1. Introduction
The behavior of
$L^1$
norms of exponential sums, i.e.
\begin{equation} \int_{0}^{1} \bigg | \sum_{n \leqslant X} a(n) e(n \alpha) \bigg | \,d \alpha , \quad \sum_{n \leqslant X} |a(n)|^2 \asymp X,\end{equation}
is a classical topic in harmonic analysis. It follows from works of Konyagin [Reference KonyaginKon81] and McGehee, Pigno, and Smith [Reference McGehee, Pigno and SmithMPS81], resolving a conjecture of Littlewood that (1) is always at least
$\log X$
if a(n) takes on a fixed set of values. This lower bound is attained for a(n) the indicator function of sequences of an additive nature, for example the indicator function of an arithmetic progression.
For multiplicative sequences such as
$\mu(n)$
or
$\lambda(n)$
, we expect the lower bound to be closer to
$\sqrt{X}$
, but known results fall short of this expectation. For
$a(n) = \lambda(n)$
and
$a(n) = \mu(n)$
, a lower bound of
$X^{c/\log\log X}$
and
$X^{1/6}$
, respectively, is due to Balog and Perelli [Reference Balog and PerelliBP98] and Balog and Ruzsa [Reference Balog and RuzsaBR01], respectively. The large discrepancy between these bounds arises because the proof of Balog and Ruzsa crucially uses that
$\mu(p^2 m) = 0$
for integers m and a sequence of primes p, while
$\lambda(n) \neq 0$
for all integers n.
Our main aim in this paper is to put both of these cases on the same footing and obtain improved lower bounds in both cases. Since both cases are similar, we present a full proof only for
$a(n) = \lambda(n)$
, the hitherto harder case.
Theorem 1.1.
For every
$\varepsilon \gt 0$
,
\begin{equation} \int_{0}^{1} \bigg | \sum_{n \leqslant X} \lambda(n) e(n \alpha) \bigg | \,d \alpha \gg_{\varepsilon} X^{1/4 - \varepsilon}. \end{equation}
The main arithmetic input in our proof is the observation that for any primitive character
$\chi$
, the Liouville function resonates with
$\chi(n) n^{-i \gamma}$
provided that
$\beta + i \gamma$
is a zero of
$L(s, \chi)$
. Ultimately, our lower bound comes from the fact that there exists a Dirichlet L-function with a zero with real part
$\geqslant 1/2$
. This readily follows for each
$L(s, \chi)$
from the functional equation and the explicit formula.Footnote
1
We now state a more precise version of Theorem 1.1.
Theorem 1.2.
Let
$\chi$
be a primitive character. Let
$\rho_{\chi} = \beta_{\chi} + i \gamma_{\chi}$
denote a zero of
$L(s, \chi)$
. Then, for every
$\varepsilon \gt 0$
,
\[ \int_{0}^{1} \bigg | \sum_{n \leqslant X} \lambda(n) e(n \alpha) \bigg | \,d \alpha \gg_{\rho, \varepsilon, \chi} X^{\beta_{\chi} / 2 - \varepsilon}. \]
We remark that with more careful tracking through the argument, the factor of
$X^{-\varepsilon}$
in Theorems 1.1 and 1.2 can be replaced with
$(\log X)^{-O(1)}$
.
Theorem 1.2 has some amusing implications. First, if Siegel zeros exist (in the very weak sense that
$\beta_{\chi} \rightarrow 1$
for a subsequence of characters
$\chi$
), then the lower bound in Theorem 1.1 can be improved to
$X^{1/2 - \varepsilon}$
. Second, an unrealistic (but not currently ruled out) upper bound for (2) of the form
$X^{1/2 - \delta}$
with
$\delta \gt 0$
implies a zero-free strip for every Dirichlet L-function. As a final consequence of Theorem 1.2, if one aims to just slightly improve Theorem 1.1, one can assume, without loss of generality, the Riemann hypothesis for every
$L(s, \chi)$
.
Currently, even conditionally on the generalized Riemann hypothesis (GRH), we do not know how to improve Theorem 1.1. We describe below a simple argument that recovers Theorem 1.1 conditionally on GRH. It follows from the work of Baker and Harman [Reference Baker and HarmanBH91] that, for all
$\varepsilon \gt 0$
,
\begin{equation} |S(\alpha; X)| \ll_{\varepsilon} X^{3/4 + \varepsilon} , \quad S(\alpha; X) := \sum_{n \leqslant X} \lambda(n) e(n \alpha).\end{equation}
We then note that
$$X = \int_{0}^{1} |S(\alpha; X)|^2 \, d \alpha \ll X^{3/4 + \varepsilon} \int_{0}^{1} |S(\alpha; X)| \, d \alpha,$$
and thus Theorem 1.1 follows conditionally on GRH. Improving Baker and Harman’s bound (3) presents significant challenges. Any method based on Type-II sums is a no-go: when the bilinear form factors into two terms of equal length, we can save at most
$X^{1/4}$
over the trivial bound. Thus, in that case, we need to exploit cancellations in a sum over low-lying zeros of different L-functions. This is, however, currently way out of reach.
Before proceeding to the proof, we give a quick survey of related results. When
$a(n) = \Lambda(n)$
, Vaughan [Reference VaughanVau88] shows that (1) is at least
$\gg\sqrt{X}$
and at most
$\ll\sqrt{X \log X}$
. The lower bound is non-trivial and comes from the fact that the behavior of the exponential sum near the major arcs can be understood; one can hope for similar approaches to work whenever the sequence a(n) is non-negative. For sequences a(n) such as the divisor function or Fourier coefficients of a cusp form, an asymptote of size
$\sqrt{X}$
was recently obtained by the first author [Reference PandeyPan21]. The main idea in that case is to use a recursive mechanism based on Voronoi’s summation formula; this fundamentally uses the automorphic provenance of the coefficients. Note that, in the case of cusp-form Fourier coefficients, positivity is not available and it is the existence of a nice summation formula that is crucial. Finally, for
$a(n) = \mu_r(n)$
, the indicator function of r-free numbers, Balog and Ruzsa [Reference Balog and RuzsaBR01] determined the order of magnitude of (1) to be
$X^{1/(r + 1)}$
, improving earlier work of Brüdern et al. [Reference Brüdern, Granville, Perelli, Vaughan and WooleyBGP+98]. Recently, a short interval generalization was considered in [Reference ShenShe22]. Finally, Balog and Ruzsa [Reference Balog and RuzsaBR99] have also obtained a lower bound for (1), valid for any sequence that vanishes on non-squarefree integers. This may be substantially improved as a corollary of the work of Konyagin [Reference KonyaginKon04]. A similar result for sequences supported on primes was recently obtained in [Reference Eckels, Jin, Ledoan and TobinEJL+23].
We mention also that in the next paper in this series, we will obtain results valid for general multiplicative functions using different methods.
1.1 Notation
We write
$e(x) := e^{2\pi i x}$
. Given a smooth function f, we define, respectively, its Fourier transform, Mellin transform, and Sobolev norm as
\[ \widehat{f}(u) := \int_{\mathbb{R}} f(x) e( - x u) \, dx, \ \widetilde{f}(s) := \int_{0}^{\infty} f(x) x^{s - 1} \,dx, \quad \| f \|_{p, r} := \sum_{i = 0}^{r} \bigg ( \int_{\mathbb{R}} |f^{(i)}(x)|^p \bigg )^{1/p}. \]
2. Proof of Theorems 1.1 and 1.2
Theorem 1.1 follows immediately from Theorem 1.2. Pick
$\chi$
to be the trivial character and note that
$L(s, \chi) = \zeta(s)$
has a zero on the half-line with imaginary part
$\gamma \approx 14.134725 \ldots$
. In turn, Theorem 1.2 follows from two main propositions that we state below.
