On a secant Dirichlet series and Eichler integrals of Eisenstein series
aa r X i v : . [ m a t h . N T ] J un ON A SECANT DIRICHLET SERIES AND EICHLERINTEGRALS OF EISENSTEIN SERIES
BRUCE C. BERNDT AND ARMIN STRAUB
Abstract.
We consider, for even s , the secant Dirichlet series ψ s ( τ ) = P ∞ n =1 sec( πnτ ) n s ,recently introduced and studied by Lal´ın, Rodrigue and Rogers. In particular,we show, as conjectured and partially proven by Lal´ın, Rodrigue and Rogers,that the values ψ m ( √ r ), with r > π m .We then put the properties of the secant Dirichlet series into context by show-ing that they are Eichler integrals of odd weight Eisenstein series of level 4.This leads us to consider Eichler integrals of general Eisenstein series and todetermine their period polynomials. In the level 1 case, these polynomials wererecently shown by Murty, Smyth and Wang to have most of their roots on theunit circle. We provide evidence that this phenomenon extends to the higherlevel case. This observation complements recent results by Conrey, Farmerand Imamoglu as well as El-Guindy and Raji on zeros of period polynomialsof Hecke eigenforms in the level 1 case. Finally, we briefly revisit results of asimilar type in the works of Ramanujan. Introduction
Our considerations begin with the secant Dirichlet series(1) ψ s ( τ ) = ∞ X n =1 sec( πnτ ) n s , which were recently introduced and studied by Lal´ın, Rodrigue and Rogers [LRR14].One of the motivations for considering these sums is their similarity in shape and,as we will see, in properties to the cotangent Dirichlet series(2) ξ s ( τ ) = ∞ X n =1 cot( πnτ ) n s . For instance, as first proved by Lerch and also recorded by Ramanujan (see Section8 as well as [Ber77] or [Ber89, p. 276]) the difference τ m ξ m +1 ( − /τ ) − ξ m +1 ( τ )is a rational function in τ . This modular functional equation and its ramifica-tions continue to inspire research to this date, including, for instance, [GMR11],[MSW11], [Riv12], [LS13].As shown in [LRR14], the secant Dirichlet series ψ m ( τ ) satisfy modular func-tional equations as well. In Section 2, we give an alternative derivation of thesefunctional equations based on the residue theorem and in the spirit of [Ber76]. Inthis way, we obtain a compact representation of the associated rational function asa certain Taylor coefficient of a quotient of trigonometric functions. We then show Date : June 7, 2014.2010
Mathematics Subject Classification.
Primary 11F11, 33E20; Secondary 11L03, 33B30.The first author’s research was partially supported by NSA grant H98230-11-1-0200. in Section 3, as conjectured and partially proven in [LRR14], that, for a positiverational number r , ψ m ( √ r ) is a rational multiple of π m .In Section 4, we observe that the 2 m th derivative of ψ m (2 τ ) is, up to a constantterm, an Eisenstein series of weight 2 m + 1, level 4 and character χ − . In otherwords, the secant Dirichlet series ψ m are Eichler integrals of Eisenstein series; thebasic theory of Eichler integrals is reviewed in Section 5. In this light, several ofthe properties of the function ψ m become natural and, in Section 6, we use themodular setting to give another derivation of the functional equation by evaluatingthe period polynomial of the corresponding Eisenstein series.In fact, the computation of the period polynomials in Section 6 is carried outmore generally for Eisenstein series corresponding to pairs of Dirichlet characters. Aspecial case of this computation allows us, for instance, to rederive the Ramanujan-style formulas for Dirichlet L -values of [Kat74]. The resulting period polynomialsmirror the well-known polynomials occuring in the level 1 case, studied, for in-stance, in [GMR11] and [MSW11], where they (more accurately, their odd parts)are referred to as the Ramanujan polynomials . In [MSW11] it was shown that theRamanujan polynomials are nearly unimodular, that is, all their nonreal roots lieon the unit circle. On the other hand, it was conjectured in [LR13], and provenin [LS13], that the full period polynomial is unimodular in the level 1 case, andthis property is also shown to hold for several related polynomials. We indicate inSection 7 that, after a linear change of variables, the period polynomials in higherlevel appear to have all or most of their roots on the unit circle as well. This ob-servation fits well with and complements the recent result of [CFI12] and [EGR13],where it is shown that the nontrivial zeros of period polynomials of modular forms,which are Hecke eigenforms of level 1, all lie on the unit circle. As an application,we derive formulas for Dirichlet L -values in terms of values of an Eichler integralat algebraic arguments of modulus 1, thus generalizing the formulas for ζ (2 k + 1)studied in [GMR11].Finally, in Section 8, we return to and discuss related entries in Ramanujan’snotebook [Ber89], including the particularly famous entry corresponding to thecotangent Dirichlet series (2), which, for odd s , are the Eichler integrals of theEisenstein series for the full modular group. In particular, we close by demonstrat-ing that the functional equation for ψ m ( τ ) is in fact a consequence of an identitystated by Ramanujan.It is on purpose, and hopefully to the benefit of some readers, that the consid-erations in this paper start out entirely explicit and elementary, with theoreticalbackground, such as Eisenstein series of higher level and Eichler integrals, beingincluded as we proceed. As a consequence of this approach, some of the earlierresults can be obtained as special cases of later results.Convergence of series such as (1) and (2), when τ is a real number, is a rathersubtle issue; see, for instance, [Riv12]. It is shown in [LRR14] that ψ s ( τ ), for s > τ with odd denominator as well as for algebraicirrational τ . On the other hand, the series converges absolutely for all nonreal τ ,and our discussion of the modular properties proceeds under the tacit assumptionthat τ is not real. In the case of evaluations of ψ m ( τ ) at real quadratic τ in Section3, one may then use limiting arguments to show that the functional equations alsohold for these arguments. N A SECANT DIRICHLET SERIES 3 The functional equation of the secant Dirichlet series viaresidues
Obviously, ψ s ( τ ) is periodic of period 2, that is ψ s ( τ +2) = ψ s ( τ ). In this section,we present an alternative proof of the additional nontrivial functional equationsatisfied by (1) in the case s = 2 m . In [LRR14] this functional equation wasestablished by skillfully and carefully convoluting partial fraction expansions. Theproof given here is based on the residue theorem and is in the spirit of the proofgiven in [Ber76] for the cotangent Dirichlet series (2). Throughout, we denote with[ z n ] f ( z ) the n th coefficient of the Taylor expansion of f ( z ). Theorem 2.1.
Let m > be an integer. Then we have (1 + τ ) m − ψ m (cid:18) τ τ (cid:19) − (1 − τ ) m − ψ m (cid:18) τ − τ (cid:19) = π m (cid:2) z m − (cid:3) sin( τ z )sin((1 − τ ) z ) sin((1 + τ ) z ) . (3) Proof.
Let s > I N = 12 πi Z C N sin (cid:0) π b − a z (cid:1) sin( πaz ) sin( πbz ) d zz s +1 , where C N is a positively oriented circle of radius R N centered at the origin. As in[Ber76], the radii R N are chosen such that the points on the circle are always atleast some fixed positive distance from any of the points na and nb , where n rangesover the integers. It is then easily seen that(5) lim N →∞ I N = 0 . The integrand of (4) has poles at z = 0 as well as at z = na and z = nb for n ∈ Z .Writing Res( α ) for the residue of the integrand at z = α , we haveRes (cid:16) na (cid:17) = a s ( − n π n s +1 sin (cid:0) πn b − a a (cid:1) sin (cid:0) πn ba (cid:1) = a s π sec (cid:0) πn b − a a (cid:1) n s +1 , where the last equality is obtained by writing ( − n = πn ) and using the trigono-metric identitycos( x ) sin( y ) − cos( y ) sin( x ) = 2 cos (cid:18) y − x (cid:19) sin (cid:18) y − x (cid:19) , with x = πn and y = πn ba . By symmetry,Res (cid:16) nb (cid:17) = − b s π sec (cid:0) πn b − a b (cid:1) n s +1 . For the residue at the origin, we find thatRes(0) = [ z s ] sin (cid:0) π b − a z (cid:1) sin( πaz ) sin( πbz ) = π s [ z s ] sin (cid:0) b − a z (cid:1) sin( az ) sin( bz ) =: π s p s ( a, b ) . We now assume that s is odd. Collecting residues and letting N → ∞ , we obtain,using (5), 0 = a s π ∞ X n =1 sec (cid:0) πn b − a a (cid:1) n s +1 − b s π ∞ X n =1 sec (cid:0) πn b − a b (cid:1) n s +1 + π s p s ( a, b ) . BRUCE C. BERNDT AND ARMIN STRAUB
Finally, let a = 1 − τ and b = 1 + τ , and write s = 2 m −
1, to obtain the functionalequation in the form(1 + τ ) m − ψ m (cid:18) τ τ (cid:19) − (1 − τ ) m − ψ m (cid:18) τ − τ (cid:19) = π m p m − (1 − τ, τ ) , as claimed. (cid:3) Note that, upon replacing τ → ττ +1 and simplifying slightly, we find that equation(3) becomes(6) (2 τ + 1) m − ψ m (cid:18) τ τ + 1 (cid:19) − ψ m ( τ ) = π m (cid:2) z m − (cid:3) sin( τ z )sin( z ) sin((2 τ + 1) z ) , which, while less symmetric, makes the modular transformation property moreapparent. Remark 2.2.
