We prove new bounds on complete sums of products of k additively shifted Kloosterman sums to odd high prime power moduli q=p^n , which feature substantially stronger power savings (about q^-1/⌈ k/2⌉ in generic configurations) and a novel quantification of the alignment among the shifts. We prove our bounds by developing a method to estimate complete exponential sums with a broad class of phases well-controlled by a specified number of initial terms in a property reflecting p-adic differentiability.
We bound the total variation distance in the Kubilius model for sequences with positive level of distribution. We obtain a result that we expect is qualitatively optimal. As a special case, it recovers a recent result of Ford on shifted primes, with a slightly simplified proof. In the classical case considered by Kubilius, our theorem gives a simple proof of the optimal bound discovered by Tenenbaum, up to factors of x^o(1) and u^o(u) .
We study an analogy between classical modular forms with Nebentypus characters and degree 2 L -functions in the Selberg class. We prove that every normalized degree 2 L -function with integer conductor, admitting a degree 2 Euler product and stable under twists by Dirichlet characters, possesses a Dirichlet character ψ that plays the role of a Nebentypus character. This result provides evidence for the general conjecture that the elements of integer degree in the Selberg class correspond to automorphic representations of arithmetic groups. Moreover, our method extends to functions in the extended Selberg class 𝒮^♯ that may lack a global Euler product but exhibit multiplicative behavior at selected primes, yielding a characterization of their local Euler factors in terms of an as- sociated Dirichlet character.
