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Mathew Rogers

Publications and source records attributed to Mathew Rogers.

10 recordsLinked to original sources
arXiv2013

Secant Zeta Functions

We study the series $ψ_s(z):=\sum_{n=1}^{\infty} \sec(nπz)n^{-s}$, and prove that it converges under mild restrictions on $z$ and $s$. The function possesses a modular transformation property, which allows us to evaluate $ψ_{s}(z)$ explicitly at certain quadratic irrational values of $z$. This supports our conjecture that $π^{-k} ψ_{k}(\sqrt{j})\in\mathbb{Q}$ whenever $k$ and $j$ are positive integers with $k$ even. We conclude with some speculations on Bernoulli numbers.

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arXiv2013

Mahler measure and the WZ algorithm

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lalín, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper "Periods".

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arXiv2013

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that $L(Δ, k)$ is a period in the sense of Kontsevich and Zagier when $k\ge12$. As an illustration, we reduce $L(Δ, k)$ to explicit integrals of hypergeometric and algebraic functions when $k\in\{12,13,14,15\}$.

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arXiv2012

A solution of Sun's $520 challenge concerning 520/pi

We prove a Ramanujan-type formula for $520/π$ conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for $1/π$.

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arXiv2012

Ramanujan series upside-down

We prove that there is a correspondence between Ramanujan-type formulas for 1/\pi, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

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arXiv2011

Modular equations and lattice sums

We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.

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arXiv2011

Variations of the Ramanujan polynomials and remarks on $ζ(2j+1)/π^{2j+1}$

We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang. Our proofs rely upon theorems of Schinzel, and Lakatos and Losonczi and some generalizations.

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arXiv2011

On the Mahler measure of $1+X+1/X+Y+1/Y$

We prove a conjectured formula relating the Mahler measure of the Laurent polynomial $1+X+X^{-1}+Y+Y^{-1}$, to the $L$-series of a conductor 15 elliptic curve.

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arXiv2010

From $L$-series of elliptic curves to Mahler measures

We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family $(1+X)(1+Y)(X+Y)-\alpha XY$, $\alpha\in\mathbb R$.

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