169 research outputs found
On a hybrid fourth moment involving the Riemann zeta-function
We provide explicit ranges for for which the asymptotic formula
\begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\;
T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds
as , when , where is the
Riemann zeta-function. The obtained ranges improve on an earlier result of the
authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An
application to a divisor problem is also givenComment: 21 page
On some problems involving Hardy's function
Some problems involving the classical Hardy function are discussed. In particular we discuss the odd moments of
, the distribution of its positive and negative values and the primitive
of . Some analogous problems for the mean square of are
also discussed.Comment: 15 page
Shifted convolution and the Titchmarsh divisor problem over F_q[t]
In this paper we solve a function field analogue of classical problems in
analytic number theory, concerning the auto-correlations of divisor functions,
in the limit of a large finite field.Comment: 22 pages, updated versio
Negative values of the Riemann zeta function on the critical line
We investigate the intersections of the curve with the real axis. We show unconditionally that the
zeta-function takes arbitrarily large positive and negative values on the
critical line.Comment: 18 pages, improved Corollary
On the Rankin-Selberg problem: higher power moments of the Riesz mean error term
Let be the error term of the first Riesz means of the
Rankin-Selberg zeta function. We study the higher power moments of
and derive an asymptotic formula for 3-rd, 4-th and 5-th
power moments by using Ivi\'c 's large value arguments.Comment: 18 Page
On the mean square of the zeta-function and the divisor problem
Let denote the error term in the Dirichlet divisor problem, and
the error term in the asymptotic formula for the mean square of
. If with , then we obtain the
asymptotic formula where is a polynomial of degree three in
with positive leading coefficient. The exponent 7/6 in the error term
is the limit of the method.Comment: 10 page
Strong coupling regime of semiconductor quantum dot embedded in the nano-cavity
Self-assembled quantum dots on semiconductor substrate have found many applications in optoelectronic devices such as single photon emitters, qubits for quantum computers, etc [1,2]. In this work, we study the interaction of the electron in nano-dot embedded in the nano-cavity with photons of a incident beam. Theoretical framework of our study is the semi-classical model Hamiltonian, which describes nano-dot interacting with the electromagnetic field. For the practical calculations we have employed rotating wave approximation. The influence of both, decay rates of cavities and quantity of coupling constant to level shift of electrons in a quantum dot have been analyzed. The boundary between strong coupling and weak-coupling regimes has been presented.V International School and Conference on Photonics and COST actions: MP1204, BM1205 and MP1205 and the Second international workshop "Control of light and matter waves propagation and localization in photonic lattices" : PHOTONICA2015 : book of abstracts; August 24-28, 2015; Belgrad
On the asymptotic formula for Goldbach numbers in short intervals
Let
, \Sing(k) = 2
\prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack{
p\mid k\\ p>2 }} \left(\frac{p-1}{p-2}\right) if is even and \Sing(k)
=0 if is odd. It is known that R(k) \sim k\Sing(k) as
for almost all and that \sum_{k\in [n,n+H)}R(k) \sim
\sum_{k\in [n,n+H)} k\Sing(k) \quad\hbox{for} \quad n\to \infty \eqno{(1)}
uniformly for . Here we prove, assuming
and , that (1) holds for
almost all
The subconvexity problem for \GL_{2}
Generalizing and unifying prior results, we solve the subconvexity problem
for the -functions of \GL_{1} and \GL_{2} automorphic representations
over a fixed number field, uniformly in all aspects. A novel feature of the
present method is the softness of our arguments; this is largely due to a
consistent use of canonically normalized period relations, such as those
supplied by the work of Waldspurger and Ichino--Ikeda.Comment: Almost final version to appear in Publ. Math IHES. References
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