Proposition 2.1. Let
$\chi$
be a primitive character of conductor q. Let w and
$\phi$
be two smooth functions, compactly supported in (0,1). Let t be a real number. Let a(n) be a sequence of real numbers. For any prime p, define a new sequence
$a_p(n) := a(n) \cdot (p \mathbf{1}_{p | n} - 1)$
. Then,
\begin{align*} &\| \phi \|_{1,2} \|w \|_{1,0} \cdot q^2(1 + t^2) \int_{0}^{1} \bigg | \sum_{n \leqslant X} a(n) e(n \alpha) \bigg | \,d \alpha \\ & \quad\gg \frac{1}{X} \sum_{p \leqslant \sqrt{X}} \bigg | \int_{p^2}^{\infty} w \bigg( \frac{y}{X}\bigg) \mathcal{S}(a_{p}, t, \chi, \phi)(y) \cdot \frac{dy}{y} \bigg |, \end{align*}
where we define, for any sequence a(n),
\begin{equation} \mathcal{S}(a, t, \chi, \phi)(y) := \sum_{n} a(n) \chi(n) n^{-it} \phi \bigg( \frac{n}{y}\bigg). \end{equation}
In the next proposition, we establish the existence of a special kernel that resonates cleanly with the partial sums of
$\lambda(n) \chi(n) n^{-i \gamma}$
when
$\gamma$
is the ordinate of a zero of
$L(s, \chi)$
. The construction of such kernels can be traced back to Pintz [Reference PintzPin83].
Proposition 2.2.
Let
$\chi$
be a primitive Dirichlet character. Let
$\rho = \beta + i \gamma$
denote a zero of
$L(s, \chi)$
. Let
$A \gt 10$
. There exists a smooth function
$w_{A, \rho,\chi}$
, compactly supported in (0,1), such that the following hold.
-
(1) For all
$0 \leqslant u \leqslant 1$
, we have
$w_{A, \rho, \chi}(u) \ll_{\rho, \chi} u^{A}$
. -
(2) For any
$Y \gt 1$
, we have (5)with
\begin{align} \int_{0}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{Y} \bigg ) \mathcal{S}(\lambda, \gamma, \chi, \phi )(y) \cdot \frac{dy}{y} = \widetilde{\phi} ( \beta ) Y^{\beta} + O_{A, \rho,\chi}(Y^{-A}), \end{align}
$\mathcal{S}(a, t, \chi, \phi)$
defined in (4).
We are now ready to conclude the proof.
Proof of Theorem 1.2. Let
$\rho = \beta + i \gamma$
be a zero of
$L(s, \chi)$
. Let
$0 \leqslant \phi \leqslant 1$
be a smooth nonzero function, compactly supported in (0, 1). The function
$\phi$
is fixed once and for all. Thus, the implicit constants in
$\gg$
will not depend on
$\phi$
. First, note that, for each
$p \leqslant X^{1/2 - \varepsilon}$
, due to the rapid decay of the kernel
$w_{A, \rho, \chi}$
(where we pick
$A = 100 / \varepsilon$
), we have
$$ \int_{p^2}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{X} \bigg ) \mathcal{S}(\lambda_{p}, t, \chi, \phi)(y) \cdot \frac{dy}{y} = \int_{0}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{X} \bigg ) \mathcal{S}(\lambda_{p}, t, \chi, \phi)(y) \cdot \frac{dy}{y} + O_{\rho, \varepsilon, \chi}(X^{-10}). $$
Since
$\lambda$
is completely multiplicative, we have
$$ \mathcal{S}(\lambda_{p}, \gamma, \chi, \phi)(y) = - p^{1-i\gamma} \chi(p) \mathcal{S}(\lambda, \gamma, \chi, \phi)(y / p) - \mathcal{S}(\lambda, \gamma, \chi, \phi)(y). $$
Therefore, by Proposition 2.2, we have
\begin{align*} & \int_{0}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{X} \bigg ) \mathcal{S}(\lambda_{p}, t, \chi, \phi)(y) \cdot \frac{dy}{y} \\ &\quad = - \widetilde{\phi} ( \beta) \cdot \bigg ( p^{-i \gamma} \chi(p) p \cdot \bigg ( \frac{X}{p} \bigg )^{\beta} + X^{\beta} \bigg ) + O_{\rho, \varepsilon, \chi}(X^{-10}). \end{align*}
In absolute value, this is
$\geqslant \widetilde{\phi}(\beta) \cdot (X^{\beta} p^{1 - \beta} - X^{\beta}) - O_{\rho ,\varepsilon, \chi}(X^{-10}) \gg_{\rho, \varepsilon, \chi} X^{\beta} p^{1 - \beta}$
for p co-prime to the conductor of
$\chi$
. Therefore, by Proposition 2.1,
$$ \int_{0}^{1} \bigg | \sum_{n \leqslant X} \lambda(n) e(n \alpha) \bigg | \,d \alpha \gg_{\rho, \varepsilon, \chi} \frac{1}{X} \sum_{\substack{p \leqslant X^{1/2 - \varepsilon}\\ p\nmid q}} X^{\beta} p^{1 - \beta} \gg X^{\beta / 2 - \varepsilon},$$
and the result follows.
3. Proof of Proposition 2.1
We first need a bound for
$L^1$
norms of exponential sums with coefficients composed of multiplicative characters. We note that the exponents in the following lemma can be sharpened, but such sharper versions are of no use to us.
Lemma 3.1. Let
$\chi$
be a primitive character of conductor q. Let
$0 \leqslant \phi \leqslant 1$
be a smooth function, compactly supported in (0, 1). Then, for all
$ y \gt 1$
, real t,
$$ \int_{0}^{1} \bigg | \sum_{n} n^{-it} \chi(n) \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \bigg | \,d\alpha \ll (1 + t^2) q^2 \cdot \| \phi \|_{1,2}. $$
Proof. Note that it is enough to bound
$$ \int_{-1/2}^{1/2} \bigg | \sum_{n} n^{-it} \chi(n) \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \bigg | \,d\alpha. $$
Furthermore, for every integer
$|a| \leqslant q$
, the contribution of all the neighborhoods
$|\alpha - a / q| \leqslant 1 / (q y)$
is trivially bounded by
$\ll 1$
. We thus assume
$1/(q y) \lt |\alpha - a / q|$
for every integer
$|a| \leqslant q$
and
$|\alpha| \leqslant \frac{1}{2}$
. Poisson summation gives
\begin{align} & \sum_{n} n^{-it} \chi(n) \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \nonumber \\ &\quad = \frac{1}{q} \sum_{\ell} \bigg ( \sum_{a \pmod{q}} \chi(a) e \bigg ( - \frac{a \ell}{q} \bigg ) \bigg ) \int_{0}^{\infty} x^{-it} \phi \bigg ( \frac{x}{y} \bigg ) e \bigg (x \alpha - \frac{\ell x}{q} \bigg ) \,d x. \end{align}
By a change of variables
$u := x / y$
, the integral is equal to
$$y^{1 - it} \int_{0}^{\infty} \phi (u) u^{-it} e \bigg (u y \bigg ( \alpha - \frac{\ell}{q} \bigg ) \bigg ) \,d u.$$
Integrating by parts twice, we see that this is
$$ \ll y (1 + |t|)^{2} \cdot (y |\alpha - \ell / q|)^{-2} \cdot \| \phi \|_{1,2}. $$
Therefore, (6) is, for
$|\alpha| \leqslant \frac{1}{2}$
,
$$ \ll \| \phi \|_{1,2} \cdot \frac{1 + t^2}{y} \bigg ( \sum_{|\ell| \leqslant q} |\alpha - \ell / q|^{-2} + \sum_{|\ell| \gt q } \frac{q^2}{\ell^2}\bigg). $$
Integrating this over
$|\alpha|\leqslant \frac{1}{2}$
, for which
$1 / (q y) \leqslant |\alpha - \ell / q|$
for every
$|\ell| \leqslant q$
, yields the final bound
$\ll (1 + t^2) q^2 \| \phi \|_{1,2}$
.
Lemma 3.2. Let
$\chi$
be a primitive character of conductor q. Let
$0 \leqslant \phi \leqslant 1$
be smooth and compactly supported in (0, 1). Let t be a real number. Then, for any
$y \leqslant X$
,
$$ \| \phi \|_{1,2} \cdot q^2(1 + t^2) \int_{0}^{1} \bigg | \sum_{n \leqslant X} \lambda(n) e(n \alpha) \bigg | \,d \alpha \gg \int_{0}^{1} \bigg | \sum_{n} \lambda(n) \chi(n) n^{-it} \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \bigg | \,d \alpha. $$
Proof. Note that
\begin{align*} &\sum_{n} \lambda(n) \chi(n) n^{-it} \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \\ &\quad = \int_{0}^{1} \bigg ( \sum_{n \leqslant X} \lambda(n) e(n (\alpha - \beta)) \bigg ) \cdot \bigg ( \sum_{n} \chi(n) n^{-it} \phi \bigg ( \frac{n}{y} \bigg ) e(n \beta) \bigg ) \,d \beta. \end{align*}
Therefore, we have
\begin{align*} &\int_{0}^{1} \bigg | \sum_{n} \lambda(n) \chi(n) n^{-it} \phi \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \bigg | \,d \alpha \\ &\quad \leqslant \int_{0}^{1} \bigg | \sum_{n \leqslant X} \lambda(n) e(n \alpha) \bigg | \, d \alpha \int_{0}^{1} \bigg | \sum_{n} \chi(n) n^{-it} \phi \bigg ( \frac{n}{y} \bigg ) e(n \beta) \bigg | \,d \beta. \end{align*}
The result now follows from Lemma 3.1.
Lemma 3.3.
There exists a smooth
$0 \leqslant \varrho \leqslant 1$
, compactly supported in
$(-1/2, 1/2)$
and such that
$\widehat{\varrho}(x) \gt 0$
for all
$x \in \mathbb{R}$
.
Proof. Let
$f(x) := 1 - |2 x|$
for
$|x| \leqslant 1/2$
, and 0 otherwise. As is well known, the Fourier transform of f is non-negative. Consider now
$$\varrho(x) := f(x) \widehat{f}(-x).$$
This is non-negative, compactly supported in
$(-1/2,1/2)$
, and bounded above by
$\widehat{f}(0) = 1/2$
. Finally,
\begin{align*} \widehat{\varrho}(u) = \int_{\mathbb{R}} \widehat{f}(u - x) f(x) \,dx. \end{align*}
Both
$\widehat{f}$
and f are non-negative, therefore, the above convolution is non-negative. In fact, it is strictly positive, since the only way it could be zero is if
$\widehat{f}$
were zero on an interval, but this is impossible because
$\widehat{f}$
is an entire function.
With these lemmas at hand, we are ready to prove Proposition 2.1.
Proof of Proposition 2.1. Fix
$0 \leqslant \varrho \leqslant 1$
, a smooth function compactly supported in
$(-1/2,1/2)$
and with
$\widehat{\varrho} \gt 0$
. Let
\begin{align*} \phi_{1}(x) := \frac{1}{m} \cdot \frac{\phi(x)}{\widehat{\varrho}(x)} , \quad m := \min_{x \in (0,1)} \widehat{\varrho}(x),\end{align*}
so that
$0 \leqslant \phi_1 \leqslant 1$
and
$\phi_1$
is compactly supported and smooth (note that
$\widehat{\varrho}$
is entire because
$\varrho$
is compactly supported). By Lemma 3.2, for any
$y \leqslant X$
we have
$$\| \phi_1 \|_{1,2} q^2(1 + t^2) \cdot \int_{0}^{1} \bigg | \sum_{n \leqslant X} a(n) e(n \alpha) \bigg | \,d \alpha \gg \int_{0}^{1} \bigg | \sum_{n} a(n) \chi(n) n^{-it} \phi_1 \bigg ( \frac{n}{y} \bigg ) e(n \alpha) \bigg | \,d \alpha.$$
