In contrast to the rest of this paper, this section does not directly relate to a regular sequence but gives a general method to derive Fourier coefficients of fluctuations.
Pseudo-Tauberian Theorem
In this section, we generalise the pseudo-Tauberian argument by Flajolet, Grabner, Kirschenhofer, Prodinger and Tichy [18, Proposition 6.4]. The basic idea is that for a 1-periodic Hölder-continuous function \(\Phi \) and \(\gamma \in {\mathbb {C}}\), there is a 1-periodic continuously differentiable function \(\Psi \) such that
and there is a straight-forward relation between the Fourier coefficients of \(\Phi \) and the Fourier coefficients of \(\Psi \). This relation exactly corresponds to the additional factor \(s+1\) when transitioning from the zeroth order Mellin–Perron formula to the first order Mellin–Perron formula.
In contrast to [18, Proposition 6.4], we allow for an additional logarithmic factor, have weaker growth conditions on the Dirichlet series and quantify the error. We also extend the result to all complex \(\gamma \). The generalisation from \(q=2\) there to our real \(q>1\) is trivial.
Proposition 14.1
Let \(\gamma \in {\mathbb {C}}\) and \(q>1\) be a real number, m be a positive integer, \(\Phi _0, \ldots , \Phi _{m-1}\) be 1-periodic Hölder continuous functions with exponent \(\alpha >0\), and \(0<\beta <\alpha \). Then there exist continuously differentiable functions \(\Psi _{-1}\), \(\Psi _{0}, \ldots , \Psi _{m-1}\), periodic with period 1, and a constant c such that
for integers \(N\rightarrow \infty \).
Denote the Fourier coefficients of \(\Phi _j\) and \(\Psi _j\) by \(\varphi _{j\ell }:=\int _0^1\Phi _j(u)\exp (-\,2\ell \pi i u)\, {\mathrm {d}}u\) and \(\psi _{j\ell }:=\int _0^1\Psi _j(u)\exp (-\,2\ell \pi i u)\, {\mathrm {d}}u\), respectively. Then the corresponding generating functions fulfil
for \(\ell \in {\mathbb {Z}}\) and \(Z\rightarrow 0\).
If \(q^{\gamma +1}\ne 1\), then \(\Psi _{-1}\) vanishes.
Remark 14.2
Note that the constant c is absorbed by the error term if \(\mathfrak {R}\gamma +1>\alpha \), in particular if \(\mathfrak {R}\gamma >0\). Therefore, this constant does not occur in the article [18].
Remark 14.3
The factor \(\gamma +1+\frac{2\ell \pi i}{\log q} + Z\) in (14.2) will turn out to correspond exactly to the additional factor \(s+1\) in the first order Mellin–Perron summation formula with the substitution \(s=\gamma +\frac{2\ell \pi i}{\log q}+ Z\) such that the local expansion around the pole in \(s=\gamma +\frac{2\ell \pi i}{\log q}\) of the Dirichlet generating function is conveniently written as a Laurent series in Z. See the proof of Theorem E for details.
Before actually proving Proposition 14.1, we give an outline.
Overview of the Proof of Proposition 14.1
We start with the left-hand side of (14.1) and split the range of summation according to \(\lfloor \log _q n \rfloor \), thereby, in terms of our periodic functions, split after each period. We then use periodicity of the \(\Phi _j\) and collect terms. This results in Riemann sums which converge to the corresponding integrals. Therefore, we can approximate these sums by the integrals.
More rewriting constructs and reveals the functions \(\Psi _j\) [of the right-hand side of (14.1)]: these functions are basically defined via the above mentioned integral. We then show that these functions are indeed periodic and that their Fourier coefficients relate to the Fourier coefficients of the \(\Phi _j\). The latter is done by a direct computation of the integrals defining these coefficients.
