Results 21 to 30 of about 891 (145)
New representations of Ramanujan’s tau function [PDF]
Proceedings of the American Mathematical Society, 1999 The author derives interesting new formulas expressing Ramanujan's tau function \(\tau(n)\) as finite sums involving sums of cubes of divisors and the number of representations as a sum of 8 squares. The proofs use special cases of Jacobi's classic triple-product identity together with more recent infinite product identities obtained by the author ...
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Rocky Mountain Journal of Mathematics, 1998 Ramanujan's function \(\tau(n)\) is defined by \[ \sum_1^\infty \tau(n) x^n= x\prod_1^\infty (1-x^n)^{24}, \qquad |x|
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Multiplicative functions and Ramanujan's τ-function [PDF]
Journal of the Australian Mathematical Society, 1981 AbstractIt is proved that (|τ(n)|n−11/2)δ has a mean-value for 0 <δ > < 2, where τ(n) is Ramanujan's function from modular arithmetic. Some further results are conjectured.
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Note on the Distribution of Ramanujan's Tau Function [PDF]
Mathematics of Computation, 1970 According to a conjecture of Sato and Tate, the angle θ \theta whose cosine is 1 2 τ ( p ) p − 11 / 2
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Character sum, reciprocity, and Voronoi formula
Bulletin of the London Mathematical Society, Volume 57, Issue 12, Page 3797-3823, December 2025.Abstract We prove a novel four‐variable character sum identity that serves as a twisted, non‐Archimedean analog of Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we provide a short spectral proof of the Voronoi formulae for classical modular forms with character twists.
Chung‐Hang Kwan, Wing Hong Leung
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Angewandte Chemie, Volume 137, Issue 5, January 27, 2025.Crystals of a BOIMPY dianion made the exploration of a catalytic process that relies on consecutive multi‐photoinduced electron transfer (multi‐PET) possible. Irradiation of the dye molecule leads to an excited species that allows the reduction of strong carbon‐halide bonds and arenes.
Amit Biswas +12 more
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A formula for Ramanujan's $\tau$-function
Illinois Journal of Mathematics, 1975 Let \(\sigma_k(n) = \sum_{d\mid n} d^k\), \(\sigma(n) = \sigma_1(n)\) and set \(\exp(2\pi iz) = e(z) =x\). Ramanujan's function \(\tau(n)\) is defined as the coefficient of \(x\) in the power series expansion of \[ \Delta(z) =x \prod_{n=1}^\infty (1 -x^n)^{24} = \sum_{n=1}^\infty \tau(n)x^n.
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Correlations of the squares of the Riemann zeta function on the critical line
Journal of the London Mathematical Society, Volume 112, Issue 3, September 2025.Abstract We compute the average of a product of two shifted squares of the Riemann zeta function on the critical line with shifts up to size T3/2−ε$T^{3/2-\varepsilon }$. We give an explicit expression for such an average and derive an approximate spectral expansion for the error term similar to Motohashi's.
Valeriya Kovaleva
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Triple sums of Kloosterman sums and the discrepancy of modular inverses
Journal of the London Mathematical Society, Volume 112, Issue 3, September 2025.Abstract We investigate the distribution of modular inverses modulo positive integers c$c$ in a large interval. We provide upper and lower bounds for their box, ball, and isotropic discrepancy, thereby exhibiting some deviations from random point sets. The analysis is based, among other things, on a new bound for a triple sum of Kloosterman sums.
Valentin Blomer +2 more
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Ramanujan’s Tau-Function and Convolution Sums
European Journal of Theoretical and Applied SciencesWe study certain type of convolution sums involving an arbitrary arithmetic function f, which it is applied to Ramanujan’s tau function when f coincides with the sum of divisors function.
R. Sivaraman +2 more
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