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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]

We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem. We include several applications. [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]

We produce nontrivial asymptotic estimates for shifted sums of the form ∑ a (h) b (m) c (2 m − h) , in which a (n) , b (n) , c (n) are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions n − h , n , n + h such that a (n − h) a (n) a (n + h) ≠ 0. [ABSTRACT FROM AUTHOR]

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

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

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]

Let f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑ n ≤ X | S f ( n ) | 2 and proved that the Classical Conjecture, that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ , holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D ( s , S f × S g ) = ∑ S f ( n ) S g ( n ) ‾ × n − ( s + k − 1 ) and D ( s , S f × S g ‾ ) = ∑ n S f ( n ) S g ( n ) n − ( s + k − 1 ) . We then prove asymptotics for the smoothed second moment sums ∑ S f ( n ) S g ( n ) ‾ e − n / X , giving a smoothed generalization of [9] . We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L -function L ( s , f × g ) and convolution sums of the coefficients of f and g . In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on | S f ( n ) | 2 is true on short intervals, and to prove sign change results on { S f ( n ) } n ∈ N . [ABSTRACT FROM AUTHOR]

Let f be a weight k holomorphic cusp form of level one, and let S f ( n ) denote the sum of the first n Fourier coefficients of f . In analogy with Dirichlet's divisor problem, it is conjectured that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ . Understanding and bounding S f ( X ) has been a very active area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − 1 2 + 1 3 ( log ⁡ X ) − 0.1185 from Wu [17] . Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for S f ( X ) holds on average over intervals of length X . Jutila [10] improved this result to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 3 4 + ϵ . Building on the results and analytic information about ∑ | S f ( n ) | 2 n − ( s + k − 1 ) from our recent work [9] , we further improve these results to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 2 3 ( log ⁡ X ) 1 6 . [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]