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Let $ Lambda $ denote von Mangoldts function, and consider the averages begin{align*} A_N f (x) &=frac{1}{N}sum_{1leq n leq N}f(x-n)Lambda(n) . end{align*} We prove sharp $ ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, Gsubset [0,N]$ there holds begin{equation*} N ^{-1} langle A_N mathbf 1_{F} , mathbf 1_{G} rangle ll frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigl( operatorname {Log} frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigr) ^{t}, end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ sup_N A_N mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
We prove a sharp, global-in-time Strichartz estimate for the Schrodinger equation on the cylinder $mathbb{R}timesmathbb{T}$.
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $ell^{p}$-improving inequality begin{e
Let $ lambda ^2 in mathbb N $, and in dimensions $ dgeq 5$, let $ A_{lambda } f (x)$ denote the average of $ f ;:; mathbb Z ^{d} to mathbb R $ over the lattice points on the sphere of radius $lambda$ centered at $x$. We prove $ ell ^{p}$ improving pr
The Zagier $L$-series encode data of real quadratic fields. We study the average size of these $L$-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.