Papers

Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\bigr)_\infty$. Our method is to regard the exponent map $m\mapsto m(\nu_2(m)+2)$ as a dynamical system on the positive integers: the factors of the associated quotient link into chains along its forward orbits, and the three orbits seeded at $1$, $4$ and $5$ are pairwise disjoint and telescope to exactly $1/\bigl((1-q)(1-q^4)(1-q^5)\bigr)$. The exponent triple $(1,4,5)$ is admissible in the sense of earlier work by Liu, which yields the desired factorization into two series with nonnegative coefficients. This orbit telescoping technique appears to be a mechanism complementary to the Pólya–Szeg\H o criterion that underlies most existing positivity results of this kind in the literature.

We prove an $S$-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let $C$ be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-$p$ positive integers $m_i$ and any finite set of closed points $S\subset C$, we prove that $C\setminus S$ meets the special divisor $Z(m_i)$ for all but finitely many $i$. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford–Tate conjecture.

We introduce a broad generalization of a recursive formula for the Möbius function $\mu(n) = 1-n-\sum_{d=2}^{n-1}\mu(d)\left[\frac{n}{d}\right]$ due to George Spencer-Brown. By replacing the greatest integer function $\left[\frac{n}{d}\right]$ in this classical recurrence with an arbitrary arithmetic function $f\left(\left[\frac{n}{d}\right]\right)$ with $f(1)=1$, we define a new generalized family of functions, denoted $\star(n)$. We prove that $\star(p) = -1$ if and only if $p$ is prime. This result yields a surprising algebraic characterization of primes and reveals a deep structural property underlying divisor sums and the greatest integer function.

We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.

We present the first method for native, continuous gradient descent for machine learning models with $p$-adic parameters. Existing native optimizers are discrete, mostly combinatorial searches, as the $p$-adic numbers $\mathbb{Q}_p$ are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with $\mathbb{Q}_p$ via its Berkovich affine line: a canonical, path-connected expansion of $\mathbb{Q}_p$ that preserves its isometries and uniquely extends its analytic maps. This hull is a metric tree with interpretable points and local derivatives, which we show enables effective optimizers and backpropagation. We formulate gradient descent and show that its approximations efficiently learn linear models with coefficients in $\mathbb{Q}_p$ to do modular arithmetic, an XOR-like task not expressible by linear models in $\mathbb{R}$. We also demonstrate momentum and Adam variants, linear regression, and classification on binary-encoded hierarchies (Quillian semantic networks), addressing open problems posed by Martins (2025). Library at https://github.com/google-deepmind/padic-ml

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are those whose coordinate gcd is not divisible by the $r$th power of any prime. Fix $1\le q\le \infty$. A lattice point is $R$-deep if every lattice point in the closed $\ell_q$-ball of radius $R$ centred there is non-$r$-free. We mark each finite nearest-neighbour component of non-$r$-free points containing an $R$-deep point by its lexicographically least such point. Uniformly for $1\le q\le\infty$, the densities of $R$-deep points and of these representatives both have the asymptotic form $\exp\left\{-\frac{v_{d,q}R^d}{\zeta(dr)}\left[d(dr-1)\log R+dr\log\log R-\Upsilon_{d,r,q}+r\frac{\log\log R}{\log R}-\frac{\Lambda_{d,r,q}}{\log R}+O_{d,r}\left(\frac{(\log\log R)^2}{(\log R)^2}\right)\right]\right\}$ as $R\to\infty$, where $v_{d,q}$ is the volume of the unit $\ell_q$-ball and $\Upsilon_{d,r,q},\Lambda_{d,r,q}$ are explicit constants. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. For $d=1$, consider the densities of $r$-free integers followed by at least $g-1$ consecutive non-$r$-free integers, or by exactly $g-1$ such integers and then another $r$-free integer. Both have the asymptotics $\exp\left\{-\frac{g}{\zeta(r)}\left[(r-1)\log g+r\log\log g-\widehat{\Upsilon}_r+r\frac{\log\log g}{\log g}-\frac{\widehat{\Lambda}_r}{\log g}+O_r\left(\frac{(\log\log g)^2}{(\log g)^2}\right)\right]\right\}$ as $g\to\infty$, where $\widehat{\Upsilon}_r=\Upsilon_{1,r,1}+(r-1)\log2$ and $\widehat{\Lambda}_r=\Lambda_{1,r,1}+r\log2$. The exact-gap expansion improves Grimmett's leading asymptotic and refines the formula in Jiang's recent preprint. For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.