The rational functions on the right-hand sides of (3) and (6) can bemade more explicit by expanding the sine functions. Namely, if we expand the twosine functions in the denominator using the defining generating function,(7) ze xz e z − X n > B n ( x ) z n n ! , of the Bernoulli polynomials B n ( x ), then the rational functions are seen to be equalto double sums such as (cid:2) z m − (cid:3) sin (cid:0) b − a z (cid:1) sin( az ) sin( bz ) = ( − m ab X k + n + r = m (cid:0) b − a (cid:1) k +1 B n ( ) B r ( )(2 a ) n (2 b ) r (2 k + 1)!(2 n )!(2 r )! , where the sum is over nonnegative integers k, n, r summing to m . We remark that B n ( ) = − (1 − − n ) B n , where B n = B n (0) is the n th Bernoulli number. Notethat this shows, in particular, that the right-hand sides of equations (3) and (6)are rational functions of the form π m p m ( τ ) / (1 − τ ) and π m q m ( τ ) / (2 τ + 1),respectively, where p m and q m are polynomials of degree 2 m + 1 with rationalcoefficients.In fact, as we will see from versions of the functional equation derived later inthis paper, the mentioned double sums can be reduced to single summations; seeRemark 6.7.3. The secant Dirichlet series at real quadratic irrationals
In [LRR14] it was conjectured and partially proven that ψ m ( √ k ) is a rationalmultiple of π m whenever m and k are positive integers. In this section, we provethis assertion and extend it to the case when k is a rational number. For the sakeof simplicity, we first prove, in Theorem 3.2, the case of integral k and then brieflyindicate how the more general case, stated in Theorem 3.4, follows in essentiallythe same way. We point out that an independent proof, based on the theoryof generalized η -functions, of Theorem 3.4 has been given by P. Charollois andM. Greenberg [CG14]. We thank M. Lal´ın, who received preprints of both [CG14]and this paper within a couple of days, for making us aware of this reference.Consider the matrices(8) A = (cid:18) (cid:19) , B = (cid:18) (cid:19) . N A SECANT DIRICHLET SERIES 5
As usual, we denote by Γ(2) the congruence subgroupΓ(2) = { γ ∈ SL ( Z ) : γ ≡ I (mod 2) } , where I is the identity matrix. The group Γ(2) is generated by A , B and − I ; see,for instance, [Yos97, Exercise II.6, p.34]. Remark 3.1.
We note that, as is well-known, Γ(2) is conjugate to Γ (4) under τ τ , and that in this setting the group h A, B i corresponds to Γ (4).In terms of the matrices A and B , we then have(9) ψ m ( Aτ ) = ψ m ( τ ) , ψ m ( Bτ ) = 1(2 τ + 1) m − ψ m ( τ ) + π m f m ( τ ) , where f m ( τ ), given in (6), is a rational function over Q .Note that the first equation in (9) simply expresses the periodicity of ψ m , that is ψ m ( τ + 2) = ψ m ( τ ), while the second one is the functional equation (6) discussedin the previous section. Theorem 3.2.
Let m and k be positive integers. Then ψ m ( √ k ) ∈ π m Q . Proof.
Following [LRR14], we observe that if the integers X and Y solve Pell’sequation(10) X − kY = 1 , then D · √ k = √ k , where D is the matrix D = (cid:18) X kYY X (cid:19) ∈ SL ( Z ) . Here, and in the sequel, we let 2 × (cid:18) a bc d (cid:19) · x = ax + bcx + d . A proof that for every positive nonsquare k there exist nontrivial solutions X , Y to Pell’s equation (10) was first published by Lagrange in 1768 [Lag92]. For furtherinformation and background on Pell’s equation we refer to [Len02].We now make the simple observation that, by (10), D = (cid:18) X kYY X (cid:19) = (cid:18) X + kY kXY XY X + kY (cid:19) ∈ Γ(2) . Hence, we can always find C ∈ h A, B, − I i = Γ(2), C = ± I , such that C · √ k = √ k .Let C = (cid:18) a bc d (cid:19) be such a matrix. Note that necessarily c = 0.Repeatedly applying the functional equations (9), we deduce that ψ m ( √ k ) = ψ m ( C · √ k ) = 1( c √ k + d ) m − ψ m ( √ k ) + π m f m,C ( √ k ) , where f m,C ( τ ) ∈ Q ( τ ) is a rational function with rational coefficients. Conse-quently, ψ m ( √ k ) = π m g k ( √ k ) BRUCE C. BERNDT AND ARMIN STRAUB for some rational function g k ( τ ) ∈ Q ( τ ). On the other hand, the above argumentapplied to −√ k shows that ψ m ( −√ k ) = π m g k ( −√ k ). Since ψ m is even, itfollows that g k ( √ k ) = g k ( −√ k ) , which implies that g k ( √ k ) is a rational number. (cid:3) Example 3.3.
We now illustrate Theorem 3.2 by evaluating ψ ( √ X, Y ) = (3 , √
2. Then C = (cid:18) (cid:19) = − AB − A, and, indeed, one easily verifies that √ AB − A . In the present case s = 1, and the transformations (9) satisfied by ψ take the form(11) ψ ( Aτ ) = ψ ( τ ) , ψ ( Bτ ) = 12 τ + 1 ψ ( τ ) + π f ( τ ) , with f ( τ ) = τ (3 τ + 4 τ + 2)6(2 τ + 1) . We therefore find that ψ ( AB − Aτ ) = ψ ( B − Aτ )= (2 B − Aτ + 1) (cid:2) ψ ( Aτ ) − π f ( B − Aτ ) (cid:3) = − τ + 3 ψ ( τ ) − ( τ + 2)(3 τ + 8 τ + 6)6(2 τ + 3) π . For the second equality we applied the second transformation of (11) with B − Aτ in place of τ , while for the third equality we use the fact that B − Aτ = − τ +22 τ +3 .For τ = √ ψ ( √
2) = (2 √ − ψ ( √
2) + 23 ( √ − π , which has the solution ψ ( √
2) = − π , in agreement with the value given in[LRR14]. Families of more general explicit evaluations of ψ m at real quadraticirrationalities are derived in Example 6.8.In fact, it is the case that ψ m ( √ r ) is a rational multiple of π m whenever r is arational number. This may be shown in essentially the same fashion, as we indicatenext. Theorem 3.4.
Let m be a positive integer and r a positive rational number suchthat √ r is not a rational number with an even denominator. Then ψ m ( √ r ) ∈ π m Q . Proof.
Write r = ab , where a and b are positive integers. Similarly to the proofof Theorem 3.2, we observe that if the integers X and Y solve Pell’s equation X − abY = 1, then D · √ r = √ r , where D is the matrix D = (cid:18) X aYbY X (cid:19) ∈ SL ( Z ) . N A SECANT DIRICHLET SERIES 7
Again, one observes that D ∈ Γ(2). Hence we can find C ∈ h A, B, − I i = Γ(2), C = ± I , such that C · √ r = √ r .It remains to proceed exactly as in the proof of Theorem 3.2. (cid:3) Remark 3.5.
We remark that (parts of) the proofs of Theorems 3.2 and 3.4 alsoapply to other functions that satisfy a modular transformation which is of a similarform as the one for the secant Dirichlet series ψ m . This includes the class of Eichlerintegrals to which, as discussed in the Section 5, ψ m belongs.4. Eisenstein series
We begin this section by observing a natural relation between the secant Dirichletseries ψ m and Eisenstein series. Specifically, we note that the 2 m th derivative of ψ m is, essentially, an Eisenstein series of weight 2 m +1 and level 4. In the languageof Section 5, this means that the secant Dirichlet series are Eichler integrals ofcertain Eisenstein series. In order to investigate Eichler integrals of Eisensteinseries in general, we recall basic facts about Eisenstein series in this section.Throughout, we write D = dd τ and q = e πiτ . For n >
0, we denote with E n the n th Euler number, defined by(12) sech x = ∞ X n =0 E n x n n ! , where | x | < π . Lemma 4.1.
