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Statement

Setting (p. 2). For large $y$ and integers $\ell\le k$, $\mathcal N(y;k,\ell)$ is the number of tuples with $p_1\cdots p_k\equiv1\pmod{q_1\cdots q_\ell}$, the $p_i$ running over all primes in $(y/2,y]$ and the $q_j$ over all primes in $(y/4,y/2]$ (display (1)). Write $\lambda=\sum_{y/4<q\le y/2}1/q\sim\log2/\log y$ (display (2)) and $P$ for the number of primes in $(y/2,y]$, so $P\sim y/(2\log y)$ (display (3)). All implied constants are absolute.

Proposition 2 (pp. 2-3). Let $y\ge10$ be real and let $\ell,k$ be integers with $1\le\ell\le k\le y^{1/3}/(\log y)^2$.

• In the range $\ell\le k/2$, $\mathcal N(y;k,\ell)=\lambda^\ell P^k\bigl(1+O(1/\log y)\bigr)$.
• In the range $k/4\le\ell\le k/2$, $\mathcal N(y;k,\ell)=\lambda^\ell P^k\bigl(1+O(1/\log y)\bigr)+O\bigl(\ell^{k-\ell}(4\lambda P)^\ell y^{k/2}\bigr)$.

The ranges are as printed. In the proof, the moduli of $t\le k/4$ primes contribute $O(P^k\lambda^\ell/\log y)$ (estimate (12), p. 6) and those of $k/4<t\le\ell$ primes contribute the second error term (estimate (13), p. 6). The paper describes the result as an average statement on the equidistribution of smooth numbers in arithmetic progressions (p. 3).

Proof pointer

Section 3, pp. 4-6. Orthogonality of Dirichlet characters writes $\mathcal N(y;k,\ell)$ as a sum over characters modulo $q_1\cdots q_\ell$; the principal character gives $(1+O(\ell/y))\lambda^\ell P^k$ (display (5)). The non-principal characters are passed to the primitive characters inducing them, grouped by moduli made of $t$ primes from $(y/4,y/2]$, and bounded by two large-sieve estimates (Lemma 3, p. 5, from orthogonality and Theorem 7.13 of Iwaniec and Kowalski): a $2t$-th moment bound (10) and a $4t$-th moment bound (11). For $t\le k/4$ the bound (11) with the trivial bound gives (12); for $k/4<t\le\ell$ Hölder's inequality between (10) and (11) gives (13).

Read depth

Claims checked: the definitions, the statement and the proof structure in Section 3 were read on the page images of arXiv:1902.07397v1. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input named by the paper: the multiplicative large sieve, Theorem 7.13 of Iwaniec and Kowalski, Analytic number theory (2004).

Source. Junsoo Ha and Kannan Soundararajan, Many solutions to the S-unit equation a + 1 = c, Acta Math. Hungar. 160 (2020), 153--160, doi:10.1007/s10474-019-00948-z; pages are those of arXiv:1902.07397v1, the edition named on the source card.

Bears on

None directly; it is the input to Theorem 1.