Borwein conjecture
WebAbstract. We prove part of a conjecture of Borwein and Choi concerning an es-timate on the square of the number of solutions to n = x2 + Ny2 for a squarefree integer N. 1. Introduction We consider the positive deflnite quadratic form Q(x;y) = x2 + Ny2 for a squarefree integer N. Let r2;N(n) denote the number of solutions to n = Q(x;y ... WebJan 28, 2024 · An asymptotic approach to Borwein-type sign pattern theorems Chen Wang, Christian Krattenthaler The celebrated (First) Borwein Conjecture predicts that for all positive integers~ the sign pattern of the coefficients of the ``Borwein polynomial'' is . It was proved by the first author in [Adv. Math. 394 (2024), Paper No. 108028].
Borwein conjecture
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WebFeb 12, 2024 · We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.... WebFeb 18, 2024 · Recently, Li (Int J Number Theory, 2024) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci China Math, 2010). In fact, Li …
WebNov 1, 1995 · We consider several conjectures by Peter Borwein concerning products of the form ∏ n=1 (1 -q n / ∏ N n=1 (1 - q pn ), p=3,5. WebAbout this book. The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely ...
WebSep 1, 2024 · A conjecture of Borwein asserts that for any positive integers n and k, the coefficient a3k of q3k in the expansion of ∏j=0n(1−q3j+1)(1−q3j+2) is nonnegative. WebFeb 5, 2024 · We note that the conditions for r and k in Conjecture 1.6 are more restrictive than those in Definition 1.3, Definition 1.4. We remark that Andrews [2] proved the identity (1.12) for η = λ + 1 using the q-difference method. Recently in [9], we proved Conjecture 1.6 for λ = 2 and k ≥ r ≥ 1 by finding a combinatorial proof of (1.11) in ...
Webconjecture can be written by three sequences of self-inversive polynomials. At last, our numerical evaluating persuade us that a few auxiliary conjectures are true We think that …
The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics. blue bean coffee coWebIn number theory the Agoh–Giuga conjecture on the Bernoulli numbers B k postulates that p is a prime number if and only if ... it has at least 13,800 digits (Borwein, Borwein, … free harvest festival clip artWebBorwein J.M., Skerritt M. and Maitland C. "Computation of a lower bound to Giuga's primality conjecture." Integers 13 (2013). Giuga G. "Su una presumibile proprietà caratteristica dei numeri primi", Ist. free harvest apron patternWebBorwein 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. Keywords Infinite Borwein product ·Sign pattern · Asymptotics · Positivity ·Circle method · Vanishing of coefficients free harvard vacationsWebFeb 18, 2024 · In 1990, Peter Borwein (see ) empirically discovered quite a number of mysteries involving sign patterns of coefficients of certain polynomials. The most easily … blue beanie boo cat slippersWebNov 20, 2024 · A Note on Giuga's Conjecture - Volume 50 Issue 1. Due to planned system work, ecommerce on Cambridge Core will be unavailable on 12 March 2024 from 08:00 – 18:00 GMT. ... Borwein, J. M., Borwein, P. B., and Girgensohn, R., Giuga's conjecture on primality. Amer.Math. blue beanie baby clubbyWebMay 23, 2013 · But my question here is not about if this conjecture true or false. ... For example, Bailey, Borwein, and Borwein pointed out the example $$ \left(\frac{1}{10^5} \sum_{n=-\infty}^\infty e^{-n^2/10^{10}}\right)^2 \approx \pi, $$ which holds to over 42 billion digits. If you know something about Poisson summation, it's pretty obvious that this is ... blue beanie baggy winter