Displaying similar documents to “Zeros of Hecke L-functions associated with cusp forms”

On some problems involving Hardy’s function

Aleksandar Ivić (2010)

Open Mathematics

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Some problems involving the classical Hardy function Z t = ζ 1 2 + i t χ 1 2 + i t - 1 1 2 2 , ζ s = χ s ζ 1 - s , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

On the riemann zeta-function and the divisor problem

Aleksandar Ivić (2004)

Open Mathematics

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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ς 1 2 + i t . If E * t = E t - 2 π Δ * t / 2 π with Δ * x = - Δ x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain ∫ 0 T E * t 4 d t ≪ e T 16 / 9 + ε . We also show how our method of proof yields the bound ∑ r = 1 R ∫ t r - G t r + G ς 1 2 + i t 2 d t 4 ≪ e T 2 + e G - 2 + R G 4 T ε , where T 1/5+ε≤G≪T, T

Nonlinear exponential twists of the Liouville function

Qingfeng Sun (2011)

Open Mathematics

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Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum ∑ X ⩽ n ⩽ 2 X λ ( n ) e 2 π i α n , 0 ≠ α ∈ ℝ The main tool we use is Vaughan’s identity for λ(n).

An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

Hideaki Ishikawa, Kohji Matsumoto (2011)

Open Mathematics

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We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.

An Alternative Form of the Functional Equation for Riemann’s Zeta Function, II

Andrea Ossicini (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for ζ ( s ) . We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau...