Let [Formula: see text] be a Galois extension of [Formula: see text] of degree [Formula: see text], and let [Formula: see text] be an automorphic cuspidal representation of [Formula: see text]. Let [Formula: see text] denote the coefficients of the [Formula: see text]-function [Formula: see text] over [Formula: see text]. If the automorphic induction of [Formula: see text] down to [Formula: see text] is known, in this paper a Voronoi summation formula for the coefficients [Formula: see text] is derived using the Meier G-function. As an application, under the assumption that the imaginary part of the sum of the Archimedean parameters of the [Formula: see text]-function over [Formula: see text] vanishes, using well-known results on base change from [1], an asymptotic expansion for the resonance sum of [Formula: see text] against [Formula: see text] is computed when [Formula: see text] is cyclic of prime degree [Formula: see text] and [Formula: see text] is Galois invariant. Consistent with previous cases as in [8] and [31], we show when [Formula: see text] the main term has size [Formula: see text] when the phase [Formula: see text] is close or equal to [Formula: see text] where [Formula: see text] is a positive integer and [Formula: see text] denotes the conductor of [Formula: see text]. When [Formula: see text] we show that the sum decays rapidly, and this exhibits the oscillatory nature of the sequence [Formula: see text] coming from an automorphic representation over a number field.
Let [Formula: see text] and [Formula: see text] denote the largest and smallest prime factors of [Formula: see text], respectively. Let [Formula: see text], [Formula: see text]. For sufficiently large [Formula: see text], we show that there exists a strictly positive constant [Formula: see text] such that [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is fixed, which improves the previous result of Lü-Wang-Chen (2019) with [Formula: see text]. Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In particular, we prove that the pattern [Formula: see text] is realized by a positive proportion of [Formula: see text], for [Formula: see text] and [Formula: see text], which also improves an implicit result of Lü-Wang-Chen (2019) by replacing [Formula: see text] with [Formula: see text].
Let I(b,d,k) be the subseries of the harmonic series keeping the integers having exactly k occurrences of the digit d in base b. We prove the existence of an asymptotic expansion to all orders in descending powers of b, for fixed d and k, of I(b,d,k) - blog(b). We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers.
Let [Formula: see text] denote the error term in the divisor problem with congruence conditions. This article examines the relationship between the discrete mean value [Formula: see text], and the continuous mean value formula [Formula: see text]. We also study the explicit representation of the following integral, which involves the error term [Formula: see text] for [Formula: see text] and [Formula: see text], inspired by the work of Lavrik-Israilov-Edgorov, Sitaramachandra, and Furuya-Tanigawa.
Let [Formula: see text] be a prime and [Formula: see text] be the polynomial, where [Formula: see text] runs over the quadratic residues modulo [Formula: see text]. In this paper, we establish an explicit formula for evaluating [Formula: see text] at an arbitrary root of unity [Formula: see text] utilizing the special values of Dirichlet [Formula: see text]-functions at [Formula: see text]. We apply our results to provide unified proofs for several conjectures of Z.-W. Sun. We also evaluate various finite trigonometric products over quadratic residues. Finally, we extend this framework to the case where the modulus is a square-free composite integer [Formula: see text].