For this proof, we use an approach via exponential generating functions. This reduces the overhead for dealing with the logarithmic factors \((\log n)^k\) in (14.1) such that we can essentially focus on the case \(m=1\). The resulting formula (14.1) follows by extracting a suitable coefficient of this power series.
There is another benefit of the generating function approach: this formulation allows to easily translate the relation between the Fourier coefficients here to the additional factors occurring when transitioning to higher order Mellin–Perron summation formulæ, in particular the factor \(s+1\) in the first order Mellin–Perron summation. \(\square \)
Proof of Proposition 14.1
We split the proof into six parts.
Notations. We start by defining quantities that are used through the whole proof.
Without loss of generality, we assume that \(q^{\mathfrak {R}\gamma +1}\ne q^{\alpha }\): otherwise, we slightly decrease \(\alpha \) keeping the inequality \(\beta <\alpha \) intact. We use the abbreviations \(\Lambda :=\lfloor \log _q N \rfloor \), \(\nu :=\{ \log _q N \}\), i.e., \(N=q^{\Lambda +\nu }\). We use the generating functions
for \(0\le u\le 1\) and \(0<|Z |<2r\) where \(r>0\) is chosen such that \(r<(\alpha -\beta )/2\) and such that \(Q(Z)\ne 1\) and \(|Q(Z) |\ne q^{\alpha }\) for these Z. (The condition \(Z\ne 0\) is only needed for the case \(q^{1+\gamma }=1\).) We will stick to the above choice of r and restrictions for Z throughout the proof.
It is easily seen that the left-hand side of (14.1) equals \([Z^{m-1}]L(N, Z)\), where \([Z^{m-1}]\) denotes extraction of the coefficient of \(Z^{m-1}\).
Approximation of the Sum by an Integral. We will now rewrite L(N, Z) so that its shape is that of a Riemann sum, therefore enabling us to approximate it by an integral.
Splitting the range of summation with respect to powers of q yields
We write \(n=q^px\) (or \(n=q^\Lambda x\) for the second sum), use the periodicity of \(\Phi \) in u and get
The inner sums are Riemann sums converging to the corresponding integrals for \(p\rightarrow \infty \). We set
It will be convenient to change variables \(x=q^w\) in I(u, Z) to get
We define the error \(\varepsilon _p(u, Z)\) by
As the sum and the integral are both analytic in Z, their difference \(\varepsilon _p(u, Z)\) is analytic in Z, too. We bound \(\varepsilon _{p}(u, Z)\) by the difference of upper and lower Darboux sums (step size \(q^{-p}\)) corresponding to the integral I(u, Z): On each interval of length \(q^{-p}\), the maximum and minimum of a Hölder continuous function can differ by at most
. As the integration interval as well as the range for u and Z are finite, this translates to the bound
as \(p\rightarrow \infty \) uniformly in \(0\le u\le 1\) and \(|Z |<2r\). This results in
$$\begin{aligned} L(N, Z)= & {} I(1, Z)\sum _{0\le p<\Lambda }Q(Z)^p + \sum _{0\le p<\Lambda }Q(Z)^p \varepsilon _{p}(1, Z) \\&+ I(\nu , Z)\,Q(Z)^{\Lambda } + Q(Z)^\Lambda \varepsilon _{\Lambda }(\nu , Z). \end{aligned}$$
If \(|Q(Z) |/q^\alpha =q^{\mathfrak {R}\gamma +1 + \mathfrak {R}Z -\alpha }<1\), i.e., \(\mathfrak {R}\gamma +\mathfrak {R}Z<\alpha -1\), the second sum involving the integration error converges absolutely and uniformly in Z for \(\Lambda \rightarrow \infty \) to some analytic function \(c'(Z)\); therefore, we can replace the second sum by
in this case. If \(\mathfrak {R}\gamma + \mathfrak {R}Z>\alpha -1\), then the second sum is
. By our choice of r, the case \(\mathfrak {R}\gamma +\mathfrak {R}Z=\alpha -1\) cannot occur. So in any case, we may write the second sum as
by our choice of r. The last summand involving \(\varepsilon _{\Lambda }(\nu , Z)\) is absorbed by the error term of the second summand. Note that the error term is uniform in Z and, by its construction, analytic in Z.
