We study sums involving multiplicative functions that take values over a nonhomogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas for quantities such as the number of integers in a Beatty sequence that are representable as a sum of two squares up to a given magnitude.2000 Mathematics subject classification: 11E25, 11B83.
We study the problem of representing integers N ≡ κ (mod 2) as a sum of κ prime numbers from the Beatty sequencewhere α, β ∈ R with α > 1, and α is irrational and of finite type.In particular, we show that for κ = 2, almost all even numbers have such a representation if and only if α < 2, and for any fixed integer κ 3, all sufficiently large numbers N ≡ κ (mod 2) have such a representation if and only if α < κ.
Let n 1 a n n −s be the L-series of an elliptic curve E defined over the rationals without complex multiplication. In this paper, we present certain similarities between the arithmetic properties of the coefficients {a n } ∞ n=1 and Euler's totient function ϕ(n). Furthermore, we prove that both the set of n such that the regular polygon with |a n | sides is ruler-and-compass constructible, and the set of n such that n − a n + 1 = ϕ(n) have asymptotic density zero. Finally, we improve a bound of Luca and Shparlinski on the counting function of elliptic pseudoprimes.
We study the one-level density for families of $L$-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $\chi _n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz–Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under the Generalized Riemann Hypothesis. This is the first result extending the support outside the trivial range $(-1, 1)$ for a family of cubic $L$-functions. This implies that a positive density of the $L$-functions associated with these characters do not vanish at the central point $s=1/2$. A key ingredient in our proof is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson [22, 23].
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.
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