Papers

In the recent work of Bergeron-Charollois-García \cite{BCG23}, a conjectural analytic expression for elliptic units for complex cubic fields was proposed: namely, as the value of a smoothed quotient of elliptic Gamma functions. The complex number depended on the conductor ideal, ray class group element, smoothing ideal, and a `torsion point' $h$. If this analytic function is to fit into the framework of explicit class field theory, then, as in the case of torsion points on CM elliptic curves, our choice of $h$ should only depend on its congruence class modulo some lattice $L$. This conjecture has been supported by much numerical evidence. In this paper, we use the cocycle relations satisfied by the smoothed elliptic Gamma function to prove that the construction is indeed independent of $h$.

For elliptic curves $E/\mathbb{Q}$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about $42.2\%$ have exactly one prime with nontrivial local Tamagawa number, and only about $6.8\%$ have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.

Levin asked what can be said about the locus of points lying on a given curve in $\mathbb{G}_m^n$ that have a non-zero integer multiple on another given curve in $\mathbb{G}_m^n$ for $n \geq 3$. We give a definite answer to the abelian analogue of Levin's question, proving what is predicted by the Zilber–Pink conjecture in this case. An important ingredient in our proof is a strengthening of a height inequality by Vojta and Rémond.

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.

We study rational approximation to logarithms of positive rational numbers and give a separated-weight determinant argument for $\mu(\log r)=2$ whenever $r\in\mathbb Q_{>0}\setminus\{1\}$. The geometric input is an interpolation theorem for logarithmic jets at points with distinct multiplicative coordinates. Its successive weight thresholds are independent of the interpolation centres; the eventual degree threshold may depend on them. A local intersection-length bound and a curve inequality lead to surjectivity through a blow-up and Serre vanishing. For a fixed rational argument, this produces a nonzero rational determinant. Clearing denominators gives an arithmetic lower bound, while an exact row translation and repeated Taylor indices yield a contradictory analytic upper bound. The consequences include nonzero rational linear combinations of such logarithms, rational affine changes, and logarithms of positive numbers with a rational positive integer power. We also give a bounded-fibre interpolation extension, an explicit bound for the final analytic degree, and a counterexample to interpolation under the volume condition alone.

In this paper we study the least positive integer $G(n)$ such that the integers $a\leq G(n)$ with $(a,n)=1$ generate $(\mathbb Z/n\mathbb Z)^\times$. We prove that, for every $\varepsilon>0$, \[ \sum_{n\leq x}G(n)\ll_\varepsilon x(\log x)^{8/3+\varepsilon}, \] improving the previously known bound $\ll x(\log x)^{97}$. We also prove that $G(n)\leq(\log n)^2$ for almost all $n$, unconditionally. Thus, for almost all $n$, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.

Given a maximal order $\mathfrak{O}$ in a rational definite quaternion algebra, and a prime $\ell$ coprime to the discriminant of $\mathfrak{O}$, this paper considers subsets of $\mathfrak{O}$ consisting of elements with $\ell$-power norms that together generate $\mathfrak{O}$ as a $\mathbb{Z}$-algebra. We prove two theorems about the existence of such sets: the first states that $\mathfrak{O}$ is generated by elements of norm $\ell^k$ for any $k$ larger than an explicit bound, and the second states that there is a generating set for $\mathfrak{O}$ consisting of at most three elements, each with norm a power of $\ell$. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders $\mathcal{O}$ that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for $\mathcal{O}$ and explicit bounds on the coefficients of the theta function of $\mathcal{O}$. Computational experiments are also discussed.

Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order $+,\times,-,\div$. We ask for the fewest moves needed to make every integer from $1$ to $N$. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least $N-1$ moves are needed. We show that $N-1$ moves are also enough for every $N\ge 9$. Conventional induction cannot work, because a division that comes right after a completed interval $\{1,\dots,P\}$ produces numbers already made. Instead we extend a completed interval $\{1,\dots,P\}$ to $\{1,\dots,3P\}$ all at once, which counting shows is the smallest multiplicative extension $P\to kP$ that can work, and then adjust the last few moves to reach every other $N$. For $9\le N\le 33$ we give explicit sequences, found by computer search.

