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Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL(2) as saying that the local components of automorphic representations lie in the principal series, and suggested this condition as a generalization of the Ramanujan–Petersson conjecture to automorphic forms on other groups. Another ...
Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.
See Winnie Li's survey on Ramanujan's conjecture and other aspects of number theory relevant to these results. [ 5 ] Lubotzky , Phillips and Sarnak [ 2 ] and independently Margulis [ 6 ] showed how to construct an infinite family of ( p + 1 ) {\displaystyle (p+1)} -regular Ramanujan graphs, whenever p {\displaystyle p} is a prime number and p ...
In mathematics, Ramanujan's congruences are the congruences for the partition function p(n) discovered by Srinivasa Ramanujan: (+) (), (+) (), (+) ().In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence
Download as PDF; Printable version; In other projects ... (1917) and the third one, called the Ramanujan conjecture, ... Ramanujan's L-function is defined by = ...
In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x 2 + y 2 + 10z 2 with integral values for x, y and z. [ 1 ] [ 2 ] Srinivasa Ramanujan considered this expression in a footnote in a paper [ 3 ] published in 1916 and briefly discussed the representability of integers in this form.
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Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f , the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)} is defined as