Linking the Circle and the Sieve: Ramanujan- Fourier Series
نویسنده
چکیده
Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be Ramanujan Fourier series.
منابع مشابه
Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series
In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering. Norbert Wiener forged a powerful tool for electrical engineers by combining two distinct branches of...
متن کاملConvolution and Cross-correlation of Ramanujan-fourier Series
where x̄ is the complex conjugate of x This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanu...
متن کاملThe Partition Function Revisited
In 1918, Hardy and Ramanujan wrote their landmark paper deriving the asymptotic formula for the partition function. The paper however was fundamental for another reason, namely for introducing the circle method in questions of additive number theory. Though this method is powerful, it is often difficult and technically complicated to employ. In 2011, Bruinier and Ono discovered a new algebraic ...
متن کاملRamanujan sums analysis of long-period sequences and 1/f noise
Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the Fourier transform....
متن کاملOFFPRINT Ramanujan sums analysis of long-period sequences and 1/f noise
Europhysics Letters (EPL) has a new online home at www.epljournal.org Take a look for the latest journal news and information on: • reading the latest articles, free! • receiving free e-mail alerts • submitting your work to EPL Abstract – Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to s...
متن کامل