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We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree |$n$| polynomials |$f$| with |$\textrm {Gal}(f) \subseteq A_n$|⁠. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree |$n$| monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree |$n$| number fields with almost prime discriminants. [ABSTRACT FROM AUTHOR]

The mean value theorem of calculus states that, given a differentiable function f on an interval [ a , b ] , there exists at least one mean value abscissa c such that the slope of the tangent line at (c , f (c)) is equal to the slope of the secant line through (a , f (a)) and (b , f (b)) . In this article, we study how the choices of c relate to varying the right endpoint b. In particular, we ask: When we can write c as a continuous function of b in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions. [ABSTRACT FROM AUTHOR]

The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF+16 ] and [ Zha ]. We extend [ BDF+16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers. [ABSTRACT FROM AUTHOR]

We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogs and the existence of "weighted harmonic functions". [ABSTRACT FROM AUTHOR]

The article presents a mathematical question and answer related to odd prime number.