Stronger arithmetic equivalence

This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence. The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.

AN ARITHMETIC EQUIVALENCE OF THE RIEMANN HYPOTHESIS

Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$, we have $\log h(n)<\sqrt{\text{li}^{-1}(n)}$ (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

Weak Arithmetic Equivalence

Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence