Sub-Weyl strength bounds for twisted GL(2)GL(2) short character sums

Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, \lambda_{f}(n) \, \chi(n),$$ where $\lambda_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $\chi$ is a primitive character of conductor $p^{r}$. In this article we prove a sub-Weyl strength bounds for $S(N)$. Indeed, we obtain $$S(N) \ll \, N^{\frac{5}{9}} \ p^{\frac{13r}{45}},$$ provided that $p^{13r/20} \leq N \leq p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\geq \left(p^{r}\right)^{\frac{2}{3}-\frac{1}{60}}$.Comment: First draf

Effect of crystal structure and cationic order on phonon modes across ferroelectric phase transformation in Pb(Fe0.5-xScxNb0.5)O3 bulk ceramics

Pb(Fe0.5-xScxNb0.5)O3 [(PFSN) (0 ≤ x ≤ 0.5)] multiferroic relaxors were synthesized and the temperature dependence of phonon modes across ferroelectric to paraelectric transition was studied. With varying Sc content from x = 0 to 0.25 the structure remains monoclinic and with further addition (x = 0.3 - 0.5) the structure transforms into rhombohedral symmetry. Structural refinement studies showed that the change in crystal structure from monoclinic to rhombohedral symmetry involves a volume increment of 34-36%. Associated changes in the tolerance factor (1.024 ≤ t ≤ 0.976) and bond angles were observed. Structure assisted B′-B″ cation ordering was confirmed through the superlattice reflections in selected area electron diffraction (SAED) pattern of Pb(Sc0.5Nb0.5)O3 (x = 0.5). Cation ordering is also evident from the evolution of Pb-O phonon mode in Raman spectra of compositions with rhombohedral symmetry (x ≥ 0.3). The high temperature Raman scattering studies show that the B-localized mode [F1u, ∼250 cm−1] and BO6 octahedral rotational mode [F1g, ∼200 cm−1], both originating from polar nano regions (PNRs) behave like coupled phonon modes in rhombohedral symmetry. However, in monoclinic symmetry they behave independently across the transition. Softening of B localized mode across the transition followed by the hardening for all compositions confirms the diffusive nature of the ferroelectric transformation. The presence of correlation between the B localized and BO6 rotational modes introduces a weak relaxor feature for systems with rhombohedral symmetry in PFSN ceramics, which was confirmed from the macroscopic dielectric studies

Sub-convexity bound for GL(3)×GL(2)GL(3) \times GL(2) LL-functions: GL(3)GL(3)-spectral aspect

Let $\phi$ be a Hecke-Maass cusp form for $SL(3, \mathbb{Z})$ with Langlands parameters $({\bf t}_{i})_{i=1}^{3}$ satisfying $$|{\bf t}_{3} - {\bf t}_{2}| \leq T^{1-\xi -\epsilon}, \quad \, {\bf t}_{i} \approx T, \quad \, \, i=1,2,3$$ with $1/2 0$. Let $f$ be a holomorphic or Maass Hecke eigenform for $SL(2,\mathbb{Z})$. In this article, we prove a sub-convexity bound $$L(\phi \times f, \frac{1}{2}) \ll \max \{ T^{\frac{3}{2}-\frac{\xi}{4}+\epsilon} , T^{\frac{3}{2}-\frac{1-2 \xi}{4}+\epsilon} \}$$ for the central values $L(\phi \times f, \frac{1}{2})$ of the Rankin-Selberg $L$-function of $\phi$ and $f$, where the implied constants may depend on $f$ and $\epsilon$. Conditionally, we also obtain a subconvexity bound for $L(\phi \times f, \frac{1}{2})$ when the spectral parameters of $\phi$ are in generic position, that is $${\bf t}_{i} - {\bf t}_{j} \approx T, \quad \, \text{for} \, i \neq j, \quad \, {\bf t}_{i} \approx T , \, \, i=1,2,3.$$Comment: First draf

The second moment of derivatives of GL(2)GL(2) LL-functions over quadratic twists

Let $f$ be an Hecke eigenform for the group $\Gamma_{0}(q)$ and $\chi_{d}$ be a primitive quadratic character of conductor $|d|$. In this article, we prove an asymptotic for the second moment of the derivative of $L(s, f \otimes \chi_{8d})$ at the central point $1/2$, which was previously known under GRH by Petrow \cite{petrow}.Comment: 19 Page