For a positive integer $t$, let $d_t$ denote the natural density of the set of $n$ for which $t$ is a sum of distinct divisors of $n$. Erdős proved that $d_t$ exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether $d_t \sim c_3/(\log t)^{c_4}$. We record the explicit bounds \[ \nu_t \le d_t \ll \frac{(\log\log t)^{\delta-3/2}}{(\log t)^{\delta}}, \qquad \nu_t = \frac{K}{\log t}\Bigl(1 + O\Bigl(\frac{\log\log t}{\log t}\Bigr)\Bigr), \] where $\delta = 0.086071\ldots$ is the Erdős–Ford–Tenenbaum constant, $K=c\,e^{-\gamma}$, and $c=1.33607\ldots$ is the practical-number constant. Consequently, if Erdős's asymptotic holds, then $\delta<c_4\le 1$. A two-prime construction, using a half-scale sumset to obtain full residue coverage, then yields the pointwise excess \[ \liminf_{t\to\infty}(\log t)\,(d_t-\nu_t) \ge KI, \] where $I=\int_0^2 G(w)\,\mathrm{d}w=0.05887\ldots$ is an explicit elementary integral. In particular $d_t\ge 0.79/\log t$ for every sufficiently large $t$, and $d_t$ is not asymptotic to $K/\log t$.

Motivated by a question of Regev arising from his improved quantum factoring algorithm, we study how many small primes are needed to generate the group $({\mathbb Z}/q{\mathbb Z})^\times$ when each prime may be used with exponent only $0$ or $1$. We prove that, for every fixed $\varepsilon>0$ and $A>0$, there is an absolute constant $C_*$ and a set of at most $(\log Q)^{1+\varepsilon}$ primes, all at most $(\log Q)^{C_*(A+1)}$, such that for all but $O(Q(\log Q)^{-A})$ (with the implied constant depending only on $\varepsilon$ and $A$) integers $q\leq Q$, every element of $({\mathbb Z}/q{\mathbb Z})^\times$ is a product of a subset of these primes modulo $q$. The exponent $1+\varepsilon$ in the number of primes is best possible up to the arbitrary $\varepsilon$ in the exponent.

We study monic irreducible polynomials over $\mathbb{F}_q[T]$ whose non-leading digits, with respect to an arbitrary polynomial base $b(T)$, avoid a prescribed set of forbidden digits. Identifying the digit set with the ring $D=\mathbb{F}_q[T]/(b)$, we obtain an asymptotic formula for the number of such irreducible polynomials under conditions given in terms of Fourier parameters of the allowed digit set. The main term contains a singular series measuring the relative density of units among the allowed digits, while the error term is controlled by both pointwise and averaged Fourier estimates. As a consequence, we obtain a general criterion depending only on the cardinality of the forbidden set, as well as stronger results for structured restrictions, including examples in which the forbidden set contains a positive proportion of all digits. In particular, we treat additive cosets, restrictions compatible with the Chinese remainder decomposition of $D$, and coefficient-wise restrictions. The proof adapts the function field circle method for restricted coefficients to arbitrary polynomial bases.

Following Erdős (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(\phi(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $$ where $\phi$ is the Euler totient function. The key ingredient is the Poisson random model for $\omega(n,T)$, the number of the prime divisors of $n$ in a given subset of primes $T$, which was introduced in a recent work of Ford.

We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along $\Omega$ converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector–disk comparison in an explicit logarithmic shrinking range, with a power-saving error in $\log N$, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and $T^{\Omega(n)} x$, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.

Let $(a_n)_{n\geq 0}$ be a sequence of numbers. We derive an expression for the Euler transform of a general binomial sum involving $a_n$ and weighted by the floor function. To demonstrate the usefulness of our approach, several examples are discussed. We rediscover some known identities and prove several new. In particular, we derive some new identities for weighted binomial sums involving harmonic numbers and central binomial coefficients. We also present new closed-forms for weighted series involving the Riemann zeta function.