We have D m [ ψ m ( τ / m )! π X ′ k,j ∈ Z χ − ( j )( kτ + j ) m +1 − ( − m E m π m m +1 , where χ − = ( − · ) is the nonprincipal Dirichlet character modulo (that is, χ − ( n ) =0 for even n , and χ − ( n ) = ( − ( n − / for odd n ).Proof. In light of the partial fraction expansion of the secant functionsec (cid:16) πτ (cid:17) = 4 π X j > χ − ( j ) jj − τ = lim N →∞ π N X j = − N χ − ( j ) τ + j , we derive that D k sec (cid:16) πτ (cid:17) = 2( − k k ! π X j ∈ Z χ − ( j )( τ + j ) k +1 . Consequently, D m X n > sec (cid:0) πnτ (cid:1) n m = 2(2 m )! π X k > X j ∈ Z χ − ( j )( kτ + j ) m +1 = (2 m )! π X ′ k,j ∈ Z χ − ( j )( kτ + j ) m +1 − m )! π L ( χ − , m + 1) , (13)where L ( χ − , s ) = ∞ X n =1 χ − ( n ) n s = ∞ X n =0 ( − n (2 n + 1) s BRUCE C. BERNDT AND ARMIN STRAUB is the Dirichlet L -series attached to χ − , also known as the Dirichlet beta function.Using Euler’s well-known evaluation [Ayo74](14) L ( χ − , m + 1) = 12 ( − m E m (2 m )! (cid:16) π (cid:17) m +1 in (13), we complete the proof of Lemma 4.1. (cid:3) Example 4.2.
In the case m = 1, we find that D ψ ( τ /
2) = 2 π X ′ k,j ∈ Z χ − ( j )( kτ + j ) − π , since L ( χ − ,
3) = π . The Eisenstein series X ′ k,j ∈ Z χ − ( j )(4 kτ + j ) = π (cid:20) − q − q + 8 q − q − q + · · · (cid:21) is a modular form of weight 3, level 4 and character χ − .Define, as in [Miy06, Chapter 7], the Eisenstein series(15) E k ( τ ; χ, ψ ) = X ′ m,n ∈ Z χ ( m ) ψ ( n )( mτ + n ) k , where k >
2, and χ and ψ are Dirichlet characters modulo L and M , respectively.As detailed in [Miy06, Chapter 7], these Eisenstein series can be used to generateall Eisenstein series with respect to any congruence subgroup. Example 4.3.
By Lemma 4.1, the secant Dirichlet series ψ is connected with thecase ψ = χ − and χ = 1, the principal character modulo 1. To be precise,(16) D m [ ψ m ( τ / m )! π [ E m +1 ( τ ; 1 , χ − ) − E m +1 ( i ∞ ; 1 , χ − )] . That the constant term on the right-hand side indeed agrees with the one stated inLemma 4.1 will become clear from the facts about the Eisenstein series E k ( τ ; χ, ψ )which we review next. Example 4.4.
As detailed in Section 8.1, the cotangent Dirichlet series ξ s ( τ ),introduced in (2), is in a similar way related to the Eisenstein series E m ( τ ; 1 , χ ( − ψ ( −
1) = ( − k since, otherwise, E k ( τ ; χ, ψ ) = 0. In order to derive the Fourier expansion of the Eisenstein series, werecall that, by the character analogue of the Lipschitz summation formula [Ber75a],for any primitive Dirichlet character ψ of modulus M , ∞ X n = −∞ ψ ( n )( τ + n ) s = G ( ψ ) ( − πi/M ) s Γ( s ) ∞ X m =1 ¯ ψ ( m ) m s − e πimτ/M . N A SECANT DIRICHLET SERIES 9
Here, and in the sequel, G ( ψ ) = P Ma =1 ψ ( a ) e πia/M denotes the Gauss sum associ-ated with ψ . If ψ is primitive, we thus find that E k ( τ ; χ, ψ ) = a ( E k ) + 2 ∞ X m =1 χ ( m ) X n ∈ Z ψ ( n )( mτ + n ) k = a ( E k ) + 2 G ( ψ ) ( − πi/M ) k Γ( k ) ∞ X m =1 χ ( m ) ∞ X n =1 ¯ ψ ( n ) n k − e πinmτ/M = a ( E k ) + A ∞ X n =1 a n ( E k ) e πinτ/M , (17)where A = 2 G ( ψ )( − πi/M ) k / Γ( k ) and a ( E k ) = (cid:26) L ( k, ψ ) , if χ = 1,0 , otherwise, a n ( E k ) = X d | n χ ( n/d ) ¯ ψ ( d ) d k − . We note that the series in (17) converges for any complex value of k , and hence(17) provides the analytic continuation of (15) to the entire complex k -plane. Example 4.5.
In light of Example 4.3, we therefore find that the q -expansion ofthe secant Dirichlet series is given by(18) ψ m (2 τ ) = 2 X n > X d | n χ − ( d ) d m q n n m . Recall that the L -function of a modular form f ( τ ) = P ∞ n =0 b ( n ) e πinτ/λ is de-fined as(19) L ( f, s ) = (2 π ) s Γ( s ) Z ∞ [ f ( iτ ) − f ( i ∞ )] τ s − d τ = λ s ∞ X n =1 b ( n ) n s . As another consequence of (17), if ψ is primitive, the L -function of E ( τ ) = E k ( τ ; χ, ψ )is given by(20) L ( E, s ) = AM s L ( χ, s ) L ( ¯ ψ, − k + s ) . Depending on the parity of s , the values of L ( E, s ) can be evaluated in terms ofgeneralized Bernoulli numbers B n,χ , which are defined by(21) ∞ X n =0 B n,χ x n n ! = L X a =1 χ ( a ) xe ax e Lx − , if χ is a Dirichlet character modulo L . We observe that, for n = 1, the classicalBernoulli numbers B n are equal to B n,χ with χ = 1. Similarly, the Euler numbersare connected with the case χ = χ − :(22) 12 E m (2 m )! = − B m +1 ,χ − (2 m + 1)! . Generalized Bernoulli numbers are intimatly related to values of Dirichlet L -series.Let n > χ a primitive Dirichlet character of conductor L such that χ ( −
1) = ( − n . Then, as detailed, for instance, in [Miy06, Thm. 3.3.4], L ( n, χ ) = ( − n − G ( χ )2 (cid:18) πiL (cid:19) n B n, ¯ χ n ! , (23) L (1 − n, χ ) = − B n,χ /n. On the other hand, if χ ( − = ( − n , then L (1 − n, χ ) = 0 unless χ = 1 and n = 1.Finally, let us recall the basic transformation properties of the Eisenstein series E k ( τ ; χ, ψ ), which are detailed, for instance, in [Miy06, Chapter 7]. Denote withΓ ( L, M ) the group of matrices γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ) such that M | b and L | c .For any such γ ,(24) E k ( τ ; χ, ψ ) | k γ = χ ( d ) ¯ ψ ( d ) E k ( τ ; χ, ψ ) . Moreover,(25) E k ( τ ; χ, ψ ) | k S = χ ( − E k ( τ ; ψ, χ ) . Review of Eichler integrals
In the language of Eichler integrals, to be reviewed next, the observation ofSection 4 becomes the simple statement that the secant Dirichlet series ψ m areEichler integrals of weight 2 m + 1 Eisenstein series of level 2. As such, it is naturalthat the ψ m satisfy modular functional equations as in Section 2. In fact, itbecomes a priori clear that the ψ m satisfy modular relations such as (3), in whichthe coefficients of the rational function on the right-hand side are determined bythe period polynomial of the Eichler integral.Lemma 4.1 shows that F m ( τ ) := ψ m ( τ ) + E m m )! ( πiτ ) m has the property that its 2 m th derivative is a modular form with respect to Γ = h A, B i of weight 2 m + 1 (with A and B as defined in (8)). In other words, F m is an Eichler integral (we adopt the common custom and also refer to ψ m as anEichler integral). In particular, for all γ = (cid:18) a bc d (cid:19) ∈ Γ,(26) ( cτ + d ) m − F m ( γτ ) − F m ( τ )is a polynomial of degree 2 m −
1, the period polynomial of F m . Rewriting (26)in terms of ψ m ( τ ), we find that ψ m ( τ ) satisfies a functional equation of the form(3), where the rational function on the right-hand side is expressed in terms of theperiod polynomial of F m . For the general theory of period polynomials we referto [PP13] and the references therein. A very brief introduction, suitable for ourpurposes, is given next. Remark 5.1.
