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2007 Zeros of Partial Summs of the Riemann Zeta Function
Peter Borwein, Greg Fee, Ron Ferguson, Alexa van der Waal
Experiment. Math. 16(1): 21-40 (2007).


The semiperiodic behavior of the zeta function $\zeta(s)$ and its partial sums $\zeta_N(s)$ as a function of the imaginary coordinate has been long established. In fact, the zeros of a $\zeta_N(s)$, when reduced into imaginary periods derived from primes less than or equal to $N$, establish regular patterns. We show that these zeros can be embedded as a dense set in the period of a surface in $\mathbb{R}^{k+1}$, where $k$ is the number of primes in the expansion. This enables us, for example, to establish the lower bound for the real parts of zeros of $\zeta_N(s)$ for prime $N$ and justifies the use of methods of calculus to find expressions for the bounding curves for sets of reduced zeros in $\mathbb{C}$.


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Peter Borwein. Greg Fee. Ron Ferguson. Alexa van der Waal. "Zeros of Partial Summs of the Riemann Zeta Function." Experiment. Math. 16 (1) 21 - 40, 2007.


Published: 2007
First available in Project Euclid: 5 April 2007

zbMATH: 1219.11126
MathSciNet: MR2312975

Primary: 11M26
Secondary: 11M41 , 11Y35 , 40A25

Keywords: Dirichlet series , series convergence , zeros of exponential sums , zeta function

Rights: Copyright © 2007 A K Peters, Ltd.


Vol.16 • No. 1 • 2007
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