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### Triple correlation of the Riemann zeros

Journal de Théorie des Nombres de Bordeaux

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function  to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating  taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here,...

### Zeros of Fekete polynomials

Annales de l'institut Fourier

For $p$ an odd prime, we show that the Fekete polynomial ${f}_{p}\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right){t}^{a}$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813>{\kappa }_{0}>0.500668$. Here ${\kappa }_{0}-1/2$ is the probability that the function $1/x+1/\left(1-x\right)+{\sum }_{n\in ℤ:\phantom{\rule{4pt}{0ex}}n\ne 0,1}{\delta }_{n}/\left(x-n\right)$ has a zero in $\right]0,1\left[$, where each ${\delta }_{n}$ is $±1$ with y $1/2$. In fact ${f}_{p}\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|{f}_{p}\left(e\left(2i\pi \left(K+\tau \right)/p\right)\right)|<ϵ\sqrt{p}$ for some $\tau \in \right]0,1\left[$ then there is a zero of $f$ close to this arc.

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