28 problems

Let $\chi_d=\left(\frac{d}{\cdot}\right)$ denote the Kronecker symbol associated with a fundamental discriminant $d$. For $m>0$, define … where $\sideset{}{^*}\sum$ denotes a sum o…

For a fundamental discriminant $d$, let $F_d(z):=\sum_{n=1}^{|d|-1}\chi_d(n)z^n$ be the Fekete polynomial associated to $d$, where $\chi_d$ is the primitive quadratic character att…

Diaconu–Goldfeld–Hoffstein conjecture. As $D\to\infty$,

Let $d\in\mathcal{D}$ be a positive discriminant, let $\chi_d=\left(\frac{d}{\cdot}\right)$ be the associated Kronecker character, and let $L(s,\chi_d)$ be its Dirichlet $L$-functi…

Let $\chi_d$ range over the real quadratic Dirichlet characters appearing in the family considered in the paper. Quadratic-character maximum-value conjecture. … Random matrix theor…

Let $\chi_d$ be a real primitive Dirichlet character modulo $d$, let $D^*=\sum_{|d|\leq D}1$, and let $a_4$ be the arithmetic factor in the conjectural moment formula. Soundararaja…

Let $\mathcal{F}$ be the set of all fundamental discriminants. For $d\in\mathcal{F}$, let $\chi_d$ be the real primitive Dirichlet character modulo $|d|$, and let $L(s,\chi_d)$ be…

Classification conjecture. The only primes $p>2$ for which $\mathfrak{p}(n,\pm(\tfrac{\cdot}{p}))$ vanishes on some arithmetic progressions modulo $2p$ are $p=5$ and $p=17$.

Central-point moment conjecture. There is a constant $C(k)$ such that

Let $k$ be a positive integer, let $r$ be a positive fundamental discriminant, and let $\chi_r$ be the primitive quadratic character modulo $r$. Define … For $y>0$, define the vari…

For $0\leq\sigma\leq1$ and $T\geq1$, let $N(\sigma,T,\chi)$ denote the number of zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, counted with multiplicity, satisfying…

Let $\EuScript F_{\EuScript D}$ be the family of real primitive quadratic Dirichlet characters $\chi_d$, with $d$ nonzero and squarefree, and let $r\geq4$. For integers $N\geq1$ an…

The logarithmic-derivative conjecture. Then

Let $\varphi(x)$ be a Schwartz class function on $\mathbb{R}$ with Fourier transform having compact support. Define the density … and let $\rho=\frac12+i\gamma$, with…

Let $\mathcal L^+$ denote the set of positive integers associated with quadratic characters whose partial sums are positive, and let $\pi(x)$ denote the prime-counting function. De…

Function-field secondary-term conjecture.

Moment asymptotic conjecture. It is conjectured that

Sun's conjecture. This sum should vanish:

Let $p_1,p_2,\ldots$ be Chua's sequence, and let $n=p_1\cdots p_k$ be the product of its first $k$ terms. The next term $p_{k+1}$ is characterized by the existence of a divisor…

Let … where the sum is over real primitive Dirichlet characters and $D$ tends to infinity. Quadratic Dirichlet moment conjecture. The $k$th moment is conjectured to satisfy … This…

Let $D$ tend to infinity, and consider the first moment of quadratic Dirichlet $L$-functions over the relevant family of quadratic characters $\chi_d$ with $0<d\leq D$. Anderson–Ru…

Chowla's omega conjecture. Chowla's omega result is conjectured to reflect the true nature of the extreme values of $L(1,\chi_d)$, rather than merely provide a lower-bound phenomen…

Average first negative value conjecture. As $x\to\infty$, one expects

Let $F(d)$ be as in Theorem 1.1. Let $\beta$ and $\beta$ lie in the rectangle … Let $l=l_1l_2$ be odd, with $l_1$ squarefree and $l_2$ a square. Define … and suppose that … where…

Let $A$ be a sequence obtained by appending an initial fraction of a rotated Legendre sequence to itself. Borwein et al.'s conjecture. This sequence family has asymptotic merit fac…