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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]

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]