Let $f$ be a primitive holomorphic cusp form of arbitrary weight, level and nebentypus. We prove $\int_0^T|L(1/2+it,f)|^6\,dt\ll_{f,\epsilon}T^{2+\epsilon}$, extending Jutila's level-one theorem. The new ingredient is a large sieve comparing the stationary phases in the Booker-Milinovich-Ng transformation at different heights. Applications include moments of order $T^{1+\epsilon}$ on every line to the right of the critical line, $\Omega(T^{4/35-\epsilon})$ simple zeros, zero density estimates, and bounds for cubic-field power sums, shifted convolution sums and the general divisor problem.
We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\...
Let $q$ be a fixed odd prime power with $q\equiv2\pmod3$. We study the first moment of primitive cubic Dirichlet $L$-functions over $\Fq(T)$, ordered by their genus. By averaging the rational sieve variable in the double Dirichlet series of Hong and Fang, we obtain meromorphic continuation beyond the secondary polar di...
We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ over $\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula w...
Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb B$ of a complex Banach space $X$ or on the unit polydisk $\mathbb U^n$ of $\mathbb{C}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fr\'echet derivatives of $F$ when $F$...
S. Panja, Sujoy Majumder, Abhijit Banerjee· 0 citations
We prove that operators of the form $A=-\Delta^{2}-V(x)$, with $|D V(x)|\leq \gamma V(x)^\frac{5}{4}+ C_\gamma$ for some $\gamma$ small enough and $C_\gamma$ positive constant, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1\leq p<\infty$ and in $C_0(\mathbb{R}^N)$. Moreover, we characterize the maximal doma...
Federica Gregorio, Chiara Spina, C. Tacelli· 0 citations
Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements:...
Xi Cen, Xi-Tao Gao, Cheng Zhang et al.· 0 citations
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