Let $\mathbb{P}$ be the set of primes. Suppose that $P_{i}$, $i=1,2,\ldots,9$ are nine subsets of $\mathbb{P}$ with relative lower density $\underline{d}_{\mathbb{P}}(P_{i})=\mu_{i}$. When $\sum_{i=1}^{9} \mu_{i}^{3}>8$, we prove that every sufficiently large odd integer can be written as $n=\sum_{i=1}^{9} p_{i}^{3}$ with $p_{i} \in P_{i}$, $i=1,2,\ldots,9$.

The Lie algebra of multiple zeta values is realized as the space $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ of alternal moulds whose swap is alternil up to a constant mould, equipped with the $ari$ bracket. In this paper, we study the larger space $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ of alternil, swap-invariant bimoulds, which is conjecturally the Lie algebra for multiple $q$-zeta values. We propose an explicit formula for a Lie bracket $uri$ on this space. Moreover, we prove that the Lie algebra $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ embeds into $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, with the $ari$ bracket translating directly into the $uri$ bracket. Finally, we examine the associated-depth graded of $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, extending the known depth-graded setup for $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$.

We show that for any number field $K$ and any rational map $f$ in $K(x)$ that is not conjugate to a power, (signed) Chebyshev or Lattès map, then $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. We also classify the pairs $(f,\alpha)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews–Petsche conjecture to rational maps over a number field and to infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with Astra. The key insight consists of applying the equidistribution results \cite{Yua08}, followed by a classification of $(f,f)$-preperiodic curves \cite{Pak23,Pak20, Bea25} followed by an additional application of equidistribution on a parametrized preperiodic curve.

We establish a function field analogue of a recent conjecture of Andrews–Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones, whenever the degree of the polynomial is smaller than $p$, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: $$\text{Abelian} \implies \text{Finite ramification} \implies \text{PCF map} \implies \text{Isotrivial map} \implies \text{Isotrivial pair}, $$ and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches $p$ there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.

Let $K=\mathbb{F}_q(\theta)$, and let $\mathcal{Z}$ be the $K$-algebra generated by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$. We prove that $\mathcal{Z}$ is a polynomial algebra and construct an explicit polynomial basis of $\mathcal{Z}$ over $K$. We also determine the transcendence degree of $K[\zeta_A(\mathfrak{s}): \operatorname{wt}(\mathfrak{s}) \leqslant w]$ over $K$ for every integer $w \geqslant 1$, generalizing a result of Ngo Dac–Nguyen Chu–Pham. The proof uses Chang's grading theorem, the linear basis theorem proved independently by Chang–Chen–Mishiba and Im–Kim–Le–Ngo Dac–Pham, and the carry relations of Im–Kim–Ngo Dac. The key step is to show that suitable derivations obtained from deconcatenation and linear functionals descend through the carry relations to $\mathcal{Z}$.

Let $\Gamma\subset\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$ be an irreducible, non-uniform lattice and define the space $X = \operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})/\Gamma$. Let $U$ be the standard horospherical subgroup in $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$. We show that for every $\delta>0$ and $x\in X$ with dense $U$-orbit, the $U$-orbit evaluated at almost squares, $\{(u_{(n^{2-\delta},m^{2-\delta})}~:~n,m\in\mathbb{N}\}\cdot x$, is dense in $X$. The main idea of the proof is to shadow large periodic $U$-orbits and reduce the statement to a density statement within the periodic $U$-orbit, i.e. a statement for the $2$-torus $\mathbb{T}^2$. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic $U$-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in $\operatorname{SL}_2(\mathbb{R})$.

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $\alpha$ for $f$. We show that the extension $K(f^{-\infty}(\alpha))/K$ generated by the backward orbit of $\alpha$ is virtually abelian if and only if $f$ is an exceptional map with CM and $\alpha$ is $f$-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over $\mathbb{C}$ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.

Transcendence questions of periods and quasi-periods of Anderson modules, as well as the hyperderivatives thereof, are of major interest in number theory over function fields. In the case of Drinfeld modules, this question was answered by the second author. In this paper, we show that these results hold for general Anderson $t$-modules under certain assumptions on their Galois representations. The proofs use the explicit description of the $\mathfrak{p}$-adic Galois representation of the first author by means of a rigid analytic trivialization of the associated $t$-motive, as well as prolongations of $t$-modules and $t$-motives. The general result is applied to determine algebraic relations between periods and quasi-periods, and their hyperderivatives, of finitely many Drinfeld modules. Along the way, our results provide more evidence for the Mumford-Tate conjecture for Anderson modules.