Thus we end up with
where
$$\begin{aligned} S(N, Z):=I(1, Z)\sum _{0\le p<\Lambda }Q(Z)^p+I(\nu , Z)Q(Z)^\Lambda . \end{aligned}$$
(14.5)
It remains to rewrite S(N, Z) in the form required by (14.1). We emphasise that we will compute S(N, Z) exactly, i.e., no more asymptotics for \(N\rightarrow \infty \) will play any rôle.
Construction of\(\Psi \). We will now rewrite the expression S(N, Z) such that the generating function \(\Psi \) [i.e., the fluctuations of the right-hand side of (14.1)] appears. After this, we will gather properties of \(\Psi \) including properties of its Fourier coefficients.
We rewrite (14.5) as
We replace \(\Lambda \) by \(\log _q N - \nu \) and use
to get
with
Periodic Extension of\(\Psi \). A priori, it is not clear that the function \(\Psi (u, Z)\) defined above can be extended to a periodic function (and therefore Fourier coefficients can be computed later on). The aim now is to show that it is possible to do so.
It is obvious that
is continuously differentiable in \(u\in [0, 1]\). We have
because \(I(0, Z)=0\) by (14.3). The derivative of
with respect to u is
which implies that
We can therefore extend
to a 1-periodic continuously differentiable function in u on \({\mathbb {R}}\).
Fourier Coefficients of\(\Psi \) Knowing that \(\Psi \) is a periodic function, we can now head for its Fourier coefficients and relate them to those of \(\Phi \).
By using equations (14.7) and (14.3), \(Q(Z)=q^{\gamma +1+Z}\), and \(\exp (-\,2\ell \pi iu)=q^{-\chi _\ell u}\) with \(\chi _\ell =\frac{2\pi i\ell }{\log q}\), we now express the Fourier coefficients of
in terms of those of
by
The second and third summands cancel, and we get
Extracting Coefficients. So far, we have proven everything in terms of generating functions. We now extract the coefficients of these power series which will give us the result claimed in Proposition 14.1.
By (14.7),
is analytic in Z for \(0<|Z |<2r\). If \(q^{\gamma +1}\ne 1\), then it is analytic in \(Z=0\), too. If \(q^{\gamma +1}=1\), then (14.7) implies that
might have a simple pole in \(Z=0\). Note that all other possible poles have been excluded by our choice of r. For \(j\ge -1\), we write
and use Cauchy’s formula to obtain
This and the properties of
established above imply that \(\Psi _j\) is a 1-periodic continuously differentiable function.
Inserting (14.6) in (14.4) and extracting the coefficient of \(Z^{m-1}\) using Cauchy’s theorem and the analyticity of the error in Z yields (14.1) with
. Rewriting (14.8) in terms of \(\Psi _j\) and \(\Phi _j\) leads to (14.2). Note that we have to add
in (14.2) to compensate the fact that we do not include \(\psi _{j\ell }\) for \(j\ge m\).
\(\square \)
We prove a uniqueness result.
Lemma 14.4
Let m be a positive integer, \(q>1\) be a real number, \(\gamma \in {\mathbb {C}}\) such that \(\gamma \notin \frac{2\pi i}{\log q}{\mathbb {Z}}\), \(c\in {\mathbb {C}}\), and \(\Psi _0, \ldots , \Psi _{m-1}\) and \(\Xi _0, \ldots , \Xi _{m-1}\) be 1-periodic continuous functions such that
for integers \(N\rightarrow \infty \). Then \(\Psi _k=\Xi _k\) for \(0\le k<m\).