Let $H_n$ be the one-dimensional Honda formal group of height $n$ over $\mathbf F_{p^n}$ and let \[ R_n=W(\mathbf F_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p). \] We prove, for every height $n$ and every prime $p$, that the prime ideals of $A_n$ stable under an open subgroup of the Morava stabilizer group are exactly the height ideals $(u_1,\ldots,u_j)$. We also prove that a stable prime of $R_n$ avoiding $p$ is zero. The resulting radical-ideal classification implies the Hovey-Strickland classification of thick tensor ideals in the category of dualizable $K(n)$-local spectra via the forward implication of Barthel-Heard-Naumann. The special-fiber argument is local and geometric. After cutting an invariant prime by a one-parameter curve, we construct from the Cartier structure equation a smooth formal quotient $\mathcal{Q}$ and a distinguished subgroup $\mathcal{H}$. The generic fiber of $\mathcal{H}$ is identified with the deformation space of connected-étale extensions. A Cartier obstruction map from the full Honda endomorphism order is compared with evaluation on Tate vectors through completed universal covers. Fargues-Fontaine vector bundles give a period-detection statement. Chai's rigidity theorem then promotes detection by homomorphisms to formal Zariski density after a renormalization of the valuation. A uniform fixed-jet argument transfers this density to Morava-stabilizer orbits. The generic-fiber assertion is proved separately from the Gross-Hopkins period map.

For $r\ge1$, let $K=2r+14$ and $d=\dim S_K$. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd–even period independence conjecture of Xue. As a consequence, the consecutive first Rankin–Cohen brackets $[E_{2r+10-2j},E_{2j+2}]_1$, $1\le j\le d$, form a basis of $S_K$, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor–tau convolution $C_r(n)=\sum_{m=1}^{n-1}\sigma_{2r+1}(m)\tau(n-m)$. Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous $f,g\in\mathbf Q[E_4,E_6]$, we prove $[f,g]_1/\Delta=-3456\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$, reducing the Cramer system to a one-variable coordinate problem in $T=E_6^2/E_4^3$.

We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form $s^5\Psi(\log_2(s^{-2}))+o(s^5)$, where $\Psi$ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota–Müllner–Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.

Using the recent proof of the Cassels–Swinnerton-Dyer conjecture for cubic surfaces, we finish the classification of $G$-unirational complex cubic threefolds. In particular, we prove that the Klein cubic threefold is $\mathsf{PSL}_2(\mathbb{F}_{11})$-unirational, establishing that the essential dimension of $\mathsf{PSL}_2(\mathbb{F}_{11})$ is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.

In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form $aX+bY=cZ+dU$ (where $a+b=c+d$). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space $\mathbb{F}_{p}^{n}$. Next, we turn to cyclic groups $\mathbb{Z}_{m}$ and construct surprisingly large sets that have no solutions for the asymmetric equation $X+5Y=3U+3Z$. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus $m$. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.

For a positive real lacunary sequence and any probability measure $\mu$ with polynomial Fourier decay, we determine, for $\mu$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo $\mathbb Z^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\log N)/N$. For independent uniform points in a fixed convex body $\Omega\subset\mathbb R^d$, the maximal volume of an empty convex body is asymptotic to $d\operatorname{vol}(\Omega)(\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.

For any non-zero periodic function f\,: Z $\rightarrow$ C of period N ___ 1 and any non-zero algebraic number z\_0 that doesn't lie on a half-line [e^2i${\ell}$$\pi$/N , e^2i${\ell}$$\pi$/N $\infty$[, 0 ___ ${\ell}$ ___ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z\_0 )-vector space spanned by the numbers L(f, i, z\_0 )\,: sum\_{m=1}^$\infty$ f(m)z\_0^m/m^i. This generalizes a result Fischler proved in 2026, corresponding to f identically equal to 1 and |z\_0| ___ 1: in this case, the numbers L(f, i, z\_0 ) = Li\_i (z\_0 ) are polylogarithm values. Except for a finite number of cuts in the complex plane, our result still holds when |z\_0 | > 1, that is in a domain where the series definition above for the numbers L(f, i, z\_0 ) doesn't converge anymore, and we make sense of these numbers through analytic continuation. To obtain this result, we construct linear combinations of the numbers L(f, i, z\_0 ) using a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on an integral representation of the polylogarithm functions and on a ”Shidlovskii lemma”.

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 4$ which defines a smooth hypersurface in $\mathbb{P}^{n-1}_{\mathbb{C}}$. For $n \geq 24 d^4 2^d$, we prove an asymptotic formula for the number of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$, provided $F$ satisfies suitable local conditions.

We prove the Gowers uniformity of the von Mangoldt function minus its Cramér model in all short intervals $[X,X+X^{3/5+\varepsilon}]$, improving on the work of the first and fourth authors with Shao and Tao, where the exponent was $5/8$. This also implies a local-to-global theorem for linear equations in primes in such short intervals. We also show that the Möbius function has cancellation in all intervals $[X,X+X^{19/35+\varepsilon}]$, improving on the work of the first and fourth authors, where the exponent was $11/20$. Both improvements are based on improved treatment of trilinear sums over short intervals which naturally reduces to estimating mean values of products of three Dirichlet polynomials. In a general setting, we reduce the task of estimating mean values of products of Dirichlet polynomials with given large value bounds to the task of showing that a certain piecewise linear function is positive. This reduction allows recent large value estimates to be incorporated. Combining this general framework with the most recent large value estimates stemming from the work of Guth and Maynard, we prove a Heath-Brown–Iwaniec type estimate for type I/II sums in intervals of length $X^{19/35+\varepsilon}$ and a Baker–Harman–Pintz parallelogram type estimate for general trilinear sums in intervals of length $X^{3/5+\varepsilon}$.