We compute an explicit contribution of supercuspidal representations arising from Yu's construction to the middle étale cohomology of local Shimura varieties. For the proof, we construct special affinoids in local Shimura varieties whose reductions are isomorphic to Heisenberg Deligne-Lusztig varieties and then apply positive-depth Deligne-Lusztig theory. Along the way, we propose a stacky analogue of special affinoids using the stack of local shtukas and observe that the `general canonical subgroup problem' can be solved purely on the stack of local shtukas, which yields the corresponding claim on Shimura varieties via the Hodge-Tate period map.

We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally $C^2$ normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to $C^{1,1}$: we construct a non-affine $C^{1,1}$ diffeomorphism which preserves normality in every integer base, and hence absolute normality.

We consider the parametrised family of sets $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ for $(x,y) \in \mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of $E(x,y)$ for $(x,y) \in \mathbb{N}^2$. Next, we collapse this topological classification to a binary one according to whether or not $E(x,y)$ has interior. When this binary classification is visualised, it reveals a two-colour tiling $T$ of the lattice $\mathbb N^2$, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family $\{E(x,y)\}_{(x,y) \in \mathbb{N}^2}$ is viewed through the theory of self-similar sets, $T$ can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that $T$ is a factor of a substitution tiling on four “hidden” arithmetically defined states.

We discuss local properties of the classical Fibonacci and Lucas sequences, characterising in both cases those primes~$p$ for which the~$p$-part satisfies the Dold congruences or is realizable. We calculate an exact `repair' multiplier in each case and determine the relative prime densities of the Dold, realizable, almost Dold, and almost realizable prime sets. In that order, the four densities are~$(\frac34,\frac13,1,\frac13)$ for the Lucas sequence and~$(\frac12,\frac12,1,1)$ for the Fibonacci sequence.

We study sums associated with the action of $PGL_2(\mathbb Z)$ on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under suitable continuity, finiteness, and convergence assumptions, we prove that these sums are well defined, bounded, $1$-periodic, and continuous on $\mathbb R$, and satisfy a reciprocal functional equation. We also extend the construction to finite families of distinct function orbits. Our framework recovers Zagier's constancy result and includes the full family of quadratic sums for which Bengoechea proved convergence.

For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed $j\geq 2$, the limit $$\lim_{n\to\infty} s_j(n)/n$$ does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets $T_j(n)$ that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of $t_j(n) := |T_j(n)|$. We show that $$\lim_{n\to\infty}\frac{t_j(n)}n$$ exists precisely for $j\in\{0,1\}$ and fails to exist for every fixed integer $j\geq2$.

We present a method for determining whether there exists a degree $D$ rational map from $X_0(N)$ to some elliptic curve $E/\mathbb{Q}$. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree $D$ maps to elliptic curves by a lifting criterion on the lattices $H_1(-,\mathbb{Z})$ As an application of this method, we determine all $D$-elliptic curves $X_0(N)$ for all $D\leq 100$. Interestingly, in some cases we found degree $D$ rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.

In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}{5}\rho\left(\frac{1}{c}\right),1-c+g(c)\right), \end{align*} where $g(c)>0$ for $11/32<c\leq1/2$, and $g(c)=0$ for $0<c\leq11/32$. In particular, we have $g(1/2)>0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-4\rho\left(\frac{1}{c}\right),1-c\right), \end{align*} for $0<c\leq 1/2$.

Let $(V,q)$ be an anisotropic quadratic space of dimension $4$ over the rationals and let $p$ be a prime such that the local quadratic space $V\otimes \mathbb{Q}_p$ is the orthogonal direct sum of two hyperbolic planes. We show that the group of $p$-adic rigid meromorphic cocycles attached to $V$ has infinite rank. A key ingredient in the proof is the computation of the group of invertible rigid analytic functions on products of certain rigid analytic subspaces of the projective line such as the $p$-adic upper half-plane.

Let \(\Kl(1;q)\) denote the normalized Kloosterman sum modulo \(q\). We prove that, for each sign, there are \(\gg X/\log X\) square-free moduli \(q\in(X,2X]\) with at most four prime factors for which \(\Kl(1;q)\) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.