arXiv subjects
Publications and source records attributed to Richard P. Brent.
Jonathan M. Borwein (1951-2016) was a prolific mathematician whose career spanned several countries (UK, Canada, USA, Australia) and whose many interests included analysis, optimisation, number theory, special functions, experimental mathematics, mathematical finance, mathematical education, and visualisation. We describe his life and legacy, and give an annotated bibliography of some of his most significant books and papers.
We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the author's own published papers.
We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\lnΓ(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks.
We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $π$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $π$, and more generally the elementary functions. These algorithms run in almost linear time $O(M(n)\log n)$, where $M(n)$ is the time for $n$-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for $π$, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for $π$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $π$, in the sense that they produce exactly the same sequence of approximations to $π$ if performed using exact arithmetic.
Consider polynomials over ${\rm GF}(2)$. We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree $r$ for all Mersenne exponents $r = \pm 3 \bmod 8$ in the range $5 < r < 10^7$, although there is no irreducible trinomial of degree $r$. We also give trinomials with a primitive factor of degree $r = 2^k$ for $3 \le k \le 12$. These trinomials enable efficient representations of the finite field ${\rm GF}(2^r)$. We show how trinomials with large primitive factors can be used efficiently in applications where primitive trinomials would normally be used.
A recent paper by Agélas [Generalized Riemann Hypothesis, 2019, hal-00747680v3] claims to prove the Generalized Riemann Hypothesis (GRH) and, as a special case, the Riemann Hypothesis (RH). We show that the proof given by Agélas contains an error. In particular, Lemma 2.3 of Agélas is false. This Lemma 2.3 is a generalisation of Theorem 1 of Vassilev-Missana [A note on prime zeta function and Riemann zeta function, Notes on Number Theory and Discrete Mathematics, 22, 4 (2016), 12-15]. We show by several independent methods that Theorem 1 of Vassilev-Missana is false. We also show that Theorem 2 of Vassilev-Missana is false. This note has two aims. The first aim is to alert other researchers to these errors so they do not rely on faulty results in their own work. The second aim is pedagogical - we hope to show how these errors could have been detected earlier, which may suggest how similar errors can be avoided, or at least detected at an early stage.
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision required increases as the computation proceeds. We give upper and lower bounds on the number of single-precision operations required to perform various multiple-precision operations, and deduce some interesting consequences concerning the relative efficiencies of methods for solving nonlinear equations using variable-length multiple-precision arithmetic. A postscript describes more recent developments.
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 53$. Our computations extend earlier results and disprove two plausible conjectures.
We consider sums of the form $\sum ϕ(γ)$, where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
We consider the sum $\sum 1/γ$, where $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth approximation $\frac{1}{4π} \log^2(T/2π),$ the sum tends to a limit $H \approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.
We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (ψ(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$.
Let $f_0(z) = \exp(z/(1-z))$, $f_1(z) = \exp(1/(1-z))E_1(1/(1-z))$, where $E_1(x) = \int_x^\infty e^{-t}t^{-1}{\,d}t$. Let $a_n = [z^n]f_0(z)$ and $b_n = [z^n]f_1(z)$ be the corresponding Maclaurin series coefficients. We show that $a_n$ and $b_n$ may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences $(a_n)$ and $(b_n)$ as $n \to \infty$, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding $(b_n)$. Let $ρ_n = a_n b_n$, so $\sum ρ_n z^n = (f_0\,\odot f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}ρ_n \sim -\sum d_k n^{-k}$ as $n \to \infty$, where the $d_k\in\mathbb Q$, $d_0=1$. We conjecture that $2^{6k}d_k \in \mathbb Z$. This has been verified for $k \le 1000$.
Let $D(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and ${\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \le n$. A relatively simple bound is \[{\mathcal R}(n) \ge \left(\frac{2}{πe}\right)^{d/2} \left(1 - d^2\left(\fracπ{2h}\right)^{1/2}\right) \;\text{ for all }\; n \ge 1.\] An asymptotically sharper bound is \[{\mathcal R}(n) \ge \left(\frac{2}{πe}\right)^{d/2} \exp\left(d\left(\fracπ{2h}\right)^{1/2} + \; O\left(\frac{d^{5/3}}{h^{2/3}}\right)\right).\] We also show that \[{\mathcal R}(n) \ge \left(\frac{2}{πe}\right)^{d/2}\] if $n \ge n_0$ and $n_0$ is sufficiently large, the threshold $n_0$ being independent of $d$, or for all $n\ge 1$ if $0 \le d \le 3$ (which would follow from the Hadamard conjecture). The proofs depend on the probabilistic method, and generalise previous results that were restricted to the cases $d=0$ and $d=1$.
Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\mathcal D}(n)$ and $\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \le n$. For example, $\mathcal R(n) > (πe/2)^{-d/2}$ if $1 \le d \le 3$, and $\mathcal R(n) > (πe/2)^{-d/2}(1 - d^2(π/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \to 0$ as $n \to \infty$, so the second bound is close to $(πe/2)^{-d/2}$ for large $n$. Previous lower bounds tended to zero as $n \to \infty$ with $d$ fixed, except in the cases $d \in \{0,1\}$. For $d \ge 2$, our bounds are better for all sufficiently large $n$. If the Hadamard conjecture is true, then $d \le 3$, so the first bound above shows that $\mathcal R(n)$ is bounded below by a positive constant $(πe/2)^{-3/2} > 0.1133$.
We give bounds on the error in the asymptotic approximation of the log-Gamma function $\lnΓ(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We show that $|R_{k+1}(z)/T_k(z)| < \sqrt{πk}$ for nonzero $z$ in the right half-plane, where $T_k(z)$ is the $k$-th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. If $k \le |z|$, then the stronger bound $|R_{k+1}(z)/T_k(z)| < (k/|z|)^2/(π^2-1) < 0.113$ holds. Similarly for the asymptotic approximation of $\lnΓ(z+\frac{1}{2})$, except that a factor $η_k = 1/(1-2^{1-2k})$ multiplies some of the bounds. We deduce similar bounds for asymptotic approximation of the Riemann-Siegel theta function $\vartheta(t)$. We show that the accuracy of a well-known approximation to $\vartheta(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp(-πt))$ to $O(\exp(-2πt))$. We discuss a similar example due to Olver (1964), and a connection with the Stokes phenomenon.
We exhibit twelve new primitive trinomials over GF(2) of record degrees $42643801$, $43112609$, and $74207281$. In addition we report the first Mersenne exponent not ruled out by Swan's theorem - namely $57885161$ - for which no primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents.
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form \[S_{α,β}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} |k^α-\ell^α|^β,\] obtaining new results in the cases $α= 1, 2$. We show that there is a close connection between these double sums in the case $α=1$ and the single centered binomial sums considered by Tuenter.
We consider discretisations of the Macdonald--Mehta integrals from the theory of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$, $\mathrm{B}_r$ and $\mathrm{D}_r$, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents $1$ and $2$. Our proofs for the exponent-$1$ cases rely on identities for classical group characters, while most of the formulas for the exponent-$2$ cases are derived from a transformation formula for elliptic hypergeometric series for the root system $\mathrm{BC}_r$. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.