A direct way to see that (26) is indeed a polynomial of degree 2 m − F and γ = (cid:18) a bc d (cid:19) ∈ SL ( R ),(27) ( D k +1 F )( γτ ) = ( cτ + d ) k +2 D k +1 (cid:2) ( cτ + d ) k F ( γτ ) (cid:3) . For the present purpose, we apply (27) with F = F m and k = 2 m − N A SECANT DIRICHLET SERIES 11
In the following we adopt the notation of [PP13]. Let A be the space of holo-morphic functions on the upper half-plane H . As usual, Γ = SL ( Z ) acts on H by linear fractional transformations, and on A via the slash operators; namely, if f ∈ A and k is an integer, then( f | k g )( τ ) = ( cτ + d ) − k f ( gτ ) , g = (cid:18) a bc d (cid:19) ∈ Γ . This action extends naturally to the group algebra C [Γ ].As usual, we denote with T , S and R the matrices(28) T = (cid:18) (cid:19) , S = (cid:18) −
11 0 (cid:19) , R = (cid:18) (cid:19) , and recall that the matrices T and S generate Γ .From now on, let f be a (not necessarily cuspidal) modular form of integralweight k > . Notethat the modularity of f implies that f | k g = f for all g ∈ Γ. In the sequel, wewill abbreviate f | g = f | k g since this is the only action of Γ on modular forms ofweight k that we consider.Throughout this section, w = k −
2. Let V w be the space of complex polynomialsof degree at most w . The (multiple) period polynomial, introduced in [PP13],attached to f is the map ρ f : Γ \ Γ → V w defined by(29) ρ f ( A )( X ) = Z i ∞ [ f | A ( t ) − a ( f | A )] ( t − X ) w d t. In the sequel, we will often omit the dependence on X and just write ρ f ( A ) forthe left-hand side. The goal of the final set of definitions is to connect these periodpolynomials, whose coefficients encode the critical L -values of f , to the transfor-mation properties of Eichler integrals of f . The (multiple) Eichler integral of f ,introduced in [PP13], is the function ˜ f : Γ \ Γ → A defined by(30) ˜ f ( A )( τ ) = Z i ∞ τ [ f | A ( z ) − a ( f | A )] ( z − τ ) w d z, with a ( f ) = f ( i ∞ ) denoting the constant term of the Fourier expansion of f . If g ∈ Γ then ˜ f | g ( A ) = ˜ f ( Ag − ) | − w g defines an action of Γ , and hence C [Γ ], onfunctions ˜ f : Γ \ Γ → A .The following result is [PP13, Proposition 8.1], which may also be found in[Wei77], where it is expressed in slightly different terms. Proposition 5.2.
With f as above, define ˆ ρ f = ˜ f | (1 − S ) . Then, for any A ∈ Γ \ Γ , ˆ ρ f ( A ) = ρ f ( A ) + ( − w a ( f | A ) w + 1 X w +1 + a ( f | AS − ) w + 1 X − . Note that matters simplify when only cusp forms are considered; in that case, ρ f and ˆ ρ f coincide. Though not necessarily a polynomial in the Eisenstein case, wewill also refer to ˆ ρ f , as well as to ˜ f | (1 − γ ) for γ ∈ Γ , as period polynomials of f . Example 5.3.
Let us make these definitions very explicit in the case where f is the Eisenstein series E ( τ ) = E k ( τ ; χ, ψ ) introduced in (15). Then E | S = χ ( − E k ( τ ; ψ, χ ) by (25), and thus(31) ˆ ρ E ( I ) = ˜ E k ( X ; χ, ψ ) − ψ ( − X k − ˜ E k ( − /X ; ψ, χ ) , assuming that χ ( − ψ ( −
1) = ( − k (since, otherwise, E = 0). Here, and in thesequel, we denote, with a slight abuse of notation,˜ E k ( τ ; χ, ψ ) = ˜ E ( I )( τ ) , where the right-hand side is defined by (30). It follows from the Fourier expansion(17) that, for primitive ψ ,(32) ˜ E k ( τ ; χ, ψ ) = − πik − G ( ψ ) M ∞ X n =1 X d | n ¯ ψ ( n/d ) χ ( d ) d − k e πinτ/M . Example 5.4.
Following Example 4.3, we observe from (16) that(33) ψ m ( τ /
2) = 2 mπ ˜ E m +1 ( τ ; 1 , χ − ) , thus making explicit the nature of ψ m as an Eichler integral. In the present context,the functional equation (6), on replacing τ with τ / E m +1 ( τ ; 1 , χ − ) | − m ( R −
1) = π m +1 m (cid:2) z m − (cid:3) sin( τ z/ z ) sin(( τ + 1) z ) , where R is as in (28).The next result allows us to express the left-hand side of (34) in terms of ˆ ρ E m +1 which, in the sense of Proposition 5.2, is a period polynomial of E m +1 . Proposition 5.5.
Let R be as in (28). Let f be a modular form for a group Γ Γ ,and let n be such that R n ∈ Γ . Then ˜ f | (1 − R n )( I ) = ˆ ρ f | (1 − R n )( I ) . Proof.
For any C = (cid:18) a bc d (cid:19) ∈ Γ, it follows from the definition of ˆ ρ f thatˆ ρ f | (1 − C ) = ˜ f | (1 − S )(1 − C ) = ˜ f | (1 − C ) − ˜ f | ( S − SC ) . It therefore suffices to show that ˜ f | S ( I ) = ˜ f | SR n ( I ). To see this, we observe that,for C ∈ Γ as above,˜ f | SC ( I ) = ˜ f ( S − ) | − w SC = ( aτ + b ) w Z i ∞ SCτ (cid:2) f | S − ( z ) − a ( f | S − ) (cid:3) ( z − SCτ ) w d z. The change of variables z = SCS − z ′ yields˜ f | SC ( I ) = τ w Z a/bSτ (cid:2) f | S − ( z ) − ( − bz + a ) − k a ( f | S − ) (cid:3) ( z − Sτ ) w d z, and the desired equality follows because, upon setting a = 1 and b = 0, the right-hand side does not depend on the value of c . (cid:3) N A SECANT DIRICHLET SERIES 13 Period polynomials of Eisenstein series
In the case χ = 1 and ψ = 1, the Eisenstein series E ( τ ) = E k ( τ ; 1 ,
1) is the usualEisenstein series of weight 2 k with respect to the full modular group. Its periodpolynomial, defined in Proposition 5.2 and made explicit in (31) for Eisensteinseries, is well-known to be(35)ˆ ρ E ( I ) = − (2 πi ) k k − " k X s =0 B s (2 s )! B k − s (2 k − s )! X k − s − + ζ (2 k − πi ) k − ( X k − − ;compare, for instance, [Zag91, (11)]. On the other hand, this evaluation is equiva-lent to the formula (48), which Ramanujan famously recorded and which we brieflydiscuss in Section 8.1. A beautiful account of this connection is contained in[GMR11].The (Laurent) polynomials on the right-hand side of (35) have interesting prop-erties, which have been studied, for instance, in [GMR11], [MSW11], and [LR13].In Section 7, we indicate that the generalized polynomials obtained in this sectionfor higher level share similar properties.In light of Example 5.4 and Proposition 5.5, the functional equations satisfiedby the secant Dirichlet series are determined by the period polynomials associatedto the Eisenstein series E k +1 ( τ ; 1 , χ − ). We next compute the period polynomialsof the Eisenstein series E k +1 ( τ ; χ, ψ ), with χ and ψ being any pair of primitiveDirichlet characters. Theorem 6.1.