Proof
If \(\mathfrak {R}\gamma <0\) and \(c\ne 0\), then (14.9) is impossible as the growth of the right-hand side of the equation is larger than that on the left-hand side. So we can exclude this case from further consideration. We proceed indirectly and choose k maximally such that \(\Xi _k\ne \Psi _k\). Dividing (14.9) by \((\log _q N)^k\) yields
for \(N\rightarrow \infty \). Let \(0< u<1\) and set \(N_j=\lfloor q^{j+u} \rfloor \). We clearly have \(\lim _{j\rightarrow \infty } N_j=\infty \). Then
$$\begin{aligned} j+u + \log _q(1-q^{-j-u}) = \log _q(q^{j+u}-1)\le \log _q N_j \le j+u. \end{aligned}$$
We define \(\nu _j:=\log _q N_j-j-u\) and see that
for \(j\rightarrow \infty \), i.e., \(\lim _{j\rightarrow \infty } \nu _j = 0\). This implies that \(\lim _{j\rightarrow \infty }\{ \log _q N_j \}=u\) and therefore
$$\begin{aligned} \lim _{j\rightarrow \infty }(\Xi _k-\Psi _k)(\log _q N_j)=\lim _{j\rightarrow \infty }(\Xi _k-\Psi _k)(\{ \log _q N_j \})=\Xi _k(u)-\Psi _k(u). \end{aligned}$$
Setting \(N=N_j\) in (14.10) and letting \(j\rightarrow \infty \) shows that
$$\begin{aligned} \Xi _k(u)-\Psi _k(u) = \lim _{j\rightarrow \infty }cN_j^{-\gamma }[ k=0 ]. \end{aligned}$$
(14.11)
If \(k\ne 0\) or \(\mathfrak {R}\gamma >0\), we immediately conclude that \(\Xi _k(u)-\Psi _k(u)=0\). If \(\mathfrak {R}\gamma <0\) we have \(c=0\), which again implies that \(\Xi _k(u)-\Psi _k(u)=0\).
Now we assume that \(\mathfrak {R}\gamma =0\) and \(k=0\). We set \(\beta :=-\frac{\log q}{2\pi i}\gamma \), which implies that \(N^{-\gamma }=\exp (2\pi i \beta \log _q N)\). We choose sequences \((r_\ell )_{\ell \ge 1}\) and \((s_\ell )_{\ell \ge 1}\) such that \(\lim _{\ell \rightarrow \infty }s_\ell =\infty \) and \(\lim _{\ell \rightarrow \infty }|s_\ell \beta - r_\ell |=0\): For rational \(\beta =r/s\), we simply take \(r_\ell =\ell r\) and \(s_\ell =\ell s\), and for irrational \(\beta \), we consider the sequence of convergents \((r_\ell /s_\ell )_{\ell \ge 1}\) of the continued fraction of \(\beta \) and the required properties follow from the theory of continued fractions; see for example [28, Theorems 155 and 164]. By using \(\log _q N_j = j+u+\nu _j\), we get
These two limits are distinct as \(\beta \notin {\mathbb {Z}}\) by assumption. Thus \(\lim _{j\rightarrow \infty }N_j^{-\gamma }\) does not exist. Therefore, (14.11) implies that \(c=0\) and therefore \(\Xi _k(u)-\Psi _k(u)=0\).
We proved that \(\Xi _k(u)=\Psi _k(u)\) for \(u\notin {\mathbb {Z}}\). By continuity, this also follows for all \(u \in {\mathbb {R}}\); contradiction. \(\square \)
Proof of Theorem E
We again start with an outline of the proof.
Overview of the Proof of Theorem E
The idea is to compute the repeated summatory function of F twice: On the one hand, we use the pseudo-Tauberian Proposition 14.1 to rewrite the right-hand side of (6.6) in terms of periodic functions \(\Psi _{aj}\). On the other hand, we compute it using a higher order Mellin–Perron summation formula, relating it to the singularities of \({\mathcal {F}}\). More specifically, the expansions at the singularities of \({\mathcal {F}}\) give the Fourier coefficients of \(\Psi _{aj}\). The Fourier coefficients of the functions \(\Psi _{aj}\) are related to those of the functions \(\Phi _j\) via (14.2). \(\square \)
And up next comes the actual proof.
