ))[ ν Γ l ( r ) AB ( r ) + d lAB ( r )( γ l + ( r ) − )] (cid:17) − γ l + ( r ) + i . Proof.
Thus from equation (18) and lemma 5.3, we obtain the following: m ( n ) X i = (cid:0) B ( x iA + B δ ( n ) ) B ( y iA + B δ ( n ) ) + b i ( n ) B ( x iA + B δ ( n ) ) (cid:1) = s µ ( s ) X l = [ − B ( x S l ( n ) A + B δ ( n ) ) B ( x S l ( n ) − A + B δ ( n ) )+ q P dm = A l , m ( r ) k m + q τ l ( r ) + γ l ( r ) + B ( x S l ( n ) A + B δ ( n ) )]+ s µ ( s ) − X l = S l ( n )+ q P dm = A l + m ( r ) k m + q τ l + ( r )+ γ l + ( r ) − X i = S l ( n )+ F ( x iA + B δ ( n ) , x i − A + B δ ( n ) ) (21)Since { F ( x iA + B δ ( n ) , x i − A + B δ ( n ) ) } S l ( n )+ ≤ i ≤ S l + ( n ) − has period q , we have thefollowing: S l ( n )+ q P dm = A l + m ( r ) k m + q τ l + ( r )+ γ l + ( r ) − X i = S l ( n )+ F ( x iA + B δ ( n ) , x i − A + B δ ( n ) )=( d X m = A l + m ( r ) k m + τ l + ( r )) S l ( n )+ q X i = S l ( n )+ F ( x iA + B δ ( n ) , x i − A + B δ ( n ) )+ S l ( n )+ γ l + ( r ) − X i = S l ( n )+ F ( x iA + B δ ( n ) , x i − A + B δ ( n ) ) (22)We note that(23) x S l ( n ) A + B δ ( n ) = ν Γ l ( r ) AB ( r ) (24) x S l ( n ) − A + B δ ( n ) = ν Γ l ( r ) − AB ( r ) (25) x S l ( n )+ γ l + ( r ) − A + B δ ( n ) = ν Γ l + ( r ) − AB ( r ) Form (19),(20) and (23)-(25), we find that (22) is equal to the follow-ing: h d X m = A l + m ( r ) k m + τ l + ( r ) i · h ( qd lAB ( r ) + ( − d lAB ( r ))[ ν Γ l ( r ) AB ( r ) + d lAB ( r ) q ] ) − q i + B ( ν Γ l + ( r ) − AB ( r )) − B ( ν Γ l + ( r ) AB ( r )) + (cid:16) ( γ l + ( r ) − ) d lAB ( r ) + ( − d lAB ( r ))[ ν Γ l ( r ) AB ( r ) + d lAB ( r )( γ l + ( r ) − )] (cid:17) − γ l + ( r ) + (cid:3) Proof of Theorem 2.4
We note that ν Γ l ( r ) AB ( r ) , ν Γ l ( r ) − AB ( r ) and d lAB ( r ) ∈ q Z . Thus for i =
0, 1, · · · , d , B iAB ( r ) ∈ q Z .From Proposition 2.3, we complete the proof of theorem 2.4. ARTIAL ZETA FUNTION 17
6. T
WO EXAMPLES
We present here two examples of ( K n , b n ) showing polynomial be-haviour of ζ q ( b n ) . For both examples, we take b n = ( C + D δ ( n )) O K n ,for C + D δ ( n ) ∈ F δ ( n ) . The first one is a family of real quadratic fieldsalready appeared in a literature dealing the associated class number oneproblem. The second family is given by a quartic polynomial .6.1. Case 1: f ( n ) = n + . The following example is one of so-calledRichad-Degert type. The quasi-linearity was studied in [ ] to solvethe class number one problem for the family. From the computation ofpartial zeta values for ray class in this article, one can recover exactlythe result in loc.cit. for partial Hecke’s L-values associated to a mod- q Dirichlet character χ .For square free f ( n ) = n +
2, let K n = Q ( p f ( n ) . We fix b n = O K n thering of integers in K . O K n = [ δ ( n )] and • δ ( n ) = p f ( n ) + n + • The continued fraction of δ ( n ) − [[ n , n ]] . • The totally positive fundamental unit ε n is n + + n p f ( n ) .One can easily check that N ( C + D δ ( n )) is invariant for n modulo q .Let n = qk + r . We can describe the orbit of C + D δ ( n ) by the action of ε n as follows: ε n ∗ ( C i ( n ) + D i ( n ) δ ( n )) = C i + ( n ) + D i + ( n ) δ ( n ) ,where C ( n ) = C , D ( n ) = DC i + ( n ) = ¬ C i ( n )( n + ) + D i ( n )( n + n + n + ) ¶ q D i + ( n ) = ¬ C i ( n ) n + D i ( n )( n + n + ) ¶ q One sees that