Let k > , and let χ and ψ be primitive Dirichlet characters modulo L and M , respectively, such that χ ( − ψ ( −
1) = ( − k . For the Eisenstein series E ( τ ) = E k ( τ ; χ, ψ ) , defined in (15), ˆ ρ E ( I ) = − ψ ( − k − k X s =0 χ ( − − s L ( s, χ ) L ( k − s, ψ ) X k − s − − ψ ( − k − πi (cid:2) ε χ L ( k − , ψ ) X k − − ε ψ L ( k − , χ ) (cid:3) . (36) Here, ε χ = 1 if χ = 1 , and ε χ = 0 otherwise.Proof. We first observe, from the definition (29), the general fact that, for a modularform E of weight k , ρ E ( A ) = ( − k − k − X s =1 (cid:18) k − s − (cid:19) Γ( s )(2 πi ) s L ( E | A, s ) X k − s − , with the L -function of E as defined in (19). Let s be an integer with 0 < s < k .We deduce from (20) and the functional equation, given, for instance, in [Apo76,Theorem 12.11, p. 263],(37) L (1 − s, ¯ ψ ) = M s − Γ( s )(2 π ) s ( e − πis/ + ψ ( − e πis/ ) τ ( ¯ ψ ) L ( s, ψ )that, for s such that χ ( −
1) = ( − s , L ( E, s ) = 4( − s (2 πi ) s Γ( k − s )Γ( k ) L ( s, χ ) L ( k − s, ψ ) . On the other hand, for s such that χ ( − = ( − s , we have L ( ¯ ψ, − k + s ) = 0,which implies that L ( E, s ) = 0 unless s = 1, χ = 1 or s = k − ψ = 1. Combiningthese, we find that ρ E ( I ) = − ψ ( − k − k − X s =1 χ ( − − s L ( s, χ ) L ( k − s, ψ ) X k − s − − ( − k πi ε χ L ( E, X k − − ( − k (2 πi ) k − Γ( k − ε ψ L ( E, k − , where the sum is over all integers s such that 0 < s < k and χ ( −
1) = ( − s . From(20), together with the functional equation (37) of the involved Dirichlet L -series,we deduce that, assuming χ = 1, L ( E,
1) = (2 πi ) k − L ( k − , ψ ) . On the other hand, if ψ = 1, then L ( E, k −
1) = − ( − πi ) k ( k − L ( k − , χ ) . It follows from Proposition 5.2 thatˆ ρ E ( I ) = ρ E ( I ) + ( − k a ( E ) k − X k − + a ( E | S − ) k − X − . The values for a ( E ) and a ( E | S − ) = ( − k a ( E | S ) are given by (17) in combi-nation with (25). Finally, using the fact that L (0 , χ ) = 0 for any even Dirichletcharacter χ = 1, we obtain (36). (cid:3) Observe that, in the case χ = ψ = 1, using Euler’s identity [Ayo74] ζ (2 m ) = ( − m +1 B m m )! (2 π ) m , we obtain from Theorem 6.1 the well-known special case (35). In the same spirit,Theorem 6.1 may always be rewritten, using (23), in terms of generalized Bernoullinumbers as we record below. For the cases where χ = 1 or ψ = 1 the appropriateextra terms need to be inserted. Corollary 6.2.
Under the assumptions of Theorem 6.1, if, additionally, χ and ψ are both nonprincipal, then ˆ ρ E ( I ) = − χ ( − G ( χ ) G ( ψ ) (2 πi ) k k − k X s =0 B k − s, ¯ χ ( k − s )! L k − s B s, ¯ ψ s ! M s X s − . Remark 6.3.
We note that similar results, based on residue calculations in thespirit of Section 2, are obtained in [Ber75b]. In fact, the Eisenstein series consideredin [Ber75b], namely X ′ m,n ∈ Z χ ( m ) ψ ( n )(( m + r ) τ + n + r ) k , have the two extra parameters r and r in comparison with E k ( τ ; χ, ψ ). How-ever, the analysis in [Ber75b] is restricted to the case when χ and ψ are primitivecharacters of the same modulus. N A SECANT DIRICHLET SERIES 15
Example 6.4.
In the case χ = ψ = 1, Theorem 6.1 reduces to Ramanujan’sidentity (35), which in particular yields an interesting formula for the odd zetavalues ζ (2 k − L -series have been derived in[Kat74] using partial fraction expansions. We will now illustrate how the resultsof [Kat74] follow from the special case ψ = 1 of Theorem 6.1, thus providing analternative proof. In the setting of Theorem 6.1, with ψ = 1 and χ = 1, we haveˆ ρ E ( I ) = 2 πik − L ( k − , χ ) − k − ⌊ k/ ⌋ X j =0 L ( k − j, χ ) ζ (2 j ) X j − . On the other hand, from (31),ˆ ρ E ( I ) = ˜ E k ( τ ; χ, − τ k − ˜ E k ( − /τ ; 1 , χ ) . Using the Fourier expansion (32), as well as G ( χ ) G ( ¯ χ ) = χ ( − L and the simplesummations ∞ X n =1 X d | n χ ( d ) d − k q n = ∞ X n =1 χ ( n ) n k − q n − q n , ∞ X n =1 X d | n χ ( n/d ) d − k q n = L X a =1 χ ( a ) ∞ X n =1 n k − q an − q Ln , we obtain ˆ ρ E ( I ) = 4 πik − (cid:20) F ( τ ) − ( − τ ) k − G ( ¯ χ ) F ( − /τ ) (cid:21) , where, similar to [Kat74], F ( τ ) = ∞ X n =1 χ ( n ) n k − e πinτ e πinτ − ,F ( τ ) = L X a =1 ¯ χ ( a ) ∞ X n =1 n k − e πianτ/L e πinτ − . Solving for L ( k − , χ ), we have arrived at12 L ( k − , χ ) = F ( τ ) − ( − τ ) k − G ( ¯ χ ) F ( − /τ ) + 1 πi ⌊ k/ ⌋ X j =0 L ( k − j, χ ) ζ (2 j ) τ j − , which is equivalent to the main result of [Kat74]. Note that this formula expressesthe L -value as a combination of two Eichler integrals and a power of π times aLaurent polynomial with rational coefficients. In Example 7.7, similar formulasfor these L -values are given, where only one Eichler integral is involved (evaluatedat two arguments) and the polynomials appear to have the additional property ofhaving all their nonreal roots on the unit circle.With Theorem 6.1 in place, it is easy to deduce functional equations such as(6), which is the case χ = 1, ψ = χ − of the next result. Note that the restrictionto ψ = 1 is just to avoid the presence of an additional term. The case ψ = 1 isdiscussed in Example 7.7, where the formulas promised at the end of Example 6.4are derived. Corollary 6.5.
Let k > , and let χ and ψ be primitive Dirichlet characters modulo L and M , respectively, such that χ ( − ψ ( −
1) = ( − k . Let R be as in (28). If ψ = 1 , then, for any integer n such that L | n , ˜ E k ( X ; χ, ψ ) | − k (1 − R n ) = − ψ ( − k − k X s =0 χ ( − − s L ( s, χ ) L ( k − s, ψ ) X k − s − (1 − ( nX +1) s − ) . Proof.
Recall from (24) that E ( τ ) = E k ( τ ; χ, ψ ) is modular with respect to Γ ( L, M ).Since R n ∈ Γ ( L, M ), we may apply Proposition 5.5 and Theorem 6.1 to obtain˜ E k ( X ; χ, ψ ) | − k (1 − R n ) = ˜ E | (1 − R n )( I ) = ˆ ρ E ( I ) | − w (1 − R n ) , with the Laurent polynomial ˆ ρ E ( I ) given explicitly in (36). Note that X j | − w (1 − R n ) = X j − ( nX + 1) w (cid:18) XnX + 1 (cid:19) j = X j (cid:16) − ( nX + 1) w − j (cid:17) . In particular, X w | − w B = X w , so that the term in (36) involving L ( k − , ψ ), if atall existent, is eliminated in ˜ E | (1 − R n )( I ). (cid:3) Example 6.6.
As indicated in Example 5.4, Corollary 6.5 specializes to a variationof Theorem 2.1 on setting χ = 1 and ψ = χ − . Namely, using (22) to relate thegeneralized Bernoulli numbers to Bernoulli and Euler numbers, we find that, forany positive integer m , the secant Dirichlet series satisfies the functional equation(2 τ + 1) m − ψ m (cid:18) τ τ + 1 (cid:19) − ψ m ( τ )= ( πi ) m m X n =0 n − B n E m − n (2 n )!(2 m − n )! τ m − n (cid:2) − (2 τ + 1) n − (cid:3) . (38)We note that (38) is the same functional equation as (6) but representing the right-hand side in a somewhat different way; see Remark 6.7. Remark 6.7.