Proof of Theorem E
Initial observations and notations. As \(\Phi _j\) is Hölder continuous, its Fourier series converges by Dini’s criterion; see, for example, [40, p. 52].
For any sequence g on \({\mathbb {Z}}_{>0}\), we set \(({\mathcal {S}}g)(N):=\sum _{1\le n< N}g(n)\). We set \(A=1 + \max \{ \lfloor \eta \rfloor , 0 \}\). In particular, A is a positive integer with \(A>\eta \).
Asymptotic Summation. We first compute the Ath repeated summatory function \({\mathcal {S}}^A F\) of F (i.e., the \((A+1)\)th repeated summatory function \({\mathcal {S}}^{A+1} f\) of the function f) by applying Proposition 14.1A times. This results in an asymptotic expansion involving new periodic fluctuations while keeping track of the relation between the Fourier coefficients of the original fluctuations and those of the new fluctuations.
A simple induction based on (6.6) and using Proposition 14.1 shows that there exist 1-periodic continuous functions \(\Psi _{aj}\) for \(a\ge 0\) and \(-\,1\le j<m\) and some constants \(c_{ab}\) for \(0\le b<a\) such that
for integers \(N\rightarrow \infty \). In fact, \(\Psi _{0j}=\Phi _j\) for \(0\le j<m\). For \(a\ge 1\) and \(-\,1\le j<m\), \(\Psi _{aj}\) is continuously differentiable. Note that the case that \(q^{\gamma +a+1}=1\) occurs for at most one \(0\le a<A\), which implies that the number of non-vanishing fluctuations increases at most once in the application of Proposition 14.1. Also note that the assumption \(\alpha >\mathfrak {R}\gamma -\gamma _0\) implies that the error terms arising in the application of Proposition 14.1 are absorbed by the error term stemming from (6.6).
We denote the corresponding Fourier coefficients by
$$\begin{aligned} \psi _{aj\ell }:=\int _{0}^1 \Psi _{aj}(u)\exp (-\,2\ell \pi i u)\,{\mathrm {d}}u \end{aligned}$$
for \(0\le a\le A\), \(-\,1\le j<m\), \(\ell \in {\mathbb {Z}}\). By (14.2) the generating functions of the Fourier coefficients fulfil
for \(0\le a<A\), \(\ell \in {\mathbb {Z}}\) and \(Z\rightarrow 0\). Iterating this recurrence yields
for \(\ell \in {\mathbb {Z}}\) and \(Z\rightarrow 0\).
Explicit Summation. We now compute \({\mathcal {S}}^{A+1} f\) explicitly with the aim of decomposing it into one part which can be computed by the Ath order Mellin–Perron summation formula and another part which is smaller and can be absorbed by an error term.
Explicitly, we have
$$\begin{aligned} ({\mathcal {S}}^{a+1}f)(N) = \sum _{1\le n_1<n_2<\cdots<n_{a+1}<N}f(n_1) = \sum _{1\le n<N}f(n)\sum _{n<n_2<\cdots<n_{a+1}<N}1 \end{aligned}$$
for \(0\le a \le A\). Note that we formally write the outer sum over the range \(1\le n<N\) although the inner sum is empty (i.e., equals 0) for \(n\ge N-a\); this will be useful later on. The inner sum counts the number of selections of a elements out of \(\{ n+1,\ldots , N-1 \}\), thus we have
$$\begin{aligned} ({\mathcal {S}}^{a+1}f)(N) = \sum _{1\le n< N}\left( {\begin{array}{c}N-n-1\\ a\end{array}}\right) f(n)=\sum _{1\le n< N}\frac{1}{a!}(N-n-1)^{\underline{a}}f(n)\nonumber \\ \end{aligned}$$
(14.14)
for \(0\le a\le A\) and falling factorials \(z^{\underline{a}}:=z(z-1)\cdots (z-a+1)\).
