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Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field
${{\mathbb{Q}}(\sqrt D)}$
. We denote by NE(σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for NE(σ1, σ2, T), obtaining a saving of a power of log T in the error term.
We study lower bounds of a general family of L-functions on the
$1$
-line. More precisely, we show that for any
$F(s)$
in this family, there exist arbitrarily large t such that
$F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$
, where m is the order of the pole of
$F(s)$
at
$s=1$
. This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the
$1$
-line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type
$L(s,f\times f)$
on the
$1$
-line.
We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of
$\operatorname{Sp}(n,\mathbb{Z})$
on
$\mathbb{C}[(L^{\prime }/L)^{n}]$
. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree
$n$
, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard
$L$
-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic
$L$
-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet
$L$
-functions modulo
$q$
weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length
$q^{1/4}$
. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic
$L$
-functions, which we also use to improve the best known subconvexity bounds for automorphic
$L$
-functions in the level aspect.
We prove that sums of length about
$q^{3/2}$
of Hecke eigenvalues of automorphic forms on
$\operatorname{SL}_{3}(\mathbf{Z})$
do not correlate with
$q$
-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.
We prove that the complete
$L$
-function associated to any cuspidal automorphic representation of
$\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$
has infinitely many simple zeros.
We show that if the zeros of an automorphic
$L$
-function are weighted by the central value of the
$L$
-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the
$L$
-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted
$L$
-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct
$p$
-adic
$L$
-functions for non-critical slope rational modular forms, the theory has been extended to construct
$p$
-adic
$L$
-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the
$L$
-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the
$p$
-adic
$L$
-function of the eigenform to be this distribution.
We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan
$\unicode[STIX]{x1D70F}$
-Dirichlet series. Invent. Math.94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form
$L$
-functions of arbitrary conductor.
Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in
$\mathbb{P}^{2}\times \mathbb{P}^{2}$
with bihomogeneous coordinates
$[x_{1}:x_{2}:x_{3}],[y_{1}:y_{2},y_{3}]$
and in
$\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$
with multihomogeneous coordinates
$[x_{1}:y_{1}],[x_{2}:y_{2}],[x_{3}:y_{3}]$
defined by the same equation
$x_{1}y_{2}y_{3}+x_{2}y_{1}y_{3}+x_{3}y_{1}y_{2}=0$
. We thus improve on recent work of Blomer et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann.370 (2018), 491–553] and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type
$\mathbf{A}_{1}$
and three lines (the other existing proof relying on harmonic analysis by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math.148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc.108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc.166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.
We study various families of Artin
$L$
-functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
The standard twist
$F(s,\unicode[STIX]{x1D6FC})$
of
$L$
-functions
$F(s)$
in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where
$F(s)$
satisfies a functional equation with the same
$\unicode[STIX]{x1D6E4}$
-factor of the
$L$
-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such
$L$
-functions. We show that the standard twist
$F(s,\unicode[STIX]{x1D6FC})$
satisfies a functional equation reflecting
$s$
to
$1-s$
, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of
$F(s,\unicode[STIX]{x1D6FC})$
.
Assuming a conjecture on distinct zeros of Dirichlet
$L$
-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of
$L$
-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.
We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.
We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at
$s=1$
. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.
Two results related to the mixed joint universality for a polynomial Euler product
$\unicode[STIX]{x1D711}(s)$
and a periodic Hurwitz zeta function
$\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$
, when
$\unicode[STIX]{x1D6FC}$
is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.
The Li coefficients
$\unicode[STIX]{x1D706}_{F}(n)$
of a zeta or
$L$
-function
$F$
provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the
$\unicode[STIX]{x1D70F}$
-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the
$\unicode[STIX]{x1D70F}$
-Li coefficients varies depending on whether the function in question has any zeros in the half-plane
$\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$
We investigate analytically and numerically the behavior of these coefficients for such functions in both the
$n$
and
$\unicode[STIX]{x1D70F}$
aspects.
We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order
$2p$
: one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime
$p$
is ‘exceptional’ if and only if it is represented by one of six specific quadratic polynomials.
We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.
We describe an algorithm for finding the coefficients of
$F(X)$
modulo powers of
$p$
, where
$p\neq 2$
is a prime number and
$F(X)$
is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic
${\it\lambda}$
-invariants attached to those cubic extensions
$K/\mathbb{Q}$
with cyclic Galois group
${\mathcal{A}}_{3}$
(up to field discriminant
${<}10^{7}$
), and also tabulate the class number of
$K(e^{2{\it\pi}i/p})$
for
$p=5$
and
$p=7$
. If the
${\it\lambda}$
-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the
$p$
-adic
$L$
-function and deduce
${\rm\Lambda}$
-monogeneity for the class group tower over the cyclotomic
$\mathbb{Z}_{p}$
-extension of
$K$
.