The Circle and Divisor Problem
نویسندگان
چکیده
منابع مشابه
The Circle and Divisor Problems, and Ramanujan’s Contributions through Bessel Function Series
The circle problem and Dirichlet divisor problem (abbreviated by divisor problem in the sequel) are two of the most difficult, long outstanding problems in analytic number theory. Roughly, the first half of this paper will be devoted to a discussion of the circle problem, an identity of Ramanujan involving a double series of Bessel functions found on page 335 of his lost notebook [35] that he c...
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One of the most famous problems of classical geometry is the Apollonius' problem asks construction of a circle which is tangent to three given objects. These objects are usually taken as points, lines, and circles. This well known problem was posed by Apollonius of Perga ( about 262 - 190 B.C.) who was a Greek mathematician known as the great geometer of ancient times after Euclid and Archimede...
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Ramanujan’s lost notebook, along with various fragments and partial manuscripts, was published in 1988. One fragment (page 335) contains two formulas involving finite trigonometric sums and doubly infinite series of Bessel functions, which are connected with the classical circle and divisor problems. For the first time, we provide a proof of the first identity in the formulation given by Ramanu...
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For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...
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Let r2(n) denote the number of representations of n as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ∑ n≤x r2(n) is known as the circle problem. Next, let d(n) denote the number of positive divisors of n. Determining the exact order of magnitude of the error term associated with the asymptotic formula for ∑ n≤x d(n) is the divisor ...
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