In 1981, Zagier conjectured that the constant term of the automorphic form
y
12
|
Δ
(
z
)
|
2
, that is,
a
0
(
y
)
:
=
y
12
∑
n
=
1
∞
τ
2
(
n
)
exp
(
-
4
π
n
y
)
,
where
τ
(
n
)
is the
n
th Fourier ...coefficient of the Ramanujan cusp form
Δ
(
z
)
, has an asymptotic expansion when
y
→
0
+
and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function
ζ
(
s
)
. This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Möbius function
μ
(
n
)
. We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of
ζ
(
s
)
, and the error term is expressed as an infinite series involving the generalized hypergeometric series
2
F
1
(
a
,
b
;
c
;
z
)
. As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.
Generalized digamma functions ψk(x), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the Hurwitz zeta function. In this paper, a ...modular relation of the form Fk(α)=Fk(1/α) containing infinite series of ψk(x), or, equivalently, between the generalized Stieltjes constants γk(x), is obtained for any k∈N. When k=0, it reduces to a famous transformation given on page 220 of Ramanujan's Lost Notebook. For k=1, an integral containing Riemann's Ξ-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as α→0 and α→∞. Carlitz-type and Guinand-type finite modular relations involving ψj(m)(x),0≤j≤k,m∈N∪{0}, are also derived, thereby extending previous results on the digamma function ψ(x). The extension of Guinand's result for ψj(m)(x),m≥2, involves an interesting combinatorial sum h(r) over integer partitions of 2r into exactly r parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for ψj′(x) is subtle and requires delicate analysis.
Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert ...series
∑
n
=
1
∞
a
f
(
n
)
ψ
(
n
)
∗
μ
(
n
)
ψ
′
(
n
)
exp
(
-
n
y
)
,
where
a
f
(
n
)
is the
n
th Fourier coefficient of a cusp form
f
over any congruence subgroup, and
ψ
and
ψ
′
are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.
The Mordell-Tornheim zeta function and the Herglotz-Zagier function \(F(x)\) are two important functions in Mathematics. By generalizing a special case of the former, namely \(\Theta(z, x)\), we show ...that the theories of these functions are inextricably woven. We obtain a three-term functional equation for \(\Theta(z, x)\) as well as decompose it in terms of the Herglotz-Hurwitz function \(\Phi(z, x)\). This decomposition can be conceived as a two-term functional equation for \(\Phi(z, x)\). Through this result, we are not only able to get Zagier's identity relating \(F(x)\) with \(F(1/x)\) but also two-term functional equation for Ishibashi's generalization of \(F(x)\), namely, \(\Phi_k(x)\) which has been sought after for over twenty years. We further generalize \(\Theta(z, x)\) by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Kumar and Choie, were known so far. The first one among the two involves the integral \(J(x)\) who special values have received a lot of attention, more recently, in the work of Muzzaffar and Williams, and in that of Radchenko and Zagier. Analytic continuation of our generalization of \(\Theta(z, x)\) is also accomplished which allows us to obtain transformations between certain double series and Herglotz-type integrals or their explicit evaluations.
Generalized digamma functions \(\psi_k(x)\), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an ...indefinite quadratic form. In this paper, a modular relation of the form \(F_k(\alpha)=F_k(1/\alpha)\) containing infinite series of \(\psi_k(x)\), or, equivalently, between the generalized Stieltjes constants \(\gamma_k(x)\), is obtained for any \(k\in\mathbb{N}\). When \(k=0\), it reduces to a famous transformation given on page \(220\) of Ramanujan's Lost Notebook. For \(k=1\), an integral containing Riemann's \(\Xi\)-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as \(\alpha\to0\) and \(\alpha\to\infty\). Carlitz-type and Guinand-type finite modular relations involving \(\psi_j^{(m)}(x), 0\leq j\leq k, m\in\mathbb{N}\cup\{0\},\) are also derived, thereby extending previous results on the digamma function \(\psi(x)\). The extension of Guinand's result for \(\psi_j^{(m)}(x), m\geq2,\) involves an interesting combinatorial sum \(h(r)\) over integer partitions of \(2r\) into exactly \(r\) parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for \(\psi_j'(x)\) is subtle and requires delicate analysis.
Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series
associated with a cusp form over the full modular group and the M\"{o}bius
function. In this paper, we investigate the Lambert ...series
$\sum_{n=1}^{\infty}a_f(n)\psi(n)*\mu(n)\psi'(n)\exp(-ny),$ where $a_f(n)$ is
the $n$th Fourier coefficient of a cusp form $f$ over any congruence subgroup,
and $\psi$ and $ \psi'$ are primitive Dirichlet characters. This extends the
earlier work to the case of higher level subgroups and also gives a character
analogue.
Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin
transform of the symmetric square $L$-function associated to the Ramanujan tau
function has an asymptotic expansion in terms ...of the non-trivial zeros of the
Riemann zeta function $\zeta(s)$. Later, Chakraborty, Kanemitsu and the second
author extended this phenomenon for any Hecke eigenform over the full modular
group. In this paper, we study an asymptotic expansion of the Lambert series
\begin{equation*} y^k \sum_{n=1}^\infty \lambda_{f}( n^2 ) \exp (- ny), \quad
\textrm{as}\,\, y \rightarrow 0^{+}, \end{equation*} where $\lambda_f(n)$ is
the $n$th Fourier coefficient of a Hecke eigen form $f(z)$ of weight $k$ over
the full modular group.
Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the M\"{o}bius function. In this paper, we investigate the Lambert ...series \(\sum_{n=1}^{\infty}a_f(n)\psi(n)*\mu(n)\psi'(n)\exp(-ny),\) where \(a_f(n)\) is the \(n\)th Fourier coefficient of a cusp form \(f\) over any congruence subgroup, and \(\psi\) and \( \psi'\) are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.
Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square \(L\)-function associated to the Ramanujan tau function has an asymptotic expansion in ...terms of the non-trivial zeros of the Riemann zeta function \(\zeta(s)\). Later, Chakraborty, Kanemitsu and the second author extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series \begin{equation*} y^k \sum_{n=1}^\infty \lambda_{f}( n^2 ) \exp (- ny), \quad \textrm{as}\,\, y \rightarrow 0^{+}, \end{equation*} where \(\lambda_f(n)\) is the \(n\)th Fourier coefficient of a Hecke eigen form \(f(z)\) of weight \(k\) over the full modular group.
