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The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $aequiv1pmod4$. Then for the determinant $$S(1,p):={rm det}bigg[left(frac{i^2+j^2}{p}right)bigg]_{1le i,jle frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun a
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptoti
This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishin
A recent paper by Agelas [Generalized Riemann Hypothesis, 2019, hal-00747680v3] claims to prove the Generalized Riemann Hypothesis (GRH) and, as a special case, the Riemann Hypothesis (RH). We show that the proof given by Agelas contains an error. In
Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $sum_{k=0}^{p-1}binom akbinom{-1-a}kfrac p{k+b}pmod {p^2}$. For $n=0,1,2,ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudil