arXiv subjects

Robert Baillie

Publications and source records attributed to Robert Baillie.

7 recordsLinked to original sources

arXiv2026

Fun With Fourier Series

By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\sin(n)/n$. One example is this classic series for $π/4$: \[ \fracπ{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = 1 \cdot \frac{\sin(1)}{1} - \frac{1}{3} \cdot \frac{\sin(3)}{3} + \frac{1}{5} \cdot \frac{\sin(5)}{5} - \frac{1}{7} \cdot \frac{\sin(7)}{7} + \dots . \] Another example is \[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n} = \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\right)^2 = \frac{π-1}{2}. \] This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.

math.CA↗

arXiv2024

Summing the curious series of Kempner and Irwin

In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved, among other things, that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428 70807 47848 31968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.00745 x 10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol. Finally, we discuss a class of related series whose summation algorithm has not yet been developed.

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arXiv2021

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.

math.NT↗

arXiv2019

Wright's Fourth Prime

Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor g_1 \rfloor = 3$, $\lfloor g_2 \rfloor = 13$, and $\lfloor g_3 \rfloor = 16381$. But with this $c$, $\lfloor g_4 \rfloor$ is a 4932-digit composite number. However, this slightly larger value of $c$, \[ c = 1.9287800 + 8.2843 \cdot 10^{-4933}, \] reproduces Wright's first three primes and generates a fourth: \[ \lfloor g_4 \rfloor = 191396642046311049840383730258 \text{ } \ldots \text{ } 303277517800273822015417418499 \] is a 4932-digit prime. Moreover, the sum of the reciprocals of the primes in Wright's sequence is transcendental.

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arXiv2015

Sinc integrals and tiny numbers

We apply a result of David and Jon Borwein to evaluate a sequence of highly-oscillatory integrals whose integrands are the products of a rapidly growing number of sinc functions. The value of each integral is given in the form $π(1-t)/2$, where the numbers $t$ quickly become very tiny. Using the Euler-Maclaurin summation formula, we calculate these numbers to high precision. For example, the integrand of the tenth integral in the sequence is the product of 68100152 sinc functions. The corresponding $t$ is approximately $9.6492736004286844634795531209398105309232 \cdot 10^{-554381308}$.

math.CA↗

arXiv2015

Fun With Very Large Numbers

We give an example of a formula involving the sinc function that holds for every N = 0, 1, 2, ..., up to about 10^102832732165, then fails for all larger N. We give another example that begins to fail after about N ~ exp(exp(exp(exp(exp(exp(e)))))). This number is larger than the Skewes numbers.

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arXiv2011

Experiments with zeta zeros and Perron's formula

Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perron's formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Euler's $ϕ$ function and other arithmetical functions. This is largely a presentation of experimental results.

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