All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 26 Next

Displaying 501 – 520 of 1782

Showing per page

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ 0 ( N ) with normalized eigenvalues λ f ( n ) and let X > 1 be a real number. We consider the sum 𝒮 k ( X ) : = ∑ n < X ∑ n = n 1 , n 2 , ... , n k λ f ( n 1 ) λ f ( n 2 ) ... λ f ( n k ) and show that 𝒮 k ( X ) ≪ f , ϵ X 1 - 3 / ( 2 ( k + 3 ) ) + ϵ for every k ≥ 1 and ϵ > 0 . The same problem was considered for the case N = 1 , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ≥ 5 . Since the result is valid for arbitrary...

Generalizing a theorem of Schur

Lenny Jones (2014)

Czechoslovak Mathematical Journal

In a letter written to Landau in 1935, Schur stated that for any integer t > 2 , there are primes p 1 < p 2 < ⋯ < p t such that p 1 + p 2 > p t . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers t ≥ k ≥ 1 and real ϵ > 0 , there exist primes p 1 < p 2 < ⋯ < p t such that p 1 + p 2 + ⋯ + p k > ( k - ϵ ) p t .

Currently displaying 501 – 520 of 1782