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Zeros of partial sums of the Riemann zeta function. (English) Zbl 1219.11126

Summary: 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 \(\mathbb C\).

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11Y35 Analytic computations