We study the convergence of this series (a preliminary step, to study Ramanujan expansions) and define Ga Ramanujan coefficient when R G (a) converges pointwise, in all natural a.
n2N be a family of complex sequences indexed by t 2 T such that for all t 2 T the series P n 0 p n.t/is convergent; then we set XT n 0 a n D lim t!l PC1 nD0 p n.t/s n PC1 D0 p n.t/ when this limit is finite, and in this case, we say that the series P n 0 a n is T-summable. A theorem of Toeplitz (Hardy 1949) gives necessary and sufficient ...
A smooth summation of Ramanujan expansions Giovanni Coppola Abstract. We studied Ramanujan series P∞ q=1 G(q)cq(a), where cq(a) is the well-known Ramanujan sum and the complex numbers G(q), as q ∈ N, are the Ramanujan coefficients; of course, we mean, implicitly, that the series converges pointwise, in all natural a, as its partial sums P
Convergence of Ramanujan expansions, I [Multiplicativity on Ramanujan ...
2020年9月29日 · Abstract: We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define $G$ a 'Ramanujan coefficient' when $R_G(a)$ converges pointwise, in all natural $a$.
Ramanujan Summation of Divergent Series | SpringerLink
This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent …
Ramanujan Summation - SpringerLink
2017年9月13日 · We call this procedure the Ramanujan summation of the series ∑ n ≥ 1 f(n) and \(\sum _{n\geq 1}^{\mathcal{R}}f(n)\) the Ramanujan constant of the series. Some properties are immediate consequences of this definition: Linearity. If a and b are complex numbers and f and g are in \(\mathcal{O}^{\pi }\), then we verify immediately that
Ramanujan starts with the Euler–Maclaurin formula (1) a(1)+a(2)+···+a(x−1) = C+ Z x 1 a(t)dt+ X k≥1 b k k! ∂k−1a(x), and remarks that the constant Centertains a mysterious relationship with the series—it is like its “center of gravity”—so that Ramanujan proposes to …
Ramanujan summation - Wikipedia
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series , for which ...
On Ramanujan sums of a real variable and a new Ramanujan …
2021年5月6日 · We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function.
[PDF] Convergence of Ramanujan expansions, I [Multiplicativity …
2020年9月29日 · We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define $G$ a 'Ramanujan coefficient' when $R_G(a)$ converges pointwise, in all natural $a$.