Zeros of the dilogarithm

O’Sullivan, Cormac

doi:10.1090/mcom/3065

Cited by 6 publications

(1 citation statement)

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“…For zeros of zeta functions in the half-plane σ < 0, we have the following research. Peyerimhoff [11] proved that for (fixed) σ < 0, the function Li σ (z) has −⌊σ⌋ simple zeros for z ∈ C \ [1, ∞) and they are all non-positive (see also [10,Section 8]). Spira [15] showed that if σ ≤ −4a−1−2[1−2a] and |t| ≤ 1, then ζ(s, a) = 0 except for zeros on the negative real line, one in each interval (−2n − 4a − 1, −2n − 4a + 1), N ∋ n ≥ 1 − 2a.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

Nakamura

2016

Journal of Mathematical Analysis and Applications

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Abstract. For 0 < a ≤ 1, s, z ∈ C and 0 < |z| ≤ 1, the Hurwitz-Lerch zeta function is defined by Φ(s, a, z) := ∞ n=0 z n (n + a) −s when σ := ℜ(s) > 1. In this paper, we show that Φ(σ, a, z) = 0 when σ ∈ (−1, 0) if and only if [I] z = 1 and (z ∈ R and 0 < a ≤ 1. In addition, we give a new proof of the functional equation of Φ(s, a, z).1. Introduction and statement of main results 1.1. Main results. The Hurwitz-Lerch zeta function is defined by as follows.We can easily see that the Riemann zeta function ζ(s) and the Hurwitz zeta function ζ(s, a) are expressed as Φ(s, 1, 1) and Φ(s, a, 1), respectively. The Dirichlet series of Φ(s, a, z) converges absolutely in the half-plane σ > 1 and uniformly in each compact subset of this half-plane. When z = 1, the function Φ(s, a, z) is analytically continuable to the whole complex plane. However, the Hurwitz zeta function ζ(s, a) is a meromorphic function with a simple pole at s = 1. In this paper, we show the following. Note that b (−1, 0). In Section 3, we give a new proof of the functional equations (3.1) and (3.3) by using the integral representations (2.4) and (2.9), and modifying the fifth method of the proof of the functional equation for the Riemann zeta function (see [16, Section 2.8]). It should be noted that in the 21st century, Knopp and Robins [7], and Navas, Ruiz and Varona [9] gave new proofs of the functional equation for ζ(s, a) by using Poisson summation of the Lipschitz summation formula (see [7, p. 1916]) and the uniqueness of Fourier coefficients (see [9, p. 190]), respectively.2010 Mathematics Subject Classification. Primary 11M35.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

Nakamura

2016

Journal of Mathematical Analysis and Applications

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Partitions and Sylvester waves

O’Sullivan

2017

Ramanujan J

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The restricted partition function p N (n) counts the partitions of the integer n into at most N parts. In the nineteenth century Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to p N (n) in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm. * 2010 Mathematics Subject Classification. 11P82, 41A60

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