The polynomials \(\frac{1}{a!}(U-1)^{{\underline{a}}}\), \(0\le a\le A\), are clearly a basis of the space of polynomials in U of degree at most A. Thus, there exist rational numbers \(b_0, \ldots , b_A\) such that
$$\begin{aligned} \frac{U^A}{A!}=\sum _{0 \le a \le A} \frac{b_a}{a!} (U-1)^{\underline{a}}. \end{aligned}$$
Comparing the coefficients of \(U^A\) shows that \(b_A=1\). Substitution of U by \(N-n\), multiplication by f(n) and summation over \(1\le n<N\) yield
$$\begin{aligned} \frac{1}{A!}\sum _{1\le n<N}(N-n)^A f(n) = \sum _{0 \le a \le A} b_a ({\mathcal {S}}^{a+1}f)(N) \end{aligned}$$
by (14.14). When inserting the asymptotic expressions from (14.12), the summands involving fluctuations for \(0\le a< A\) are absorbed by the error term
of the summand for \(a=A\) because \(\mathfrak {R}\gamma - \gamma _0 < 1\). Thus there are some constants \(c_b\) for \(0\le b<A\) such that
for integers \(N\rightarrow \infty \).
If \(\gamma +A=b+\chi _{\ell '}\) for some \(0\le b<A\) and \(\ell '\in {\mathbb {Z}}\), then we assume without loss of generality that \(c_{b}=0\): Otherwise, we replace \(\Psi _{A(m-1)}(u)\) by \(\Psi _{A(m-1)}(u) + c_{b}\exp (-\,2\ell '\pi i u)\) and \(c_{b}\) by 0. Both (14.15) and (14.13) remain intact: the former trivially, the latter because the factor for \(a=A-b\) in (14.13) equals \(\gamma +A-b-\chi _{\ell '} + Z=Z\) which compensates the fact that the Fourier coefficient \(\psi _{A(m-1)(-\,\ell ')}\) is modified.
Mellin–Perron summation. We use the Ath order Mellin–Perron summation formula to write the main contribution of \({\mathcal {S}}^{A+1} f\) as determined above in terms of new periodic fluctuations \(\Xi _j\) whose Fourier coefficients are expressed in terms of residues of a suitably modified version of the Dirichlet generating function \({\mathcal {F}}\).
Without loss of generality, we assume that \(\sigma _{\mathrm {abs}}>0\): the growth condition (6.8) trivially holds with \(\eta =0\) on the right of the abscissa of absolute convergence of the Dirichlet series. By the Ath order Mellin–Perron summation formula (see [18, Theorem 2.1]), we have
$$\begin{aligned} \frac{1}{A!}\sum _{1\le n<N}(N-n)^A f(n) = \frac{1}{2\pi i}\int _{\sigma _{\mathrm {abs}}+1-i\infty }^{\sigma _{\mathrm {abs}}+1+i\infty } \frac{{\mathcal {F}}(s) N^{s+A}}{s(s+1)\cdots (s+A)}\,{\mathrm {d}}s \end{aligned}$$
with the arbitrary choice \(\sigma _{\mathrm {abs}}+1>\sigma _{\mathrm {abs}}\) for the real part of the line of integration.
The growth condition (6.8) allows us to shift the line of integration to the left such that
The summand for a in the second term corresponds to a possible pole at \(s=-a\) which is not taken care of in the first sum; note that \({\mathcal {F}}(s)\) is analytic at \(s=-a\) in this case by assumption because of \(\gamma _0<-a\).