We construct regular integral models for a class of Shimura varieties of symplectic and orthogonal type with maximal parahoric level at an odd prime $p$. These models are defined over the ring of integers of the reflex field and have special fiber with normal crossings and irreducible components of multiplicity one or two. Our construction uses explicit equivariant modifications of the corresponding canonical local models for symplectic and split even orthogonal similitude groups, with the minuscule cocharacters corresponding to maximal isotropic Grassmannians. These modifications are obtained by successively blowing up Schubert varieties in the special fiber. This proves the equivariant modification conjecture of Pappas at maximal parahoric level in the above cases.

Let $p\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y^2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2^\infty$-Selmer corank one for $p\equiv 3,5\pmod{8}$ and corank zero for $p\equiv 7\pmod{8}$. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class $p\equiv 1\pmod{8}$. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.

We extend the quadrature method developed in [10] to Dirichlet $L$-functions associated with primitive characters. The approximations combine finite Dirichlet sums with correction terms whose quadrature coefficients can be precomputed for each character. Numerical experiments demonstrate high accuracy in the critical strip and in other vertical strips. We also describe how to combine these approximations with direct summation and Euler–Maclaurin summation to evaluate $L(s,\chi)$ throughout the complex plane.

We give two explicit reformulations of a Balazard–Saias–Yor type criterion for the Riemann hypothesis. First, Jensen's formula is applied to $u(s)=(s-1)\zeta(s)$ after mapping the unit disk onto $Re s<1/2$. For the normalization whose center corresponds to $s=-1/2$, the images $z_n=(4n-1)/(4n+3)$ of the trivial zeros yield a telescoping Jensen product. This isolates, at finite radius, the complete divergent contribution of the trivial zeros; the remaining defect is a nonnegative integrated count of mapped nontrivial zeros off the critical line. A one-parameter version gives a gamma-quotient product and an explicit finite part. Second, after passing through the established boundary identity, we combine two Poisson weights to obtain an absolutely convergent logarithmic integral of the Riemann $\Xi$-function. From the random variable $Z=|\Xi(T)/\Xi(0)|^2$ we define its law $\mu$ and the size-biased law $\nu(\,\mathrm d z)=z\mu(\,\mathrm d z)/M$. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for $D_{\rm KL}(\mu\|\nu)$. We also record the associated Donsker–Varadhan and moment forms and express the target constant through Riemann's positive Fourier kernel. The analytic boundary criterion and Jensen–Hardy mechanism used here are known. The purpose of the paper is an explicit finite-radius treatment of the trivial zeros and an information-theoretic reformulation, not a new proof of the Riemann hypothesis or an independent boundary-integral criterion.

Let $p_n$ be the $n$th prime and $g_n=p_{n+1}-p_n$. Following Gafni and Tao, an integer $m$ with $p_n<m<p_{n+1}$ is rough for the gap if its least prime factor is at least $g_n$; let $M_n$ be the number of such integers. Gafni and Tao proved that the number of $n$ with $X\le p_n\le 2X$ and $M_n=0$ is $O(X/\log^2 X)$, and that it is asymptotic to $cX/\log^2 X$ under a form of the Dickson-Hardy-Littlewood prime tuples conjecture. We study $N_r(X)$, the number of $n$ with $X\le p_n\le 2X$ and $M_n<r$. For every $r\ge 1$ we define a constant $c_r$ by an endpoint-conditioned local sieve model, with $c_1=c$, and we prove unconditionally that $c_r=h_r^*+O_\varepsilon(r^{1/2}(\log r)^{1+\varepsilon})$ for $r\ge 3$, where $h_r^*$ is the least even $h\ge 2$ with $h\prod_{p<h}(1-1/p)\ge r$. Consequently $c_r=H_r(1+O(e^{-\kappa\sqrt{\log r}}))$ for an absolute constant $\kappa>0$, where $H_r>e$ solves $e^{-\gamma}H_r/\log H_r=r$ ($\gamma$ is Euler's constant), and $c_r=e^\gamma r(\log r+\log\log r+\gamma+(\log\log r+\gamma)/\log r+O((\log\log r)^2/(\log r)^2))$. The proof combines the Montgomery-Vaughan moment bounds for reduced residues in short intervals with an exact removal of the endpoint conditioning. For each fixed $r$, the argument of Gafni and Tao extends to give $N_r(X)\ll_r X/\log^2 X$ unconditionally, and $N_r(X)\sim c_rX/\log^2 X$ under their conjecture. Exact enumeration of the local model gives $c\ge 2.7413$.