Furthermore, in the above we can isolate integral neighborhoods of
$$\bigg ( \frac{b}{p} - \frac{1}{2 y}, \frac{b}{p} + \frac{1}{2 y} \bigg )$$
with
$(b,p) = 1$
and
$p \leqslant \min(\sqrt{y}, \sqrt{X})$
. We note that these are disjoint. Therefore,
\begin{align*} &\int_{0}^{1} \bigg | \sum_{n} a(n) \phi_1 \bigg ( \frac{n}{y} \bigg) e(n \alpha) \bigg | \,d \alpha\\[8pt] &\quad\geqslant \sum_{\substack{(b,p) = 1 \\ p \leqslant \min(\sqrt{y}, \sqrt{X})}} \int_{\mathbb{R}} \varrho(y \theta) \bigg | \sum_{n} a(n) \chi(n) n^{-it} \phi_1 \bigg ( \frac{n}{y} \bigg ) e \bigg ( \frac{n b}{p} + n \theta \bigg ) \bigg | \,d \theta.\end{align*}
In turn, the above is greater than
\begin{align*} &\sum_{p \leqslant \min(\sqrt{y}, \sqrt{X})} \bigg | \sum_{n} a(n) \chi(n) n^{-it} \bigg ( \sum_{(b,p) = 1} e \bigg ( \frac{n b}{p} \bigg ) \bigg ) \phi_1 \bigg ( \frac{n}{y} \bigg ) \int_{\mathbb{R}} \varrho(\theta y) e(n \theta) \,d \theta \bigg | \\[8pt] &\quad = \sum_{p \leqslant \min(\sqrt{y}, \sqrt{X})} \bigg | \sum_{n} a(n) \chi(n) n^{-it} \cdot ( p \mathbf{1}_{p | n} - 1 ) \phi_1 \bigg ( \frac{n}{y} \bigg ) \cdot \frac{1}{y} \cdot \widehat{\varrho} \bigg ( \frac{n}{y} \bigg ) \bigg | .\end{align*}
We recall that, by definition,
$\phi_{1} \widehat{\varrho} = ({1}/{m}) \phi$
, so we have obtained the lower bound
\begin{align*} \| \phi_1 \|_{1,2} \cdot q^2(1 + t^2) \int_{0}^{1} \bigg | \sum_{n \leqslant X} a(n) e(n \alpha) \bigg | d \alpha \gg \sum_{p \leqslant \min(\sqrt{y}, \sqrt{X})} \bigg | \mathcal{S}(a_p, t, \chi, \phi)(y) \cdot \frac{1}{y} \bigg | .\end{align*}
We now integrate both sides with respect to
$|w|(y / X)$
, yielding
\begin{align*}& X \| w \|_{1, 0} \| \phi_1 \|_{1,2} \cdot q^2(1 + t^2) \int_{0}^{1} \bigg | \sum_{n \leqslant X} a(n) \chi(n) n^{-it} e(n \alpha) \bigg | \, d \alpha \\&\quad \gg \sum_{p \leqslant \sqrt{X}} \bigg | \int_{p^2}^{\infty} w \bigg ( \frac{y}{X} \bigg ) \mathcal{S}(a_p, t, \chi, \phi) \cdot \frac{dy}{y} \bigg |.\end{align*}
It remains to note that
$\| \phi_1 \|_{1,2} \ll \| \phi \|_{1,2}$
, since
$\widehat{\varrho}$
is fixed once and for all.
4. Proof of Proposition 2.2
Let
$\gamma$
be the ordinate of a zero of
$L(s, \chi)$
with real part
$\beta \geqslant \frac{1}{2}$
. Consider the kernel
$$w_{A, \rho, \chi}(u) := \frac{1}{2\pi i} \int_{(2)} \frac{(s + i\gamma - 1)L(s + i \gamma, \chi)}{L(2 \beta + 2 i \gamma, \chi) (\beta + i\gamma - 1) \cdot (s - \beta)} \cdot u^{-s} \cdot \bigg ( \frac{1 - e^{-{(s - \beta)}}}{s - \beta} \bigg )^{2A} \,ds,$$
where we note that
$L(2 \beta +2 i \gamma)(\beta + i \gamma - 1) \neq 0$
, since
$\beta \geqslant 1/2$
and because there are no zeros with
$\beta = 1$
. We summarize its properties in the following lemma.
Lemma 4.1.
