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Register a new userSome applications of smooth bilinear forms with Kloosterman sums
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We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.
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We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
In this paper, we study the Newton polygons for the $L$-functions of $n$-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top dimensional Dwork cohomology. Using Wans decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for $L$-function of $bar{F}(bar{lambda},x):=sum_{i=1}^nx_i^{a_i}+bar{lambda}prod_{i=1}^nx_i^{-1}$.
The paper compares the asymptotic of the expressions $frac {1} {x} sumlimits_{n leq x} {f(n)}$ and $sumlimits_{n leq x} {frac {f(n)} {n}}$, $frac {1} {x} sumlimits_{p leq x} {f(p)}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$. The asymptotic of sums $sumlimits_{n leq x} {frac {f(n)} {n}}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $sumlimits_{n leq x} {f(n)}$,$sumlimits_{p leq x} {f(p)}$.
In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperbers construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dworks deformation equation for our family. Furthermore, with the help of Dworks dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dworks conjecture.
We produce nontrivial asymptotic estimates for shifted sums of the form $sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions $n-h, n, n+h$ such that $a(n-h)a(n)a(n+h) eq 0$.
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