Squarefree polynomials and Mobius values in short intervals and arithmetic progressions

Abstract

We calculate the mean and variance of sums of the Möbius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting.

Added references and corrected misprints

References in corpus (1)

• The variance of the number of prime polynomials in short intervals and in residue classes

Cited by in corpus (19)

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• Arithmetic functions in short intervals and the symmetric group
• On the variance of squarefree integers in short intervals and arithmetic progressions
• Higher moments of arithmetic functions in short intervals: a geometric perspective
• Some problems in analytic number theory for polynomials over a finite field
• Mean values of arithmetic functions in short intervals and in arithmetic progressions in the large-degree limit
• Correlations of multiplicative functions in function fields
• Chebotarev density theorem in short intervals for extensions of ${\mathbb{F}}_q(T)$
• Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in ${\mathbb{F}}_q[t]$
• Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in ${\mathbb{F}}_q(t)$
• Magic squares, the symmetric group and Möbius randomness
• The variance of the Euler totient function
• Prime and Möbius correlations for very short intervals in ${\mathbb{F}}_q[x]$
• Arithmetic correlations over large finite fields
• Sums of divisor functions and von Mangoldt convolutions in ${\mathbb{F}}_q[T]$ leading to symplectic distributions
• Asymptotic For Primitive Roots Producing Polynomials
• Squarefrees are Gaussian in short intervals
• Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor
• Variance of sums in short intervals and $L$-functions in ${\mathbb{F}}_q[t]$