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On zeros of functions satisfying certain differential-difference equations. (English) Zbl 0619.10043

The largest real zero \(\rho _{\kappa}\) of the function \[ q_{\kappa}(s)=(\Gamma (2k)/2\pi i)\int z^{-2\kappa}\exp (zs+\kappa \int ^{z}_{0}\frac{1-e^ u}{u}du)\quad dz,\quad s>0,\quad \kappa \geq 1 \] plays a fundamental role in Rosser’s sieve of dimension \(\kappa\) [H. Iwaniec, Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)] and in improvements of Selberg’s sieve of dimension \(\kappa >1\) [H. Iwaniec, J. van de Lune and H. J. J. te Riele, Indagationes Math. 42, 409-417 (1980; Zbl 0445.10035); H. Diamond, H. Halberstam and H.-E. Richert, Combinatorial sieves of dimension exceeding one. I (to appear)].
Sharp (upper and lower) bounds for \(\rho _{\kappa}\) are given by the largest real zero of certain polynomials. The construction of these polynomials is based on the Laplace representation of \(q_{\kappa}(s)\) and the differential-difference equation \(s q_{\kappa}(s))'=\kappa q_{\kappa}(s)+\kappa q_{\kappa}(s+1).\) A generalization to functions satisfying similar differential-difference equations is given.

MSC:

11N35 Sieves
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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