Let us indicate how to see, in a direct fashion, that (6) and (38)represent the same functional equation. Note that (38) can be expressed as(39) ψ m ( τ ) | − m (1 − B ) = τ m − (cid:2) h m (cid:0) τ (cid:1) − h m (cid:0) τ (cid:1)(cid:3) , where h m ( τ ) is the rational function(40) h m ( τ ) = ( πi ) m m X n =0 B n E m − n (2 n )!(2 m − n )! (2 τ ) n − . By the definitions of the Bernoulli and Euler numbers, (7) and (12), we find that h m ( τ ) = π m [ z m − ] cot( τ z ) sec( z ) . Hence, the right-hand side of (38) and (39) equals π m [ z m − ] sec( τ z ) [cot( z ) − cot((2 τ + 1) z )] . The equivalence of (6) and (38) then follows fromsec( τ z ) [cot( z ) − cot((2 τ + 1) z )] = sin( τ z )sin( z ) sin((2 τ + 1) z ) , which is obtained from basic trigonometric identities. N A SECANT DIRICHLET SERIES 17
As an application of Corollary 6.5, we now derive formulas for the values of ψ m at families of real quadratic irrationalities, thus complementing and illustrating theresults of Section 3 in an explicit fashion. Example 6.8.
Since Corollary 6.5 applies to all powers of R , which lie in theappropriate modular subgroup, we find that, for positive integers m and integers µ , (2 µτ + 1) m − ψ m (cid:18) τ µτ + 1 (cid:19) − ψ m ( τ )= ( πi ) m m X n =0 n − B n E m − n (2 n )!(2 m − n )! τ m − n (cid:2) − (2 µτ + 1) n − (cid:3) . (41)We now demonstrate how to use these functional equations to obtain families ofexplicit evaluations of ψ m at certain real quadratic irrationalities. Let τ be fixedby A λ B µ A ν , that is A λ B µ A ν τ = τ . A brief calculation shows that τ = λ − ν ± r ( λ + ν ) (cid:16) µ + ( λ + ν ) (cid:17) . Denote with T m,µ = ψ m | − m ( B µ − I ) the right-hand side of (41). It follows from ψ m ( A λ B µ A ν τ ) = ψ m ( A ν τ ) + T m,µ ( A ν τ )(2 µA ν τ + 1) m − , together with the fact that τ is fixed by A λ B µ A ν , that ψ m ( τ ) = T m,µ ( τ + 2 ν )(2 µ ( τ + 2 ν ) + 1) m − − . A straightforward, but slightly tedious, calculation using (41) and the explicit valueof τ shows that(42) ψ m ( τ ) = − ( πi ) m m X n =0 n − µ m − n B n E m − n (2 n )!(2 m − n )! r n r m , where r n are the rational numbers r n = 12 (cid:2) (1 + √ α ) n − + (1 − √ α ) n − (cid:3) = ( − α , if n = 0 , P n − j =0 (cid:0) n − j (cid:1) α j , if n > , and α = µ ( ν + λ ) + 1. Note that this is an explicit illustration of the general fact,proved in Theorem 3.4, that ψ m ( τ ) is a rational multiple of π m whenever τ is areal quadratic irrationality.We note that the right-hand side of (42) only depends on κ = ν + λ but not on ν and λ individually. For the left-hand side, this follows from the obvious periodicityrelation ψ m ( τ + 2) = ψ m ( τ ). The first two cases of (42) can thus be stated, inequivalent forms, as ψ κ + r κ (cid:16) µ + κ (cid:17)! = π (cid:18) κ µ (cid:19) ,ψ κ + r κ (cid:16) µ + κ (cid:17)! = π (cid:18) κ µ − κ (16 µ − µ (4 κµ + 3) (cid:19) , where κ and µ are integers and µ = 0. The first of these, in the special case κ = 2 λ ,is also given in [LRR14]. Remark 6.9.
Using the methods of [Ber73], [Ber75b] or, alternatively, [Raz77],one can derive the general transformation laws of the Eichler integrals ˜ E k ( τ ; χ, ψ ),and, in particular, ψ m ( τ ), under arbitrary elements of the full modular group.Here, we do not, however, pursue this further.7. Zeros of generalized Ramanujan polynomials
It has recently been shown in [CFI12] and [EGR13] that the nontrivial zeros ofperiod polynomials of modular forms, which are Hecke eigenforms of level 1, all lieon the unit circle. In this section, we consider the Eisenstein case of higher level byinvestigating the zeros of the period polynomials calculated in the previous section.We again find that, at least conjecturally, most of the roots lie on a circle in thecomplex plane. The observations suggest that the problem solved by [CFI12] and[EGR13] is interesting in the higher level case as well.An application of these considerations is that knowledge of the location of thezeros of the period polynomials calculated in the previous section gives rise toexplicit formulas for Dirichlet L -values of “wrong” parity (that is, values at integersof parity opposite to the Dirichlet character) in terms of Eichler integrals. This ismade explicit in Example 7.7. The special case of the principal character is detailedin [GMR11], in which case odd zeta values are expressed in terms of the differenceof two Eichler integrals at algebraic argument of modulus 1.For positive integer k , and Dirichlet characters χ and ψ modulo L and M , wedefine the generalized Ramanujan polynomial (43) R k ( X ; χ, ψ ) = k X s =0 B s,χ s ! B k − s,ψ ( k − s )! (cid:18) X − M (cid:19) k − s − (1 − X s − ) . Note that this is a polynomial if χ and ψ are both nonprincipal, and a Lau-rent polynomial otherwise. Further note that, if χ ( − ψ ( − = ( − k , then R k ( X ; χ, ψ ) = 0, unless ψ = 1 in which case R k ( X ; χ, ψ ) = 12 B k − ,χ ( k − − X k − ) . In the sequel, we will therefore often assume, without loss of generality and as wedid in previous sections, that χ ( − ψ ( −
1) = ( − k .In [MSW11] the Ramanujan polynomials are, essentially, defined as(44) R k ( X ) = k X s =0 B s s ! B k − s ( k − s )! X s − , where k is an even integer (in [MSW11] the index k is shifted by 1, X appears withexponent s , and the definition differs for k = 2). The next result shows that thegeneralized Ramanujan polynomials, despite their different appearance, reduce tothe Ramanujan polynomials when χ = 1 and ψ = 1. Proposition 7.1.
For k > , R k ( X ; 1 ,
1) = R k ( X ) . N A SECANT DIRICHLET SERIES 19
Proof.
As evidenced by (35), the polynomial R k ( X ) is the odd part of the periodpolynomial of the level 1 Eisenstein series E k ( τ ; 1 , R k ( X ) | − k (1 + S ) = R k ( X ) | − k (1 + U + U ) = 0 . Here, U = T S with T and S as defined in (28). On the other hand, by construction(43), R k ( X ; 1 ,
1) = R k ( X ) | − k (1 − R ) T − . A brief calculation reveals that (1 + U + U ) T = T + R + S . Hence, using bothrelations (45), we find R k ( X ) | − k (1 − R ) = R k ( X ) | − k (1 + T + S ) = R k ( X ) | − k T, which proves the claim. (cid:3) The next example indicates that the definition (43) of the generalized Ramanujanpolynomials is natural, by connecting them to period polynomials of generalizedEisenstein series studied in Section 6.
Example 7.2.
As in Corollary 6.5, let k >
3, and let χ and ψ = 1 be primitiveDirichlet characters modulo L and M , respectively, such that χ ( − ψ ( −
1) = ( − k .Then, with R as in (28),˜ E k ( X ; χ, ψ ) | − k (1 − R L ) = − χ ( − G ( χ ) G ( ψ ) (2 πi/L ) k k − LM R k ( LX + 1; ¯ χ, ¯ ψ ) . In other words, up to some scaling and a linear change of variables, the polynomial R k ( X ; ¯ χ, ¯ ψ ) is a period polynomial of the Eisenstein series E k ( τ ; χ, ψ ). Conjecture 7.3.
For nonprincipal real Dirichlet characters χ and ψ , the polyno-mial R k ( X ; χ, ψ ) is unimodular, that is, all its roots lie on the unit circle. We have verified Conjecture 7.3 numerically for all k
50 and all characters ofmodulus up to 100. We note that it follows from Proposition 7.1 and the resultsin [MSW11] that all nonreal zeros of R k ( X ; 1 ,
1) lie on the unit circle. On theother hand, it is conjectured (in equivalent form) in [LRR14] that all roots of R k ( X ; 1 , χ − ) lie on the unit circle. Before giving further evidence in support ofConjecture 7.3 as well as an application, we indicate the conjectural situation in thecases χ = 1 or ψ = 1, which is not included above. In vague summary, it appearsthat still most of the roots lie on the unit circle. Example 7.4.
