Identities for the Ramanujan zeta function
نویسندگان
چکیده
منابع مشابه
Identities for the Ramanujan Zeta Function
We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(∆, k) is a period in the sense of Kontsevich and Zagier when k ≥ 12. As an illustration, we reduce L(∆, k) to explicit integrals of hypergeometric and algebraic functions when k ∈ {12, 13, 14, 15}.
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In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by αk(s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the...
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Several identities for the Riemann zeta-function ζ(s) are proved. For example, if φ1(x) := {x} = x− [x], φn(x) := ∫ ∞ 0 {u}φn−1 ( x u ) du u (n ≥ 2), then ζn(s) (−s) = ∫ ∞ 0 φn(x)x −1−s dx (s = σ + it, 0 < σ < 1) and 1 2π ∫ ∞ −∞ |ζ(σ + it)| (σ + t) dt = ∫ ∞ 0 φ n (x)x dx (0 < σ < 1). Let as usual ζ(s) = ∑ ∞ n=1 n −s (Re s > 1) denote the Riemann zeta-function. This note is the continuation of t...
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2013
ISSN: 0196-8858
DOI: 10.1016/j.aam.2013.04.001