Explicit Error Term Research Articles

Higher order Turán inequalities for the distinct partition function

We prove that the number q(n) of partitions of n with distinct parts is log-concave for n≥33 and satisfies the third-order Turán inequalities for n≥121 conjectured by Craig and Pun. In doing so, we establish explicit error terms for q(n) and for q(n−1)q(n+1)/q(n)2 based on Chern's asymptotic formulas for η-quotients.

Computing error bounds for asymptotic expansions of regular P-recursive sequences

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

On the Divergence of the Sum of Prime Reciprocals

The sum of the reciprocals of all prime numbers diverges but the divergence is very slow. We propose effective lower and upper bounds for partial sums under the Riemann hypothesis. We give an explicit error term for all Mertens’ theorems.

Atkinson's formula for the mean square of ζ(s) with an explicit error term

We provide an explicit O(log2⁡T)-term of the celebrated Atkinson's formula for the error term E(T) of the second power moment of the Riemann zeta-function on the critical line. As an application, we obtain an explicit version of well-known estimate E(T)≪εT13+ε.

Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis

We prove an explicit error term for the ψ ( x , χ ) \psi (x,\chi ) function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.

Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

Local Moderate and Precise Large Deviations via Cluster Expansions

We consider a system of classical particles confined in a box \(\varLambda \subset {\mathbb {R}}^d\) with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature–low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with the infinite volume versions of the above are also provided.

Uniformly counting primes with a given primitive root and in an arithmetic progression

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin’s primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.

On partial sums of arithmetical functions of two variables with absolutely convergent Ramanujan expansions

In this article, we derive an asymptotic formula for the sums of the form $${\sum }_{n_{1},n_{2}\le N}f(n_1,n_2)$$ with an explicit error term, for any arithmetical function f of two variables with absolutely convergent Ramanujan expansion and Ramanujan coefficients satisfying certain hypothesis.

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.

Slicing the stars: counting algebraic numbers, integers, and units by degree and height

Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.

On approximation of ultraspherical polynomials in the oscillatory region

For k≥2 even, and α≥−(2k+1)∕4, we provide a uniform approximation of the ultraspherical polynomials Pk(α,α)(x) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g(x)=cb(x)(1−x2)(α+1)∕2Pk(α,α)(x)=cosB(x)+r(x). Here the constant c=c(k,α) is defined by the normalization of Pk(α,α)(x), B(x)=∫0xb(x)dx, and the functions b(x) and B(x), as well as bounds on the error term r(x), are given by some rather simple elementary functions.

Large Deviations for the Macroscopic Motion of an Interface

We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale. Furthermore, using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time, we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough.

Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We rst provide a detailed analysis of Duistermaat-Heckman measures in the context of test congurations, characterizing in par- ticular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kahler geometry. These functionals are dened on the space of test congurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and sin- gularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.

Finite Ramanujan expansions and shifted convolution sums of arithmetical functions

For two arithmetical functions f and g, we study the convolution sum of the form ∑n≤Nf(n)g(n+h) in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup.

A note on arithmetical functions with absolutely convergent Ramanujan expansions

Arithmetical functions with absolutely convergent Ramanujan expansions have been recently studied in certain contexts, by the present author, Murty and many others. In this article, we aim to weaken some of their hypotheses and derive asymptotic formulas with an explicit error term for sums of the form [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two arithmetical functions with absolutely convergent Ramanujan expansion.

Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions

For an arithmetical function f with absolutely convergent Ramanujan expansion, we derive an asymptotic formula for the sum \({\sum }_{n \le N} f(n)\) with explicit error term. As a corollary we obtain new results about sum-of-divisors functions and Jordan’s totient functions.

Monte Carlo with determinantal point processes

We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical N^{−1/2} , where N is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as N^{−(1+1/d)/2} , where d is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition; a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which, at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.

Schanuel's theorem for heights defined via extension fields

Let k be a number field, let ✓ be a nonzero algebraic number, and let H(·) be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of ↵ 2 k with H(↵✓)  X, and we analyze the leading constant in our asymptotic formula. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of k with a fairly explicit error term. This provides a conceptual framework for Loher and Masser’s problem and generalizations thereof. Finally, we establish asymptotic counting results for varying ✓, namely, for the number of pp↵ of bounded height, where ↵ 2 k and p is any rational prime inert in k.

ON THE DISTRIBUTION OF THE GREATEST COMMON DIVISOR OF GAUSSIAN INTEGERS.

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm x or less as x goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.