Let χ be a nonprincipal real Dirichlet character. Computationsshow that, at least for k
50 and χ of modulus at most 100, the polynomials R k ( X ; χ,
1) are unimodular, except when χ takes the same values as χ , the uniquecharacter of conductor 3. In the case χ = 1, it was shown in [MSW11] that R k ( X ; 1 , k >
2, has exactly four distinct real roots (approaching ± ± )and that the remaining roots lie on the unit circle. On the other hand, it appearsthat R k +1 ( X ; χ , k >
3, has exactly three distinct real roots ( − − ± ) and that the remaining roots again lie on the unitcircle.In the next example, we restrict to primitive characters for expositional reasonsand make use of the fact [ZR76] that, for a given conductor M , there is at mostone primitive real Dirichlet character modulo M of each parity. We label even such characters as M + and odd ones as M − . For instance, the label 8+ refers to theeven real Dirichlet character of conductor 8. Example 7.5.
The situation in the case of R k ( X ; 1 , ψ ), with ψ a real primitivecharacter, is slightly more varied. For certain characters ψ , such as3 − , − , , ± , − , , , − , , , . . ., the polynomial R k ( X ; 1 , ψ ) appears to again be unimodular. For certain othercharacters ψ , such as 1+ , − , − , , − , − , − , . . ., we observe, at least for small k , that all nonreal roots lie on the unit circle. Onthe other hand, there remains a third group of exceptional characters ψ , namely35 − , − , − , − , − , − , . . . (we observe that in each listed case ψ is odd),for which R k ( X ; 1 , ψ ) can have nonreal zeros off the unit circle. Consider, forinstance, the unique real primitive Dirichlet character χ of conductor 35. Thenthe polynomial R ( X ; 1 , χ ) has the seven roots (given to three decimal digits)1 , . ± . i, ( − . ± . i ) ± . While the first three listed roots have absolute value 1, the last four have absolutevalue 1 . ± . In each of the exceptional cases, we observed, as in the example of R ( X ; 1 , χ ), at most four nonreal zeros off the unit circle (in light of Proposition7.6, such zeros necessarily come in groups of four).In order for all zeros of a polynomial p ( X ) = a + a X + · · · + a n X n , a n = 0, to lieon the unit circle, it is a necessary condition [Coh22], [LS13], that the polynomialis self-inversive, that is, for some ε with | ε | = 1, a k = εa n − k for k = 0 , , . . . , n . Insupport of Conjecture 7.3, we now observe that, for real characters, R k ( X ; χ, ψ ) isself-inverse with ε = ±
1. In other words, R k ( X ; χ, ψ ) is reciprocal or anti-reciprocaldepending on the parity of ψ . Proposition 7.6.
Let χ and ψ be real Dirichlet characters. If χ ( − ψ ( −
1) =( − k , then R k ( X ; χ, ψ ) = ψ ( − X k − R k ( X − ; χ, ψ ) . Proof.
Temporarily, denote with p s ( X ) = ( X − k − s − (1 − X s − )one of the terms in (43) contributing to R k ( X ; χ, ψ ). It is simple to check that X k − p s ( X − ) = ( − k − s p s ( X ) . On the other hand, recall that, for any Dirichlet character χ , B s,χ = 0 if χ ( − =( − s , unless χ = 1 and s = 1. It follows that B k − s,ψ = 0 if ψ ( − = ( − k − s ,unless ψ = 1 and s = k −
1. In the latter case, when ψ = 1 and s = k −
1, we have B s,χ = B k − ,χ = 0 because χ ( −
1) = ( − k , unless χ = 1 and k = 2, which may bechecked separately. We have thus shown that X k − p s ( X − ) = ψ ( − p s ( X ) for all s that have a nonzero coefficient in (43). (cid:3) Example 7.7.
The case ψ = 1 is of special interest, because it yields explicitformulas for Dirichlet L -values at integral arguments (of parity opposite to theDirichlet character) in terms of Eichler integrals. Let χ be a primitive Dirichlet N A SECANT DIRICHLET SERIES 21 character and k > χ ( −
1) = ( − k . Applying Theorem 6.1 as inCorollary 6.5 yields k − πi ˜ E k ( X ; χ, | − k (1 − R L ) = G ( χ ) ( − πi/L ) k − R k ( LX + 1; ¯ χ, L ( k − , χ ) (cid:16) − ( LX + 1) k − (cid:17) . Solving for L ( k − , χ ), we obtain formulas for these L -values in the spirit of [Kat74];see Example 6.4. On the other hand, suppose that α , with Im( α ) >
0, is a root of R k ( α ; ¯ χ,
1) = 0, which is not a ( k − L ( k − , χ ) = k − πi (1 − α k − ) (cid:20) ˜ E k (cid:18) α − L ; χ, (cid:19) − α k − ˜ E k (cid:18) − /αL ; χ, (cid:19)(cid:21) , thus explicitly linking the L -value to values of the Eichler integral at algebraicpoints, as is studied for χ = 1 in [GMR11]. Note that by (32), as in Example 6.4,˜ E k ( τ ; χ,
1) = 4 πik − ∞ X n =1 χ ( n ) n k − e πinτ e πinτ − . Hence (46) takes the entirely explicit form L ( k − , χ ) = 21 − α k − ∞ X n =1 χ ( n ) n k − (cid:20) − e πin (1 − α ) /L − α k − − e πin (1 /α − /L (cid:21) , which expresses the L -value as a combination of two Lambert-type series. Thespecial case χ = 1, which is a consequence of Ramanujan’s identity (35), has recentlybeen studied in [GMR11]. Specifically, using the notation F k ( z ) := ki π ˜ E k +1 ( z ; 1 ,
1) = ∞ X n =1 n k e πinτ − e πinτ , it is shown in [GMR11, Theorem 1.1] that the numbers F k +1 ( β ) − β k F k +1 ( − /β ) = (2 k + 1) i π h ˜ E k +2 ( β ; 1 , − β k ˜ E k +2 ( − /β ; 1 , i are transcendental for every algebraic β ∈ H with at most 2 k +2+ δ exceptions. Here δ = 0 , , , k,
6) = 1 , , β = i , if k is even, and β = e πi/ , e πi/ , if k isdivisible by 3. In each of these three cases, however, the combination F k +1 ( β ) − β k F k +1 ( − /β ) vanishes. As pointed out in [GMR11], the existence of furtherexceptional β is very unlikely and would imply that ζ (2 k + 1) is an algebraic linearcombination of 1 and π k +1 . This conclusion relies on the results of [MSW11], bywhich the only roots of unity that are zeros of the corresponding period polynomialare ± i , if k is even, and ± e πi/ , ± e πi/ , if k is divisible by 3. The above resultsindicate that similar transcendence results should hold for the more general Eichlerintegrals ˜ E k ( τ ; χ,
1) in place of ˜ E k ( τ ; 1 , L -values of “wrong” parity is concerned. Remark 7.8.
It appears that the observations in this section can be extendedfurther. For instance, in the case of imaginary Dirichlet characters, one finds thatmost of the zeros lie on the unit circle, if one considers instead of R k ( X ; χ, ψ ) itsreal or imaginary part (that is, the polynomial with coefficients which are the real or imaginary parts of the coefficients of R k ( X ; χ, ψ )). Finally, generalizing (43),one may consider the polynomial k X s =0 B s,χ s ! B k − s,ψ ( k − s )! (cid:18) X − m (cid:19) k − s − (1 − X s − ) , where m = aM for some integer a . In the spirit of Example 7.2, this polynomialcorresponds to the period polynomial in Corollary 6.5 when n = aL . It againappears that most roots of these polynomials lie on the unit circle.As a second, and possibly more direct, generalization of the Ramanujan polyno-mials (44), we briefly also consider(47) S k ( X ; χ, ψ ) = k X s =0 B s,χ s ! B k − s,ψ ( k − s )! (cid:18) LXM (cid:19) k − s − , where χ and ψ are Dirichlet characters modulo L and M . The two generalizations(43) and (47) are related by S k ( X ; χ, ψ ) | − k (1 − R L ) = R k ( LX + 1; χ, ψ ) . By (31) and Corollary 6.2, we have that, for k > χ and ψ such that χ ( − ψ ( −
1) = ( − k ,˜ E k ( X ; χ, ψ ) − ψ ( − X k − ˜ E k ( − /X ; ψ, χ ) = − χ ( − G ( χ ) G ( ψ ) (2 πi/L ) k k − S k ( X ; ¯ χ, ¯ ψ ) , which expresses the S k ( X ; χ, ψ ) as period polynomials as well. In general, thesepolynomials are not self-inverse and therefore cannot be unimodular. Conjecture 7.9.
