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Register a new userOn the infinite Borwein product raised to a positive real power
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In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $prod_{nge 1}[(1-q^n)/(1-q^{pn})]^delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.
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We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_infty^m(q^t;q^t)_infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)leq 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.
We consider some of Jonathan and Peter Borweins contributions to the high-precision computation of $pi$ 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 $pi$, 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 $pi$, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for $pi$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $pi$, in the sense that they produce exactly the same sequence of approximations to $pi$ if performed using exact arithmetic.
The Choix de Bruxelles operation replaces a positive integer n by any of the numbers that can be obtained by halving or doubling a substring of the decimal representation of n. For example, 16 can become any of 16, 26, 13, 112, 8, or 32. We investigate the properties of this interesting operation and its iterates.
Let $pi$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $pi$ twisted by a character. The bulk of this article focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature.
We prove an isomorphism between the finite domain from 1 up to the product of the first n primes and the new defined set of prime modular numbers. This definition provides some insights about relative prime numbers. We provide an inverse function from the prime modular numbers into this finite domain. With this function we can calculate all numbers from 1 up to the product of the first n primes that are not divisible by the first n primes. This function provides a non sequential way for the calculation of prime numbers.
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