A Density of Ramified Primes

Let K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

A Note on the Primitive Roots and the Golomb Conjecture

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p 1 / 2 + ɛ ≤ n < p , there must exist two primitive roots α and β modulo p with 1 < α , β ≤ n − 1 such that the equation n = α + β holds, where 0 < ɛ < 1 / 2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p .

On Pythagorean Triples and the Primitive Roots Modulo a Prime

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

Explicit upper bound on the least primitive root modulo p2

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.

The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$ -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.