14 problems

Fejér's conjecture. For every $n\geq 1$ and $0<x<\pi$,

Let $q$ and $n$ be positive integers, and let $p_1,\ldots,p_n$ be distinct integers relatively prime to $q$ such that $q>(7/4)^n$ and … Assume that for every $k\in\{1,\ldots,n\}$,…

Zahorski's logarithmic bound conjecture. The integral is $\textbf{O}(\log n_k)$, with the bound depending on $n_k$. The conjecture arose from a question posed by Prof. Zahorski con…

Harshitha's conjectures. The limits satisfy

Let $t_n$ and $t_n'$ be the quantities defined earlier in the paper, and let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol. The tangent permanent conjecture. (i) For ev…

Let $p$ be an odd prime, and let $c_p$ and $c_p'$ be the quantities defined earlier in the paper. The trigonometric permanent sign conjecture. … and … The claim is motivated by the…

Alzer's conjecture. For every positive integer $n$ and every $0<x<\pi$,

Let $\mathcal B$ be a collection of finite sets of positive integers. For each $B\in\mathcal B$, associate a weight $M(B)=(m_b)_{b\in B}$ with $m_b=1$ or $2$. Asymmetric sum of sin…

For positive integers $a_1<a_2<a_3<a_4$, let … and let $\lambda(4)$ be defined by $-\lambda(4)=\sup L(a_1,a_2,a_3,a_4)$. Four-term cosine-sum conjecture. … This would determine the…

Ankeny–Chowla conjecture. The function $\lambda(n)$ approaches infinity as $n$ approaches infinity.

Let $C(m,n)$ and $S(m,n)$ denote the basic trigonometric power sums defined in the paper. Powers-of-two reducibility conjecture. Multiplying and dividing by $2^n$ in either…

For each positive integer $n$, let … where the infimum is over sets of distinct integers $a_1<\cdots<a_n$. Chowla's conjecture. As $n$ tends to infinity, … It is known that…

Let $k$ be a positive integer, and let the trigonometric constructs discussed in the paper be the finite sums arising from powers of circulant boolean matrices, including sums of t…

Generalized positivity conjecture. The first sum is a polynomial in $1+\cos(x)$ with nonnegative integral coefficients, and the second satisfies