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### Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac{1}{2}$.

Annals of Mathematics. Second Series

### Maximal sets of integers with distinct divisors.

The Electronic Journal of Combinatorics [electronic only]

### The second moment of quadratic twists of modular L-functions

Journal of the European Mathematical Society

We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.

### 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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