Zeroes of partial sums of the zeta-function


Timothy Trudgian


Australian National University


Wed, 18/11/2015 - 2:30pm


K-J17-101 (Ainsworth Building 101) UNSW


Take the Riemann zeta-function, $\zeta(s) = 1 + 2^{-s} + 3^{-s} + \ldots$, which converges whenever $\Re(s)>1$. Now chop the series after $N$ terms and call the finite piece $\zeta_{N}(s)$. In 1948 Turan made the striking observation that if for sufficiently large $N$ the functions $\zeta_{N}(s)$ did not vanish for $\Re(s)>1$, then the Riemann hypothesis would be true. In 1983 Montgomery showed that this is false: for all sufficiently large $N$ there are such zeroes. I shall discuss this problem and recent work with Dave Platt (at Bristol), that finally resolves the values of $N$ for which zeroes do, and do not, exist.

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