Search ResultsShowing 1-10 of 10
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Secant Zeta Functions
Comments: 10 pagesCategories: math.NTWe study the series $\psi_s(z):=\sum_{n=1}^{\infty} \sec(n\pi z)n^{-s}$, and prove that it converges under mild restrictions on $z$ and $s$. The function possesses a modular transformation property, which allows us to evaluate $\psi_{s}(z)$ explicitly at certain quadratic irrational values of $z$. This supports our conjecture that $\pi^{-k} \psi_{k}(\sqrt{j})\in\mathbb{Q}$ whenever $k$ and $j$ are positive integers with $k$ even. We conclude with some speculations on Bernoulli numbers.
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arXiv:1210.2373 (Published 2012-10-08)
A solution of Sun's $520 challenge concerning 520/pi
Comments: 17 pagesCategories: math.NTKeywords: challenge concerning, appropriate modular parameters, proof begins, hypergeometric representation, relevant double seriesTags: journal articleWe prove a Ramanujan-type formula for $520/\pi$ 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 $1/\pi$.
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Identities for the Ramanujan zeta function
Comments: 10 pagesWe prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that $L(\Delta, k)$ is a period in the sense of Kontsevich and Zagier when $k\ge12$. As an illustration, we reduce $L(\Delta, k)$ to explicit integrals of hypergeometric and algebraic functions when $k\in\{12,13,14,15\}$.
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arXiv:1206.3981 (Published 2012-06-18)
Ramanujan series upside-down
Comments: 28 pagesCategories: math.NTKeywords: ramanujan series upside-down, epstein zeta function, dirichlet l-values, ramanujan-type formulas, rapidly converging hypergeometric functionsTags: journal articleWe 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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Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi
Comments: 12 pagesWe evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers--Ramanujan continued fraction play central roles in our evaluations
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arXiv:1106.1189 (Published 2011-06-06)
Variations of the Ramanujan polynomials and remarks on $ζ(2j+1)/π^{2j+1}$
Comments: 20 pagesCategories: math.NTWe 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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On the Mahler measure of $1+X+1/X+Y+1/Y$
Comments: 18 pages, Fixed typos in Lemmas 3 and 4Journal: Intern. Math. Research Notices 2014:9 (2014) 2305--2326DOI: 10.1093/imrn/rns285Categories: math.NTTags: journal articleWe 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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From $L$-series of elliptic curves to Mahler measures
Comments: 33 pagesJournal: Compositio Mathematica 148 (2012), no. 2, 385-414Keywords: mahler measure, cm elliptic curves, conductors, functional equation, conjectural relationsTags: journal articleWe 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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Mahler measure and the WZ algorithm
Comments: 15 pagesCategories: math.NTWe 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 Lal\'{i}n, 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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Modular equations and lattice sums
Comments: 12 pages, Rewritten and updatedCategories: math.NTWe 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.