Or b ( C + D δ ( n )) depends only on C , D and r . For the sakeof simplicity and from the periodicity of the orbit, we may well denote Or b ( C + D δ ( n )) by Or b
C D ( r ) .Now we can express the partial zeta values at s = ζ q ( ( C + D δ ( n )) O K n ) = A ( r ) + A ( r ) k + · · · A d ( r ) k d ,where A i ( r ) = X ( A , B ) ∈ Or b CD ( r ) B iAB ( r ) , for B AB ( r ) = ( − ν Γ ( r ) AB ( r ))( ν Γ ( r ) − AB ( r ) − ) + r + ( ν Γ ( r ) AB ( r ) − ν Γ ( r ) AB ( r ) + )+ τ ( r ) ( qd AB ( r ) − qd AB ( r ) + qd AB ( r ) − q )+ ( ν Γ ( r ) − AB ( r ) − ν Γ ( r ) − AB ( r ) + ) − ( ν AB ( r ) − ν AB ( r ) + )+ ( γ ( r ) − ) d AB ( r ) + − d AB ( r ) ( d AB ( r ) γ ( r ) − d AB ( r )+ ν AB ( r ) − ν Γ ( r ) AB ( r )) − γ ( r ) −
112 , B AB ( r ) = q ( ν Γ ( r ) AB ( r ) − ν Γ ( r ) AB ( r ) + )+ q ( qd AB ( r ) + qd AB ( r ) − qd AB ( r ) − q ) and d AB ( r ) = 〈 ( r + ) B + A 〉 q q ν AB ( r ) = 〈 B 〉 q q ν Γ ( r ) AB ( r ) = (cid:10) ( r − r ) B + ( r − ) A (cid:11) q q ν Γ ( r ) − AB ( r ) = (cid:10) ( r − r − ) B + ( r − ) A (cid:11) q q γ ( r ) = 〈 r + 〉 q τ ( r ) = r + − 〈 r + 〉 q q .6.2. Case 2: f ( n ) = n + n + n + . For square free f ( n ) = n + n + n + n +
3, let K n : = Q ( p f ( n )) . Let us fix b n = O K n .If O K n is [ δ ( n )] for δ ( n ) described as before. For this family, we have: • δ ( n ) = p f ( n ) + [ p f ( n )] + • δ ( n ) − = [[ n + n +
2, 2 n + ]] . • The totally positive fundamental unit ε n is ( n + ) + + ( n + ) p f ( n ) . ARTIAL ZETA FUNTION 19
We can again easily check that N ( C + D δ ( n )) is invariant modulo q for n = qk + r .For C + D δ ( n ) ∈ F δ ( n ) , we have ε n ∗ ( C i ( n ) + D i ( n ) δ ( n )) = C i + ( n ) + D i + ( n ) δ ( n ) ,where C ( n ) = C , D ( n ) = DC i + ( n ) = 〈 ( n + n + n + n + n + ) D i ( n )+( n + n + n + ) C i ( n ) 〉 q D i + ( n ) = ¬ ( n + n + n + ) D i ( n ) + ( n + ) C i ( n ) ¶ q Let n = qk + r for 0 ≤ r < q . Denoting Or b ( C + D δ ( n )) by Or b
C D ( r ) ,the partial zeta value at 0 is ζ q ( ( C + D δ ( n )) O K n ) = A ( r ) + A ( r ) k + · · · + A d ( r ) k d ,where A i ( r ) = P ( A , B ) ∈ Or b CD ( r ) B iAB ( r ) , for B AB ( r ) =( − ν Γ ( r ) AB ( r ))( ν Γ ( r ) − AB ( r ) − )+ r + r + ( ν Γ ( r ) AB ( r ) − ν Γ ( r ) AB ( r ) + )+ τ ( r ) ( qd AB ( r ) − qd AB ( r ) + qd AB ( r ) − q )+ ( ν Γ ( r ) − AB ( r ) − ν Γ ( r ) − AB ( r ) + ) − ( ν AB ( r ) − ν AB ( r ) + ) + ( γ ( r ) − ) d AB ( r ) + − d AB ( r ) ( d AB ( r ) γ ( r ) − d AB ( r ) + ν AB ( r ) − ν Γ ( r ) AB ( r )) − γ ( r ) −
112 , B AB ( r ) = q + q r ( ν Γ ( r ) AB ( r ) − ν Γ ( r ) AB ( r ) + )+ q ( qd AB ( r ) + qd AB ( r ) − qd AB ( r ) − q ) , B AB ( r ) = q ( ν Γ ( r ) AB ( r ) − ν Γ ( r ) AB ( r ) + ) , and d AB ( r ) = (cid:10) ( r + r + ) B + A (cid:11) q q ν AB ( r ) = 〈 B 〉 q q ν Γ ( r ) AB ( r ) = (cid:10) ( r + r + ) B + rA (cid:11) q q ν Γ ( r ) − AB ( r ) = (cid:10) r B + ( r − ) A (cid:11) q q γ ( r ) = 〈 r + 〉 q τ ( r ) = r + − 〈 r + 〉 q q . Remark 6.1.
The family of 6.2 has not been touched in literature in thecontext of class number problem or other particular problems in arithmetic.Especially, most known families of real quadratic fields, where class numberproblems are solved, are Richaud-Decherd type which is generated by somequadratic polynomials. It would be highly interesting if one could answerthose questions for other types of families than R-D types. R EFERENCES [ ] Barnes, E.W.,
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