Let
$\chi$
be a primitive character. Let
$\rho = \beta + i \gamma$
be a zero of
$L(s, \chi)$
with
$\gamma \gt 0$
. Let
$A \geqslant 10$
. The function
$w_{A, \rho, \chi} : [0, \infty) \mapsto \mathbb{C}$
is smooth and compactly supported in [0,1) such that
$$ w_{A, \rho, \chi}(u) \ll_{A, \rho, \chi} u^{A} $$
for all
$0 \leqslant u \leqslant 1$
. Finally, suppose
$0 \leqslant \phi \leqslant 1$
is a smooth function, compactly supported in (0, 1). Then, for all Y and any
$A \gt 10$
, we have
\begin{align*} \int_{0}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{Y} \bigg ) \cdot \bigg ( \sum_{n} \lambda(n) \chi(n) n^{- i \gamma} \phi \bigg ( \frac{n}{y} \bigg ) \bigg ) \frac{d y}{y} = \widetilde{\phi} (\beta) Y^{\beta} + O_{A, \rho, \chi}(Y^{-A}). \end{align*}
Proof. For the first claim, note that we have
$$ w_{A, \rho, \chi}(u) = \frac{1}{C} \sum_{n \geqslant 1} \chi(n) n^{-i \gamma} \cdot \frac{1}{2\pi i} \int_{(2)} \frac{s + i\gamma - 1}{s - \beta} \cdot \bigg ( \frac{1 - e^{-(s - \beta)}}{s - \beta} \bigg )^{2A} \cdot (n u)^{-s} \,d s , $$
with
$C := L(2 \beta + 2 i \gamma, \chi)(\beta + i\gamma - 1)$
. If
$u \gt 1$
, then for every
$n \geqslant 1$
, upon shifting the contour to the right, we see that
\begin{align*} \frac{1}{2\pi i} \int_{(2)} \frac{s + i\gamma - 1}{s - \beta} \cdot \bigg ( \frac{1 - e^{-(s - \beta)}}{s - \beta} \bigg )^{2A} \cdot (n u)^{-s} \, ds = 0.\end{align*}
On the other hand, for the bound for
$0 \leqslant u \leqslant 1$
, we shift the contour to the left, to the line
$\Re s = -A$
, noting that the function
$$ \frac{(s + i\gamma - 1)L(s + i \gamma, \chi)}{s - \beta} $$
is entire and grows no faster than
$\ll (q (\gamma + |s|)^{1/2 - \Re s}$
.
For the last claim we note that, by a change of variable,
\begin{align*} &\int_{0}^{\infty} w_{A, \rho, \chi} \bigg ( \frac{y}{Y} \bigg )\bigg ( \sum_{n} \lambda(n) \chi(n) n^{- i \gamma} \phi \bigg ( \frac{n}{y} \bigg ) \bigg ) \frac{d y}{y} \\ &\quad = \int_{0}^{\infty} \phi \bigg ( \frac{1}{y} \bigg ) \bigg ( \sum_{n} \lambda(n) \chi(n) n^{-i \gamma} w_{A, \rho, \chi} \bigg ( \frac{n y}{Y} \bigg ) \bigg ) \frac{d y}{y}. \end{align*}
Opening up the definition of
$w_{A, \rho, \chi}$
and using the fact that
$$ \sum_{n \geqslant 1} \frac{\lambda(n) \chi(n)}{n^{s}} = \frac{L(2s, \chi)}{L(s, \chi)}, $$
we get
\begin{align*} & \sum_{n} \lambda(n) \chi(n) n^{-i \gamma} w_{A, \rho, \chi} \bigg ( \frac{n y}{Y} \bigg ) \\ &\quad = \frac{1}{2\pi i} \int_{(2)} \frac{Y^s}{y^s} \cdot \frac{(s + i\gamma - 1)L(2 s + 2 i \gamma, \chi)}{L(2 \beta + 2 i \gamma, \chi)(\beta + i\gamma - 1) \cdot (s - \beta)} \cdot \bigg ( \frac{1 - e^{-(s - \beta)}}{s - \beta} \bigg )^{2A} \,ds \\ &\quad = \bigg ( \frac{Y}{y} \bigg )^{\beta} + O_{A, \rho, \chi}(Y^{-A}), \end{align*}
again by shifting to the line
$\Re s = - A$
. Integrating the above with respect to
$\phi(1/y) \, d y / y$
yields the claim.
Conflicts of interest
None.
Financial support
The second author acknowledges support from NSF DMS-2401106.
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