A -logarithmic analogue of Euler’s sine integral
Let be a normalized primitive holomorphic cusp form of even integral weight for the full modular group , and denote its th Fourier coefficient by . We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).
Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves for x → ∞ asymptotically like . We prove, among other results, that for all integers n₁,n₂ with 1 < n₁|n₂.
Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves, for x → ∞, asymptotically like . In this article, it is proved that for every prime p, , and it is also proved that if and m is large enough. In particular, it is shown that for...
Let be an elliptic curve defined over a number field, and let be a point of infinite order. It is natural to ask how many integers fail to occur as the order of modulo a prime of . For , a quadratic twist of , and as above, we show that there is at most one such .