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We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding q-supercongruence. Similar q-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the Cyclic Sieving Phenomenon, involving the q-Lucas numbers.

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as possible. We study this problem in the function field setting, and prove for a wide class of arithmetic functions (namely factorization functions), that such an asymptotic result holds, allowing the size of the short interval to be as small as a square-root of the size of the full interval, and analogously for arithmetic progressions. For instance, our results apply for the indicator function of polynomials with a divisor of given degree, and are much stronger than those known for the analogous function over the integers. As opposed to many previous works, our results apply in the large-degree limit, where the base field F q is fixed. Our proofs are based on relationships between certain character sums and symmetric functions, and in particular we use results from symmetric function theory due to Egecioglu and Remmel. We also use recent bounds of Bhowmick, Lê and Liu on character sums, which are in the spirit of the Drinfeld-Vlȃduţ bound.

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We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.