We now compute the residue at \(s=\gamma +\chi _\ell \). We use
$$\begin{aligned} N^{s+A} = N^{\gamma +A+\chi _\ell }\sum _{k\ge 0}\frac{(\log N)^k}{k!} (s-\gamma -\chi _\ell )^k \end{aligned}$$
to split up the residue as
with
for \(j\ge -1\). Note that we allow \(j=-1\) for the case of \(\gamma \in -a+\frac{2\pi i}{\log q}{\mathbb {Z}}\) for some \(1\le a\le A\) when \({\mathcal {F}}(s)/\bigl (s\cdots (s+A)\bigr )\) might have a pole of order \(m+1\) at \(s=-a\). Using the growth condition (6.8) and the choice of A yields
for \(|\mathfrak {I}s |\rightarrow \infty \) and s which are at least a distance \(\delta \) away from the poles \(\gamma +\chi _\ell \). By writing the residue in (14.16) in terms of an integral over a rectangle around \(s=\gamma +\chi _\ell \) (distance again at least \(\delta \) away from \(\gamma +\chi _\ell \)), we see that (14.17) implies
for \(|\ell |\rightarrow \infty \). Moreover, by (14.17), we see that
Thus we proved that
for
$$\begin{aligned} \Xi _j(u) =\sum _{\ell \in {\mathbb {Z}}}\xi _{j\ell } \exp (2\ell \pi i u) \end{aligned}$$
(14.20)
where the \(\xi _{j\ell }\) are given in (14.16). By (14.18), the Fourier series (14.20) converges uniformly and absolutely. This implies that \(\Xi _j\) is a 1-periodic continuous function.
Fourier Coefficients. We will now compare the two asymptotic expressions for \({\mathcal {S}}^{A+1} f\) obtained so far to see that the fluctations coincide. We know explicit expressions for the Fourier coefficients of the \(\Xi _j\) in terms of residues, and we know how the Fourier coefficients of the fluctuations of the repeated summatory function are related to the Fourier coefficients of the fluctuations of F. Therefore, we are able to compute the latter.
By (14.15), (14.19), elementary asymptotic considerations for the terms \(N^b\) with \(b>\mathfrak {R}\gamma +A\), Lemma 14.4 and the fact that \(c_{b}=0\) if \(b\in \gamma +A+\frac{2\pi i}{\log q}{\mathbb {Z}}\) for some \(0\le b<A\), we see that \(\Xi _j=\Psi _{Aj}\) for \(-\,1\le j<m\). This immediately implies that \({\mathcal {F}}(0)=0\) if \(\gamma _0<0\) and \(\gamma \notin \frac{2\pi i}{\log q}{\mathbb {Z}}\).
To compute the Fourier coefficients \(\psi _{Aj\ell }=\xi _{j\ell }\), we set \(Z:=s-\gamma -\chi _\ell \) to rewrite (14.16) using (6.7) as
$$\begin{aligned} \psi _{Aj\ell }=[Z^{-1}] \frac{\sum _{b\ge 0}\varphi _{b\ell }Z^{b-j-1}}{\prod _{1 \le a \le A} (\gamma +a+\chi _\ell +Z)} =[Z^{j}] \frac{\sum _{b\ge 0}\varphi _{b\ell } Z^{b}}{\prod _{1 \le a \le A} (\gamma +a+\chi _\ell +Z)} \end{aligned}$$
for \(-\,1\le j<m\) and \(\ell \in {\mathbb {Z}}\). This is equivalent to
for \(\ell \in {\mathbb {Z}}\) and \(Z\rightarrow 0\). Clearing the denominator and using (14.13) as announced in Remark 14.3 lead to
for \(\ell \in {\mathbb {Z}}\) and \(Z\rightarrow 0\). Comparing coefficients shows that \(\psi _{0j\ell }=\varphi _{j\ell }\) for \(0\le j<m\) and \(\ell \in {\mathbb {Z}}\). This proves (6.9). \(\square \)