On the non-vanishing of certain Dirichlet series

# On the non-vanishing of certain Dirichlet series

Sandro Bettin DIMA - Dipartimento di Matematica, Via Dodecaneso, 35, 16146 Genova - ITALY  and  Bruno Martin Unité Mixte Internationale 3457 - CNRS & CRM - Université de Montréal - CANADA.
###### Abstract.

Given , we study the vanishing of the Dirichlet series

 Dk(s,f):=∑n≥1dk(n)f(n)n−s

at the point , where is a periodic function modulo a prime . We show that if or and , then there are no odd rational-valued functions such that , whereas in all other cases there are examples of odd functions such that .

As a consequence, we obtain, for example, that the set of values , where ranges over odd characters mod , are linearly independent over .

###### Key words and phrases:
Chowla’s problem, non-vanishing of Dirichlet series, Special values of Dirichlet L-series
###### 2010 Mathematics Subject Classification:
11M41, 11L03, 11M20 (primary), 11R18 (secondary)

## 1. Introduction

Let be prime and let be a number field. For a function which is periodic modulo , let be the Dirichlet series

 L(s,f):=∞∑n=1f(n)ns,

which is absolutely convergent for . Since , where is the Hurwitz zeta-function which is meromorphic in with a pole of residue at only, one has that admits meromorphic continuation to with (possibly) a simple pole at only of residue with

In particular, if then is entire.

In the papers [Cho64, Cho70] Chowla asked whether it is possible that for some rational-valued periodic function satisfying and with not identically zero. Following an approach outlined by Siegel, Chowla solved the problem in the case where is odd by showing that in this case is never zero. Later, Baker, Birch and Wirsing [BBW73] used Baker’s theorem on linear forms in logarithms to give a complete answer to Chowla’s question showing that whenever , where with . In the following years Chowla’s problem was considered and generalized by several other authors, for example we mention the work of Gun, Murty and Rath [GMR12] where other points besides were considered (and where the condition on was slightly relaxed) and the works of Okada [Oka82] and of Chatterjee and Murty [CM14], who gave equivalent criteria for the vanishing of when no condition on is imposed. See also [MM11] for a variation of the proof of the result by Baker, Birch and Wirsing.

In this paper we consider the analogue of Chowla’s problem for

 Dk(s,f):=∞∑n=1dk(n)f(n)ns=∞∑n1,…,nk=1f(n1⋯nk)(n1⋯nk)s,

where . As for , is absolutely convergent for and, expressing each of the series in the second expression for in terms of Hurwitz zeta-functions, one obtains analytic continuation for to . In the case where , the analyticity of at is equivalent to having and (see Lemma 5). Notice that if is odd, then both conditions are automatically met.

If is not odd, then one can easily see that by appealing to Schanuel’s conjecture. We remind the reader that Schanuel’s conjecture predicts that for any which are linearly independent over the transcendence degree of over is at least .

###### Proposition 1.

Let be prime and let . Let be -periodic with Then, under Schanuel’s conjecture we have that if then is odd.

Proposition 1 is an easy consequence of the fact that for odd is an algebraic number whereas and the values , as ranges over even non-principal Dirichlet character mod , are known to be algebraically independent under Schanuel’s conjecture. In fact the full Schanuel’s conjecture is not needed here, an analogue of Baker’s theorem for linear forms in -th powers of logarithms would suffice for Proposition 1.

Thus, at least conditionally, to determine whether can be zero we just need to consider the case of odd. The case is completely analogous to the case and one has that if . If then the situation changes drastically and already for and we can find non-trivial functions such that . Indeed, if is the odd -periodic function such that , , then . Indeed,

(cf. (1.2) below), where the sums have to be interpreted as the limits as of their truncations at . Similarly, if is the odd -periodic function such that

 f(1)=18a, f(4)=18b, f(3)=18c f(2)=19a+11b+4c, f(8)=−4a+19b+11c, f(6)=−11a−4b+19c

for any , then . Notice the pattern and that the ordering we chose is not casual: indeed mod we have and , with a primitive root mod .

The above examples are far from being unique. Indeed, if , then one has no non-trivial solutions to , with odd and periodic mod , if and only if and . We classify the possible cases in the following Theorem, which generalizes the result of Chowla corresponding to the case .

###### Theorem 1.

Let , be an odd prime and let be a number field with . Let be the vector space over consisting of odd -periodic functions and let be the subspace . Then,

 (1.1) dimK(V0)≥{dimK(V)r−1rif v2(p−1)>v2(k),dimK(V)r−2rif v2(p−1)≤v2(k),

where and denotes the -adic valuation of . Moreover, the equality holds if or if and . In particular, if and only if or if and .

