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In mathematics, a character sum is a sum () of values of a Dirichlet character χ modulo N, taken over a given range of values of n.Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N.
Srinivasa Ramanujan mentioned the sums in a 1918 paper. [1] In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes. [2]
Ramanujan mentioned four order-3 mock theta functions in his letter to Hardy, and listed a further three in his lost notebook, which were rediscovered by G. N. Watson. [5] The latter proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appell–Lerch ...
For k ∈ and n ∈ >0, the Divisor function σ k (n) is the sum of the k th powers of the divisors of n. The tau function satisfies several congruence relations; many of them can be expressed in terms of σ k ( n ) .
BOSTON (Reuters) -McKinsey & Co has agreed to pay $650 million to resolve a U.S. Department of Justice investigation into the consulting firm's work advising opioid manufacturer OxyContin maker ...
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 ...
The death of her mother when she was just 5 years old hit Madonna hard.. In archival clips and audio featured in the new Sky documentary Becoming Madonna, the Queen of Pop, 66, opens up about how ...
Agents on patrol discovered two backpacks stuffed with more than $1.1 million worth of cocaine in Washington state near the border with Canada, U.S. Customs and Border Protection said Monday.