Search ResultsShowing 1-7 of 7
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arXiv:2309.13070 (Published 2023-09-18)
Sum of Distinct Biquadratic Residues Modulo Primes
Comments: Submitted to Integers: Electronic Journal of Combinatorial Number TheoryCategories: math.NTTwo conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo any prime $p\ge 7$. Here, $c$ is an integer not divisible by $p$ and $e$ is any integer. The formula splits into eight cases, depending on the remainder of $p$ modulo $8$ and whether $c$ is a quadratic residue modulo $p$. The formula quickly extends to the non-monic case. We then apply the formula to prove a classification of the set of such sums in terms of the sets of squares and fourth powers, when $c$ in $x^4 + cx^2$ is varied over all integers with a fixed prime modulus $p\ge 7$. The sum and the set of sums are manually computed for the excluded prime moduli $p=3,5$.
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arXiv:2211.00305 (Published 2022-11-01)
Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields
Comments: 20 pagesCategories: math.NTIn this paper, we show nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$. As an application, we show that for each isogeny factor of the Jacobian of the $p$-th Fermat curve where $2$ is a quadratic residue modulo $p$, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the $11$-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.
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arXiv:2108.10624 (Published 2021-08-24)
Trinomial coefficients and a determinant of Sun
Categories: math.NTIn this paper, by using the tool of trinomial coefficients we study some determinant problems posed by Zhi-Wei Sun. For example, given any odd prime $p$ with $p\equiv 2\pmod 3$, we show that $2\det[\frac{1}{i^2-ij+j^2}]_{1\le i,j\le p-1}$ is a quadratic residue modulo $p$. This confirms a conjecture of Zhi-Wei Sun.
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arXiv:1701.02286 (Published 2017-01-09)
When the number of divisors is a quadratic residue
Comments: 9 pagesCategories: math.NTLet $q > 2$ be a prime number and define $\lambda_q := \left( \frac{\tau}{q} \right)$ where $\tau(n)$ is the number of divisors of $n$ and $\left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When $\tau(n)$ is a quadratic residue modulo $q$, then $\left( \lambda_q \star \mathbf{1} \right) (n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function $\lambda_q \star \mathbf{1}$ to the well known average order of $\tau$. The proof reveals that the results depend heavily on the value of $\left( \frac{2}{q} \right)$. A bound for short sums in the case $q=5$ is also given, using profound results from the theory of integer points close to certain smooth curves.
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arXiv:1510.05998 (Published 2015-10-20)
Extractors in Paley graphs: a random model
Comments: 11 pagesA well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset\mathbb{Z}/p\mathbb{Z}$, each of size at least $\log ^C p$ (for some constant $C$) we have that for $(\frac12+o(1))|X||Y|$ of the pairs $(x,y)\in X\times Y$, $x+y$ is a quadratic residue modulo $p$. We address the probabilistic analogue of this question, that is for every fixed $\delta>0$, given a finite group $G$ and $A\subset G$ a random subset of density $\frac12$, we prove that with high probability for all subsets $|X|,|Y|\geq \log ^{2+\delta} |G|$ for $(\frac12+o(1))|X||Y|$ of the pairs $(x,y)\in X\times Y$ we have $xy\in A$.
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arXiv:1301.0035 (Published 2013-01-01)
On the Product of Small Elkies Primes
Given an elliptic curve $E$ over a finite field $\F_q$ of $q$ elements, we say that an odd prime $\ell \nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\ell$, where $t_E = q+1 - #E(\F_q)$ and $#E(\F_q)$ is the number of $\F_q$-rational points on $E$. These primes are used in the presently most efficient algorithm to compute $#E(\F_q)$. In particular, the bound $L_q(E)$ such that the product of all Elkies primes for $E$ up to $L_q(E)$ exceeds $4q^{1/2}$ is a crucial parameter of this algorithm. We show that there are infinitely many pairs $(p, E)$ of primes $p$ and curves $E$ over $\F_p$ with $L_p(E) \ge c \log p \log \log \log p$ for some absolute constant $c>0$, while a naive heuristic estimate suggests that $L_p(E) \sim \log p$. This complements recent results of Galbraith and Satoh (2002), conditional under the Generalised Riemann Hypothesis, and of Shparlinski and Sutherland (2012), unconditional for almost all pairs $(p,E)$.
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arXiv:1112.2398 (Published 2011-12-11)
Chebyshev's bias and generalized Riemann hypothesis
Comments: 9 pagesJournal: Journal of Algebra, Number Theory: Advances and Applications 8, 1-2 (2013) 41-55Categories: math.NTKeywords: generalized riemann hypothesis, chebyshevs bias, quadratic residue modulo, riemann zeta function, littlewoods theoremTags: journal articleIt is well known that $li(x)>\pi(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $\gamma$ of the Riemann zeta function $\zeta(s)$, encoded by the equation $li(x)-\pi(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_{\gamma}\frac{\sin (\gamma \log x)}{\gamma}]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[\psi(x)]>\pi(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $\pi(x;4,3)-\pi(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $\pi(x;k,l)$ means the number of primes up to $x$ and congruent to $l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$ \cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[\phi(k)*\psi(x;k,l)]-\phi(k)*\pi(x;k,l)$ is a counting function introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.