In the cases and or and we shall also show that whenever has support entirely contained in the set of square residues mod (or, analogously, of square non-residues). As a consequence of this and of Theorem 1 we will deduce the following.

###### Theorem 2.

Let , be an odd prime with either or and . Then the set of values are linearly independent over for that runs through the odd Dirichlet characters mod . Moreover, under Schanuel’s conjecture the same result holds true also when varies among all non-principal Dirichlet characters mod .

It seems likely that the equality in (1.1) (as well as a suitable modification of Theorem 2) holds true with no conditions on ; in order to prove this one would need to show that certain explicit linear combinations of -th powers of Dirichlet -functions are non-zero.

At first sight Theorem 1 doesn’t seem to say anything about the interesting case of the odd part of the Estermann function at , , where . Indeed, the number field generated by has a non-trivial intersection with . However in fact one has (see [CW35] Hilfsatz 14 or [dlBT04] Theorem 4.4) with for and otherwise, where denotes the fractional part of . Thus, the non-vanishing of follows directly from Chowla’s result (i.e. Theorem 1 with ).

The proof of Theorem 1 is in fact a variation of Chowla’s proof in [Cho70]. In this proof he showed that the values are linearly independent over by proving that if is a generator of then the determinant of the matrix is a non-zero multiple of the relative class number . One then obtains the result on the non-vanishing of from the fact that can be written as a linear combination in .

In our case, the analogue of the cotangent function is given by the sums

for and , and where indicates that the sum is restricted to coprime moduli mod . Note that is odd. We notice that is reminiscent of several other arithmetic objects. For example in the case (and ignoring the difference in the normalizations) if we replace by , we obtain the Dedekind sum, whereas if we replace one of the cotangents by its discrete Fourier transform (i.e. essentially the fractional part ), then we obtain the Vasyunin sum (for which see e.g. [Vas95, BC13]). The closest analogy, however, is with the hyper-Kloosterman sum , which is obtained by replacing by . Indeed, for even both and take values in the real cyclotomic field , where , and behave in the same way with respect to the action of the Galois group (for odd ). More precisely, and analogously to what happen for the Kloosterman sums, it is easy to see (c.f. Corollary 3) that if is the subgroup of order of , then is in , the subfield fixed by . If one could show that for all , than one would obtain that each of the values for generates the aforementioned fixed fields. We refer to [Fis92, Wan95] for some results on the algebraic properties of related to this and to Theorem 3 below.

###### Theorem 3.

Let be a prime, let and let be a number field such that . Then the values are linearly independent over if and only if or if and . If and or if and , then the values of each of the sets are linearly independent over , where is the Legendre symbol.

We also mention that, as all the other aforementioned sums, has some nice arithmetic features. For example, for , and one has

where is the class number of (c.f. Corollary 4).

We conclude this introduction by giving an alternative “analytic” expression of , which is what will allow us to prove the above Theorems. Also, from this formula one can easily deduce the asymptotic for the moments of .

###### Proposition 2.

Let , be a prime and with . Then

 (1.2)

In particular, if is odd and periodic modulo , then

 (1.3)
###### Corollary 1.

Let with be prime. Then for even we have

whereas for odd the left hand side is trivially .

### Acknowledgement

This works was started when both authors were visiting the Centre de Recherches Mathématiques in Montréal. Both authors want to thank this institution for providing excellent working conditions. Also, we thank Stéphane Louboutin for useful suggestions. The work of the first author is partially supported by PRIN “Number Theory and Arithmetic Geometry”. The second author was a member of UMI 3457 supported by CNRS funding and is also supported by the ANR (grant ANR-14CE34-0009 MUDERA).

## 2. The sum xk(r;p)

###### Lemma 2.

Let be prime and let . Then, . More precisely, if is even then , whereas if is odd then . Moreover, for all , let be the automorphism of sending . Then, for all .

###### Proof.

By definition we have

and so . Now we have and so, since , we have for even and for odd. Also,

by making the change of variables for each . ∎

###### Corollary 3.

Let with prime and let . Then, , where is the subgroup of order of .

###### Proof.

It’s well known that is cyclic so is well defined, the Corollary then follows immediately from Lemma 2. ∎

We now give a proof of Proposition 2 and then compute the trace of .

###### Remark 1.

For the sake of simplicity we shall ignore convergence issues when manipulating the order of summation of conditionally convergent series. One could make every step rigorous in several ways, for example by some analytic continuation arguments, or by using the “approximate” functional equations for the various series (in the form of exact formulae).