The following article attempts to define a new digital signature scheme based on a variant of the MinRank problem under appropriate parameters. First, the key generation process will be defined, followed by the message signing and verification processes. The speed of signing and verification will be analyzed, and finally, the signature will be evaluated in terms of security and robustness against known attacks, which will lead to a set of requirements that we will consider when selecting certain parameters related to the size and number of variables. It should be noted that this signature, although simple in design, can serve as the foundation for an entire family of digital signatures based on the central idea that inspired its creation.

This article analyzes the idempotence of Giuga numbers and Carmichael numbers. The first sections will lead us to define an arithmetic function through which we will find a new characterization of Giuga numbers. The following sections will lead us to generalize Carmichael numbers, finding numbers that are Fermat pseudoprimes in certain specific cases, and to formulate various statements and open questions regarding their properties. Finally, we will find a new alternative condition, based on idempotence, that a number must satisfy to be a counterexample to the Agoh-Giuga conjecture and which, therefore, presumably, no number will ever satisfy. We will almost always work under the premise that n is square-free, since both numbers always have that property, but in certain situations we will extend some of the results to the general case.

We prove that every affine coordinate subspace of positive dimension and positive codimension fails to be of strong Khintchine type for convergence. Combining this with Huang's criterion for Khintchine-type affine subspaces gives a sufficient Diophantine condition for an affine coordinate subspace to be of Khintchine type but not of strong Khintchine type for convergence.

We prove that for a geometrically connected modular curve $X_H$ of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of $X_H$. The proof relies on a general criterion involving the gonality of $X_H$ and the dimension of the CM part of the Jacobian of $X_H$. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.

Let N(T) count the nontrivial zeros of the Riemann zeta function up to height T with multiplicity, N_d(T) the distinct ones, and N_0^s(T) those that are simple and on the critical line. We prove liminf N_d(T)/N(T) >= 1645064/1960733 = 0.83900...; earlier work proves 0.83699..., and a report we have not verified claims 0.83805.... Hence more than 67.80% are simple. Also liminf N_0^s(T)/N(T) >= 0.67353...; earlier work proves 0.67250..., and reports we have not verified claim 0.673492. Two estimates of Lamzouri are improved: at least 88.93% of the zeros are simple or on the critical line, and the proportions of simple zeros and of zeros on the line average at least 83.67%. By the unconditional form of Montgomery's pair-correlation theorem the energy, a sum of a test function over pairs of zeros, is known asymptotically. A zero of multiplicity d contributes d^2 to it and nearby zeros contribute too, so a lower bound for such pairs leaves less energy for multiple zeros. The proof has three steps. First, for zeros at least a fixed fraction of the mean spacing apart, a large sieve inequality lets their pairs be used in full, with multiplicities. Second, their contribution is bounded below by a computer-assisted inequality for seven or eight consecutive zeros which distinguishes simple and double zeros. Its correction terms telescope, as increments of a storage function. Third, the test function is chosen to make this contribution large, at the cost of more energy. We also prove upper limits for what such inequalities can give with the two main test functions. The lower bounds and these limits are proved in Lean 4, analytic inputs included. The proofs use three axioms beyond those of Mathlib, one per inequality. Each records that a search program returned true; Lean's kernel does not check the run itself. This work is an experiment in AI-assisted mathematical research.

Consider placing the numbers $1, \ldots, n$ consecutively around an $n$-cycle. We ask for a given $n$, which numbers $k$ can be written as a sum of consecutive numbers on the $n$-cycle. If all numbers between $1$ and $T_n$, the $n$th triangular number, can be written as a sum this way we say the $n$-cycle is complete. Our main result shows that for $n\geq 8$, $n$-cycles are incomplete.

Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^{2\pi it}, \] where the inner sum is over primes. For $\alpha=\frac{a}{q}+\frac{\theta}{q^2}$, $a\in\mathbb Z$, $q\in\mathbb N$, $(a,q)=1$, $|\theta|\le1$, we establish the bound \[ \begin{aligned} V_n(K,x,y) &\ll Ky\Biggl[ \frac{1}{\sqrt{K\log(2y)}}+ \min\left\{ \Delta^{\frac{1}{2^n}} \bigl(\log(2y)\bigr)^{\frac{n^2-1}{2^n}}, \Delta^{\frac{1}{3\cdot 2^{n-2}}} \bigl(\log(2y)\bigr)^{\frac{n^3-1}{3\cdot 2^{n-1}}} \right\} \Biggr], \end{aligned} \] where \[ \Delta= \frac{1}{q}+\frac{1}{y}+\frac{q}{Ky^n}. \] The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fejér kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.