Calculation of the Ramanujan $\tau $-Dirichlet series
نویسندگان
چکیده
منابع مشابه
On the zeros of the Ramanujan tau-Dirichlet series in the critical strip
We describe computations which show that each of the first 12069 zeros of the Ramanujan τ -Dirichlet series of the form σ + it in the region 0 < t < 6397 is simple and lies on the line σ = 6. The failures of Gram’s law in this region are also noted. The first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated. The distribution of the normalized spacing of t...
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In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247-249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled "On certain arithmetical functions" (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ram...
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We describe computations which show that each of the first 12069 zeros of the Ramanujan τ -Dirichlet series of the form σ + it in the region 0 < t < 6397 is simple and lies on the line σ = 6. The failures of Gram’s law in this region are also noted. The first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated. The distribution of the normalized spacing of t...
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We prove that there is a correspondence between Ramanujan-type formulas for 1/π and formulas for Dirichlet L-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.
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Using the theory of Calabi–Yau differential equations we obtain all the parameters of Ramanujan–Sato-like series for 1/π2 as q-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new examples of series of non-hypergeometric type. To motivate our theory we begin with the simpler case of Ramanujan–Sato series for 1/π .
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1973
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1973-0326995-4