###### Proof of Proposition 2.

We have

 (2.1)

Now, if is a primitive odd character, then we have

 L(1,χ)

where is the Gauss sum (see [Was97], p. 36). Now, for we have

and so, after making the change of variables , we have

 (2.2)

Noticing that the right hand side is zero if is even, we then have

The result then follows by expanding the power and exploiting the orthogonal relation for Dirichlet characters. ∎

###### Corollary 4.

Let and be prime; let with . Then if , then . Otherwise, we have

In particular, if and , then

###### Proof.

By Lemma 2 and Proposition 2, given a generator of we have

Now, if , writing and making the change of variables we have that the condition on the sum over becomes . Thus, making the change we obtain the opposite of the original sum and so . Now assume , notice that in particular is even. By (2.1) we have

Taking the sum over inside, we obtain unless , i.e. if is a character of order dividing . Thus,

If and , then the only character contributing to the sum is the quadratic character and so

by the class number formula. ∎

We now give a proof of Corollary 1. We don’t give many details as the proof is very similar (and actually a bit simpler than) to the proof of Theorem 1 of [Bet15].

###### Proof of Corollary 1.

Let be an odd character. Using the functional equation

and proceeding as in [Bet15] we obtain that for every

 (2.3) L(1,χ)k=∞∑n=1dk(n)χ(n)ng(n/p2k)+Oε,k(p−5k/2).

where is a smooth function such that for , for , and for . Now, applying (1.2), (2.1) and (2.3) and then going back to the congruence condition we have

Thus, using the bound , we have

where . The contribution of the terms with for some is trivially

It follows that

Since , we can remove the contribution of the functions in at a negligible cost and we obtain the claimed result. ∎

## 3. The vanishing of Dk(1,f)

###### Lemma 5.

Let be prime and let . Let be periodic mod . Then, is entire if and only if and Also, in this case we have

 (3.1)

where .

###### Proof.

For we have

 Dk(s,f) (3.2)

where is the principal character mod . Now,

and so

In particular, is meromorphic on with possibly a pole in only. Moreover, is entire if and only if

has a zero of order at . Thus, since we have that has discriminant equal to zero. Now, the discriminant of is

and so we must have that either or the term in the big brackets is zero. Then, imposing has a zero at we find that both and the term in the brackets need to be zero, as desired. Equation (3.1) then follows immediately from (3.2). ∎

We now prove Proposition 1.

###### Proof of Proposition 1.

For , the result was proven in [BBW73] so assume . Let and so that . Then, one easily checks that for odd and for even. Thus, since for odd (see formula (2.2)), then by (3.1) also . Now, by Schanuel’s conjecture we have that and the values of for even are algebraically independent over (this is stated in the paragraph after Corollary 2 in [MM11], and essentially proved in Section 4 therein, without including ; however the same proof allows one to include since when choosing the branch for the logarithm suitably). Thus we could have only if for all even, i.e. if and so if is odd. ∎

By Proposition 1, at least conditionally, in order to find functions such that we need to take odd. Then, for odd with by (1.3) we have

 (3.3) (p−1)/2∑r=1f(r)xk(r;p)=12p−2∑j=0f(gj)xk(gj;p)=0,

where is any generator of and where we used that . If with (if is even would suffice), then we can extend the automorphism defined in Lemma 2 to an automorphism of such that acts trivially on (see [Rom06] Corollary 6.5.2 p. 161). By a slight abuse of notation we still indicate the automorphism by . Then, multiplying (3.3) by and applying we obtain new conditions for :

 (3.4) p−2∑j=0f(gj)xk(gkℓ+j;p)=0

for all . It is clearly sufficient to take where and in fact, if then we can take since the following equations are just the negative of the first ones. Thus we have a system of linear equations in the values of . We now study the determinants of the relevant matrices for such system.

###### Lemma 6.

Let . For , let

 A+(v):=⎛⎜ ⎜ ⎜ ⎜⎝v0v1…vm−2vm−1v1v2…vm−1v0⋮⋮⋮⋮vm−1v0…vm−3vm−2⎞⎟ ⎟ ⎟ ⎟⎠A−(v):=⎛⎜ ⎜ ⎜ ⎜⎝v0v1…vm−2vm−1v1v2…vm−1−v0⋮⋮⋮⋮vm−1−v0…−vm−3−vm−2⎞⎟ ⎟ ⎟ ⎟⎠.

Then,

Also, for and , let

 v±j=v±j(v):=(vj,…,vm−1,±v0,…,±vj−1),

and let for . Also, let with . Then , where is a matrix defined in the proof.

###### Proof.

One can easily check that , where

 C+(v):=⎛⎜ ⎜ ⎜ ⎜⎝v0vm−1…v2v1v1v0…v3v2⋮⋮⋮⋮vm−1vm−2…v1v0⎞⎟ ⎟ ⎟ ⎟⎠C−(v):=⎛⎜ ⎜ ⎜ ⎜⎝v0−vm−1…−v2−v1v1v0…−v3−v2⋮⋮⋮⋮vm−1vm−2…vm−1v0⎞⎟ ⎟ ⎟ ⎟⎠.

