An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions.

Masanori Katsurada

Displaying similar documents to “An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions.”

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It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in $1$ and $ζ (4) = π^{4} / 90$ yielding a conditional upper bound for the irrationality measure of $ζ (4)$ ; (2) a second-order Apéry-like recursion for $ζ (4)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group...

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For a fixed integer $k \geq 3$ , and fixed $\frac{1}{2} < σ < 1$ we consider $\int_{1}^{T} {|ζ (σ + i t)|}^{2 k} d t = \sum_{n = 1}^{\infty} d_{k}^{2} (n) n^{- 2 σ} T + R (k, σ; T),$ where $R (k, σ; T) = 0 (T) (T \to \infty)$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R (k, σ; T)$ are derived in the range min $(β_{k}, σ_{k}^{*}) < σ < 1$ . We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

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Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include...