Note on a Diophantine inequality in several variables

Abstract

Abstract. We establish estimates for the number of points that belong to an aligned box in (R/Z) N in terms of certain exponential sums. These generalize previous results that were known only in case N = 1.

On modular signs

“…for all n 1, where τ (n) is the divisor function (this is the Ramanujan-Petersson conjecture, proved by Deligne). In particular, we have 2] for p H N , and hence there exists a unique angle…”

On modular signs of half integral weights

“…It is well known that the Chebychev functions X n , n 0, defined by Let z 2 be a parameter to be determined later and L ≡ 3 (mod 4) be a positive integer. According to [1,Theorem 7] with the choice of parameters N = π(z) (the number of primes p z) and u n = 0, v n = 1 4 for all n π(z), we can get two explicit trigonometric polynomials on…”

“…Let k 3 be an integer and denote by S k+1/2 = S k+1/2 (4) the space of cusp forms of half-integral weight k + 1 2 on the congruence subgroup Γ 0 (4). Let S + k+1/2 be Kohnen's plus space in S k+1/2 and S +, * k+1/2 be a basis of Hecke eigenforms of S + k+1/2 .…”

A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

“…Other classical applications of the solutions of these problems to analytic number theory include Hilbert-type inequalities [8,16,24,29,35], Erdös-Turán discrepancy inequalities [8,21,35], optimal approximations of periodic functions by trigonometric polynomials [2,8,9,35] and Tauberian theorems [16]. The extremal problem in higher dimensions, with applications, is considered in [1,17]. Approximations in L p -norms with p ̸ = 1 are treated, for instance, in [14].…”