Similarly, one shows that the identity holds. The eigenvectors of are for where indicates the transpose, whereas the eigenvectors of are with and odd. The eigenvalues are given by

The statement on the determinants then follows since . ∎

###### Lemma 7.

Let be prime. Let and assume . Write with and let (so that is even). For , let be the matrix where is a generator of . Then

for a generator of the group of characters mod . Moreover, for all we have with and as in Lemma 6.

###### Proof.

Writing we have that is also a primitive root mod . Thus, by Lemma 2, we have and since

 xk(rtu(ℓ+v2);p)=xk(rtuℓ+p−12;p)=xk(−rtuℓ;p)=−xk(rtuℓ;p)

we have , with

 x=(xk(r;p),xk(rtu;p),…,xk(rtp−1−u;p)).

Thus, by Lemma 6 we have . Also,

 det(Mg,r)=(sin(πv4)+cos(πv4))v∏ℓ=0ℓ odd(v/2−1∑j=0xk(rtju;p)ξjℓv).

Now,

where is the minimum non-negative integer such that . Then, writing if and otherwise, we have that is a primitive odd character modulo . Also, generates the group of characters mod . Then, we re-write the above as

By Proposition 2 with for odd this is

Finally, we have

 v∏ℓ=0ℓ oddη(r)ℓ=η(r)∑vℓ=0(1−(−1)ℓ)ℓ/2=η(r)v2/4

and the result follows. ∎

###### Corollary 8.

With the same notation and conditions of Lemma 7, if we have

where is the relative class number of the field , that is where and are the class numbers of and respectively.

###### Proof.

First we observe that for a generator of the group of characters we have . Then, by the proposition we have

Now, referring to [Was97] (Chapters 3 and 4) for the basic results on cyclotomic fields, we have for ,

where denotes the Dedekind the zeta-function corresponding to the field . Then by the class number formula we have (c.f. [Was97] p. 41-42)

The Corollary then follows. ∎

###### Corollary 9.

With the same notation and conditions of Lemma 7, if and we have

###### Lemma 10.

Let be prime. Let and assume . Write with and let . For , let be the matrix where is a generator of . Then

for a generator of the group of characters mod . Moreover, for all we have with and as in Lemma 6.

###### Proof.

We proceed as in the proof of Lemma 7 setting and Then, since we have , with

 x=(xk(r;p),xk(rtu;p),…,xk(rtp−1−u;p)).

Thus, by Lemma 6 we have and

 det(Mg,r)=(sin(πv2)−cos(πv2))v−1∏ℓ=0(v−1∑j=0xk(rtju;p)ξjℓv).

Now, since is odd then . Also, as in the proof of Lemma 7 we have

By symmetry the innermost sum is zero if is even and otherwise it is by Proposition 2. Thus,

where if and otherwise, and the lemma follows. ∎

###### Corollary 11.

With the same notation and conditions of Proposition 10, if and we have

###### Proof.

One proceeds as for Corollary 8. ∎

###### Corollary 12.

With the same notation and conditions of Proposition 10, if and we have

###### Proof of Theorem 1 and of Theorem 2.

First we show that the equality in (1.1) and Theorem 2 hold if or if and .

Let us begin with the case and assume with odd. As explained above, if , then we have the system of equations (3.4). Also, if then is also a primitive root and so after a change of variable we can rewrite (3.4) as

 12p−2∑j=0f(tj)xk(tℓ+j;p)=(p−3)/2∑j=0f(tj)xk(tℓ+j;p)=0.

for . Equivalently, where . Thus, since by Corollary 8 we have , then the only solution to this system is , i.e. is identically zero. Thus we have proven that the equality holds in (1.1) in this case.

Now, we prove Theorem 2 for the case . Actually in this case we prove more generally that given a number field such that then the values of when runs over odd Dirichlet characters mod are linearly independent over . Assume , with and if even. Then, writing we have . Notice that is odd and takes values in the field . Thus, by Theorem 1 in the case we have and so for all , as desired.

By a similar argument as above, and by Proposition 1, we can see that the statement in Theorem 2 under Schanuel’s conjecture follows from the unconditional case.

Next we prove the equality in (1.1) and Theorem 2 when , . Let so that is a generator of . Since is a quadratic non-residue mod , spans all residues of . Thus we can rewrite the system (3.4) as

 p−32∑j=0f(t2j)xk(t2(j+ℓ);p)=0

for . Then we conclude as above, the only difference is that in this case the system is