For nonprincipal real Dirichlet characters χ , all nonzero roots ofthe polynomial S k ( X ; χ, χ ) lie on the unit circle. We have verified this conjecture numerically for k
50 and characters χ ofmodulus at most 100. In the case χ = 1, we recall that it was shown in [MSW11]that the polynomials S k ( X ; 1 ,
1) = R k ( X ) have all their nonreal zeroes on theunit circle. On the other hand, for χ = 1, the polynomial S k ( X ; 1 ,
1) is only theodd part of the period polynomial and it was recently proved in [LS13] that the fullperiod polynomial (35) is indeed unimodular.8.
Relation to sums considered by Ramanujan
Sums of level 1.
Ramanujan famously recorded (see [Ber77] or [Ber89, p.276], as well as the references therein) the formula α − m ( ζ (2 m + 1)2 + ∞ X n =1 n − m − e αn − ) = ( − β ) − m ( ζ (2 m + 1)2 + ∞ X n =1 n − m − e βn − ) − m m +1 X n =0 ( − n B n (2 n )! B m − n +2 (2 m − n + 2)! α m − n +1 β n , (48)where α and β are positive numbers with αβ = π and m is any nonzero integer.Rewriting equation (48) using 1 e x − (cid:16) x (cid:17) − , N A SECANT DIRICHLET SERIES 23 slightly shifting the value of m and setting β = πiτ , and therefore α = − πi/τ , weobtain, as in [Ber76], for integers m = 1, τ m − ξ m − ( − τ ) − ξ m − ( τ ) = ( − m (2 π ) m − m X n =0 B n (2 n )! B m − n (2 m − n )! τ n − , where ξ s is the cotangent Dirichlet series defined in (2).As in the case of the secant Dirichlet series, discussed in Sections 4 and 6, thecotangent Dirichlet series is essentially an Eichler integral. Indeed, one easily checksthat, for integral s , i ξ s ( τ ) = 12 ζ ( s ) + ∞ X n =1 σ s ( n ) n s q n . When s = 2 k − k Eisenstein series E k ( τ ; 1 ,
1) = 2 ζ (2 k ) + 2(2 πi ) k Γ(2 k ) ∞ X n =1 σ k − ( n ) q n , which is modular with respect to the full modular group. Remark 8.1.
Proceeding as in Section 3, though matters simplify because thelevel is 1, we conclude that ξ m +1 ( √ r ) is a rational multiple of π m +1 √ r whenever r is a positive rational number (assuming, for convergence, that √ r is irrational).Explicit special cases of this observation may be found, for instance, in [Ber76].8.2. Sums of level 4.
Ramanujan also found [Ber89, Entry 21(iii), p. 277] theidentity α − m +1 / ( L ( χ − , m ) + ∞ X n =1 χ − ( n ) n m ( e αn − ) = ( − m β − m +1 / m +1 ∞ X n =1 sech( βn ) n m + 14 m X n =0 ( − n n E n (2 n )! B m − n (2 m − n )! α m − n β n +1 / , (49)which was first proved in print by Chowla [Cho28], where α and β are positivenumbers with αβ = π and m is any integer. The goal of this section is to relate(49) to the present discussion of the modular properties of the secant Dirichletseries. It will transpire that (49) is an explicit version of Theorem 6.1 in the case χ = 1 and ψ = χ − . In other words, equation (49) encodes how ψ m transformsunder S , as defined in (28). From here, we can then work out the exact wayin which ψ m transforms under any transformation of SL ( Z ), though we do notdevelop the details here (but see Remark 8.2 for indications). On the other hand,we demonstrate that we may use (49) as the basis for yet another derivation of thefunctional equation (6).With α = πiτ , and proceeding as in the case of (48), we can write (49) as(50) τ m − ψ m ( − τ ) = ˆ ψ m ( τ ) − h m ( τ ) , where ˆ ψ m ( τ ) = ∞ X n =1 χ − ( n ) cot( πn τ )( n/ m and h m ( τ ) is the rational function defined in (40). Since ˆ ψ m ( τ + 2) = ˆ ψ m ( τ ), wefind that ψ m | − m ST − = ˆ ψ m − h m | − m T − . Using further that B = ST − S − , with B as in (8), we find that ψ m | − m B = ψ m − h m | − m ( S − + T − S − )= ψ m + h m | − m ( S + T − S ) . (51)which, in expanded form, is precisely the functional equation (38). Remark 8.2.
The functional equation satisfied by ψ m is given in [LRR14] insomewhat different form. Indeed, comparing the rational functions involved in thefunctional equations, we observe that( πi ) m m X n =0 n − B n E m − n (2 n )!(2 m − n )! τ m − n (cid:2) − (2 τ + 1) n − (cid:3) = ( πi ) m m X n =0 n − B n ( ) E m − n (2 n )!(2 m − n )! ( τ + 1) m − n h − (2 τ + 1) n − i , (52)where the first sum is the right-hand side of (38) and the second sum is, up tonotation, the one derived in [LRR14]. To see that these two rational functionsindeed coincide, one may proceed as in Remark 6.7. On the other hand, in order tofurther illustrate how ψ m transforms under the full modular group, we now sketcha proof of their equivalence, which, ultimately, derives from B = ST − S − =( T ST ) .We can easily check that ψ m | − m T is given by ψ m ( τ + 1) = ∞ X n =1 ( − n sec( πnτ ) n m = 12 m − ψ m (2 τ ) − ψ m ( τ ) . It follows that ψ m | − m T S = ˆ ψ m ( τ / − ˆ ψ m ( τ ) − h m ( τ /
2) + h m ( τ ) . Since cot ( z/ − cot ( z ) = csc( z ), this equals ψ m | − m T S = ∞ X n =1 χ − ( n ) csc( πn τ )( n/ m − h m ( τ /
2) + h m ( τ ) , which then implies that ψ m | − m T ST = − ∞ X n =1 χ − ( n ) csc( πn τ )( n/ m − h m ( τ / h m ( τ + 2)= − ψ m | − m T S + [ h m ( τ ) − h m ( τ / | − m (1 + T ) . Since B = ( T ST ) = ( T ST ) ST , we thus arrive at(53) ψ m | − m B = ψ m + [ h m ( τ ) − h m ( τ / | − m ( ST + T ST ) . Comparing (53) with (51), we have shown that h m ( τ ) | − m ( S + T − S ) = [ h m ( τ ) − h m ( τ / | − m ( ST + T ST ) , which, upon expanding and using the relation B n ( ) = − (1 − − n ) B n , results inthe desired equality (52). N A SECANT DIRICHLET SERIES 25
Having thus come full circle, we close with remarking that a number of furtherinfinite sums, similar in shape to (1) and (2), with trigonometric summands andmodular properties are discussed in [Ber89, Chapter 14] and the references therein.9.
Conclusion
We have reviewed the well-known fact that, among similar sums, the cotangentand secant Dirichlet series of appropriate parity are Eichler integrals of Eisen-stein series of level 1 and 4, respectively. Their functional equations, recorded byRamanujan in his notebooks, are thus instances of the modular transformationproperties of Eichler integrals in general, with their precise form determined bythe corresponding period polynomials. This has lead us to explicitly compute pe-riod polynomials of Eisenstein series of higher level. Motivated by recent results[GMR11], [MSW11], [LR13], [LS13] on the zeros of Ramanujan polynomials, whicharise in the level 1 case, we observe that the generalized Ramanujan polynomialsappear to also be (nearly) unimodular. On the other hand, it was recently shownin [CFI12] and [EGR13] that the nontrivial zeros of period polynomials of modularforms, which are Hecke eigenforms of level 1, all lie on the unit circle. Our obser-vations for Eisenstein series of higher level suggest that it could be interesting toextend the results in [CFI12], [EGR13] to the case of higher level.
Acknowledgements.
We thank Matilde Lal´ın, Francis Rodrigue and MathewRogers for sharing the preprint [LRR14], which motivated the present work. We arevery grateful to Alexandru Popa for making us aware of the recent paper [PP13] andfor very helpful discussions, as well as to Bernd Kellner for comments on an earlierversion of this paper. Finally, the second author would like to thank the Max-Planck-Institute for Mathematics in Bonn, where part of this work was completed,for providing wonderful working conditions.
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Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W.Green St, Urbana, IL 61801, United States
E-mail address : [email protected] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W.Green St, Urbana, IL 61801, United States
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