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Register a new userDeterminants concerning Legendre symbols
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Hai-Liang Wu
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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.
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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 and investigate some related topics. For instance, given any integers $c,d$ with $d e0$ and $c^2-4d e0$, we show that there are infinitely many odd primes $p$ such that $$detbigg[left(frac{i^2+cij+dj^2}{p}right)bigg]_{0le i,jle p-1}=0,$$ where $(frac{cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
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 asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
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 vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C^* algebras related to shift invariant sets.
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 particular, Lemma 2.3 of Agelas is false. This Lemma 2.3 is a generalisation of Theorem 1 of Vassilev-Missana [A note on prime zeta function and Riemann zeta function, Notes on Number Theory and Discrete Mathematics, 22, 4 (2016), 12-15]. We show by several independent methods that Theorem 1 of Vassilev-Missana is false. We also show that Theorem 2 of Vassilev-Missana is false. This note has two aims. The first aim is to alert other researchers to these errors so they do not rely on faulty results in their own work. The second aim is pedagogical - we hope to show how these errors could have been detected earlier, which may suggest how similar errors can be avoided, or at least detected at an early stage.
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-Zudilin numbers, respectively. We also establish congruences for $$sum_{n=0}^{p-1}frac{D_n}{16^n},quad sum_{n=0}^{p-1}frac{D_n}{4^n}, quad sum_{n=0}^{p-1}frac{b_n}{(-3)^n},quad sum_{n=0}^{p-1}frac{b_n}{(-27)^n}pmod {p^2}$$ in terms of certain binary quadratic forms.
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