Fermat’s quotients

For a prime ${p}$ and an integer ${a}$ the Fermat quotient is defined as

$\displaystyle q_{p}(a)=\frac{a^{p-1}-1}{p}$

Let ${\ell_{p}}$ be the smallest valueof ${a}$ for which ${q_{p}(a) \not \equiv 0(\bmod p).}$ We want to understand how ${\ell_p}$ looks.

For instance, Granville conjectured ${\ell_{p}=o\left((\log p)^{1 / 4}\right)}.$ Lenstra conjectures ${\ell_p \le 3}$

Lenstra proved

$\displaystyle \ell_{p} \leq\left\{\begin{array}{ll} 4(\log p)^{2} & \mbox { if } p \geq 3 \\ \left(4 e^{-2}+o(1)\right)(\log p)^{2} & \mbox { if } p \rightarrow \infty \end{array}\right. \ \ \ \ \ (1)$

Granville showed that

${\ell_{p} \leq(\log p)^{2}}$ for ${p \ge 5}$ .

How do we get upper bounds on ${\ell_p}$ ?

Main ideas: (Smooth numbers+ Equidistribution of multiplicative subgroups)
If ${\ell_p \ge k}$ is large, a lot of small numbers ${\le k}$ satisy ${a^{p-1} \equiv 1\left(\bmod p^{2}\right)}$ . Then in fact, numbers with all prime factors less than ${k}$ (smooth numbers ${\psi(x, k)}$ ) have to satisfy ${a^{p-1} \equiv 1\left(\bmod p^{2}\right)}$

But numbers which satisfy ${a^{p-1} \equiv 1\left(\bmod p^{2}\right)}$ belong to the subgroup of ${p}$ -th powers inside ${\mathbb{Z}_{p^{2}}^{*}}$ of order ${p-1}$ . Thus we produce lot of elements of this subgroup inside an interval say ${[1,x]}$ . If ${x}$ is not too small enough compared to ${p}$ , we can see that the subgroup is almost equidistributed an contains the right number of elements ${\frac{x}{p}}$ in ${[1,x]}$ . This allows to get a control on how large ${k}$ can be. To get good bounds to need to take ${x}$ small. But then to establish equidistribution in very small intervals is hard. The quality of the result depends on how small ${x}$ we can afford to have equidistribution. Writing in terms of the Fourier transform it’s equivalent to get cancellation in short exponential sums over the subgroups of ${p}$ th powers. The degree ${p}$ of the exponential sum is quite big for standard finite field RH to gives cancellation. Hence one has to deal with exponentials sums like

$\displaystyle H_{p}(\lambda)=\sum_{b=1}^{p} \mathbf{e}_{p^{2}}\left(\lambda b^{p}\right)$

These are called Heilbronn sums and can be estimated by using “polynomial method” type techniques (HeathBrown, Konyagin). Further improvements can be obtained by studying the distribution of multipilcaitve subgroups in short intervals (Bourgain, Konyagin, and Shparlinsk): By counting solutions to various restricted congruences equations.

$\displaystyle w \equiv z u(\bmod n), \quad 1 \leq|z| \leq Z, u \in \mathcal{G}$

$\displaystyle u x \equiv y(\bmod n), \quad \mbox { where } 0<|x|,|y| \leq Z, \mbox { and } u \in \mathcal{G}$

These in turn can be studied using a variety of techniques: Double character sums (Friedlander Iwaniec), character sums twisted with divisor functions (Chang, Karatsuba), additive combinatorics (Sum-Product phenomenon-Bourgain)

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