We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by \(s,u\in[0,1/2]\). The error is \(O_g(Y/(\log Y)^2)\), uniformly for \(Y^{1/5}\leq R\leq Y^{1/3}\), including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.
Let \(\Kl(1;q)\) denote the normalized Kloosterman sum modulo \(q\). We prove that, for each sign, there are \(\gg X/\log X\) square-free moduli \(q\in(X,2X]\) with at most four prime factors for which \(\Kl(1;q)\) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for K...
We construct a meromorphic Wilf--Zeilberger pair for the hypergeometric series for \(3\ln 2\), completing the WZ proof left open by Guillera and Rogers. The same pair yields a one-parameter identity and, after a half-integer shift and differentiation, an inverse-binomial series for \(\zeta(3)\). We also study the finit...
We prove Breiman's conjecture under the first-moment assumption. Let $Y_1,Y_2,\ldots$ be iid nonnegative random variables with $\mathbb P\{Y_1>0\}>0$, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail o...
We prove the global Mizohata-Takeuchi estimate for bounded complexity weights in $\mathbb{R}^2$ and an arbitrary $C^2$ uniformly convex curve on the Fourier side, with an implied constant depending only on the complexity and the curve. A version with a logarithmic loss follows from a short argument using the $T^*T$ met...
Denote by $\mathrm{Kl}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $|\mathrm{Kl}(1,n)|$ on average over $n\leqslant X$, and show that \begin{align*} \sum_{n\leqslant X}|\mathrm{Kl}(1,n)|\asymp X(\log X)^{\frac{8}{3\pi}-1}. \end{align*} The lower bound is proven by considering...
We prove a polylogarithmic exceptional-set estimate for symmetric five-term sums of roots of unity of arbitrary order. More precisely, at most $O((\log(2N))^{11})$ points of the $N$-grid satisfy $|1+2\cos(2\pi x)+2\cos(2\pi y)|<\exp(-C(\log(2N))^{13})$. This implies the corresponding bound for small conjugates at the r...
Qi Cheng· 0 citations
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