Search ResultsShowing 1-20 of 25
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arXiv:2407.01101 (Published 2024-07-01)
Packing density of sets with only two non-mixed gaps
Comments: 12 pagesFor a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the corresponding optimal density, conjecture its tightness, and prove it in case one of the gap lengths, $a$ or $b$, appears only once. This is equivalent to a Motzkin problem on the independence ratio of certain integer distance graphs.
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arXiv:2406.13013 (Published 2024-06-18)
An uniform lower bound for classical Kloosterman sums and an application
Comments: 11 pagesCategories: math.NTWe present an elementary uniform lower bound for the classical Kloosterman sum $S(a,b;c)$ under the condition of its non-vanishing and $(ab,c)=1$, with $c$ being an odd integer. We then apply this lower bound for Kloosterman sums to derive an explicit lower bound in the Petersson's trace formula, subject to a pertinent condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, wherein the parameters $k$ and $N$ are permitted to vary independently.
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arXiv:2404.06951 (Published 2024-04-10)
Explicit lower bound for large gaps between some consecutive primes
Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and $X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n}, \ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there exists an effective abosolute constant $c_{LG}>0$ such that \[ G_{k}(X)\geq \frac{c_{LG}}{k^{2}}\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X} \] holds for any sufficiently large $X$. The main purpose of this paper is to determine the constant $c_{LG}$ above. We see that $c_{LG}$ is determined by several factors related to analytic number theory, for example, the level of distribution of primes, the ratio of integrals of functions in the multidimensional sieve of Maynard, the distribution of primes in arithmetic progressions to large moduli, and the coefficient of upper bound sieve of Selberg. We prove that the above inequality is valid at least for $c_{LG}\approx 3.0\times 10^{-17}$.
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arXiv:2311.10285 (Published 2023-11-17)
Zeros of linear combinations of Dirichlet $L$-functions on the critical line
Categories: math.NTLet $F$ be a linear combination of $N\geq 1$ Dirichlet $L$-functions attached to even (or odd) primitive characters. Selberg proved that a positive proportion of non-trivial zeros of $F$ lie on the critical line. Our work here is to provide an explicit lower bound for this proportion. In particular, we show that the lower bound $2.16\times 10^{-6}/(N\log N)$ is admissible for large $N$.
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arXiv:2303.17950 (Published 2023-03-31)
Explicit spectral gap for Schottky subgroups of $\mathrm{SL} (2,\mathbb{Z})$
Comments: 31 pagesLet $\Gamma$ be a Schottky subgroup of $\mathrm{SL} (2,\mathbb{Z})$. We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface $\Gamma \backslash \mathbb{H}^2$ provided the limit set of $\Gamma$ is thick enough.
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arXiv:2303.05890 (Published 2023-03-10)
On abelian cubic fields with large class number
Journal: Proc. Amer. Math. Soc. 152 (2024), 3255-3264DOI: 10.1090/proc/16827Categories: math.NTKeywords: abelian cubic fields, large class number, abelian cubic number fields, explicit lower bound, arbitrary long sequencesTags: journal articleWe investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an explicit lower bound for extreme values of class numbers of abelian cubic fields.
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arXiv:2301.02391 (Published 2023-01-06)
On effective irrationality exponents of cubic irrationals
Comments: 17 pagesCategories: math.NTWe provide an upper bound on the efficient irrationality exponents of cubic algebraics $x$ with the minimal polynomial $x^3 - tx^2 - a$. In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville in the case $|t| > 19.71 a^{4/3}$. Moreover, under the condition $|t| > 86.58 a^{4/3}$, we provide an explicit lower bound on the expression $||qx||$ for all large $q\in\mathbb{Z}$. These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.
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arXiv:2212.14778 (Published 2022-12-30)
Generic diagonal conic bundles revisited
We prove a stronger form of our previous result that Schinzel's Hypothesis holds for $100\%$ of $n$-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.
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arXiv:2206.00412 (Published 2022-06-01)
Quaternary quadratic forms with prime discriminant
Comments: 20 pagesCategories: math.NTLet $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson norm $\langle C, C \rangle$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $\langle C, C \rangle$ that depends on the smallest positive integer not represented by the dual form $Q^{*}$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.
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arXiv:2203.00731 (Published 2022-03-01)
A lower bound on the proportion of modular elliptic curves over Galois CM fields
Categories: math.NTWe calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $\zeta_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}(\zeta_n)$ with $5\nmid n$, this proportion is at least $1-\varepsilon$ with only finitely many exceptions of $n$, for any choice of $\varepsilon > 0$.
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A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd
Categories: math.NTLet $l$ be a positive odd integer. Using Cilleruelo's method, we establish an explicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$, $\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine all values of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain values of $l$.
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Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers
In this paper, we study linear forms \[\lambda = \beta_1\mathrm{e}^{\alpha_1}+\cdots+\beta_m\mathrm{e}^{\alpha_m},\] where $\alpha_i$ and $\beta_i$ are algebraic numbers. An explicit lower bound for the absolute value of $\lambda$ is proved, which is derived from "th\'eor\`eme de Lindemann--Weierstrass effectif" via constructive methods in algebraic computation. Besides, the existence of $\lambda$ with an explicit upper bound is established on the result of counting algebraic numbers.
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arXiv:2104.06974 (Published 2021-04-14)
On the local constancy of certain mod $p$ Galois representations
Categories: math.NTIn this article we study local constancy of the mod $p$ reduction of certain $2$-dimensional crystalline representations of $\mathrm{Gal}\left(\bar{\mathbb{Q}}_p/\mathbb{Q}_p\right)$ using the mod $p$ local Langlands correspondence. We prove local constancy in the weight space by giving an explicit lower bound on the local constancy radius centered around weights going up to $(p-1)^{2} +3$ and the slope fixed in $(0, \ p-1)$ satisfying certain constraints. We establish the lower bound by determining explicitly the mod $p$ reductions at nearby weights and applying a local constancy result of Berger.
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arXiv:2006.00549 (Published 2020-05-31)
On Proinov's lower bound for the diaphony
Comments: 25 pages, 2 figuresCategories: math.NTIn 1986, Proinov published an explicit lower bound for the diaphony of both finite and infinite sequences of points in the s dimensional unit cube. To the best of our knowledge, the proofs of these results were so far only available in a monograph of Proinov written in Bulgarian. The first contribution of our paper is to give a self contained version of Proinov's proof in English. Along the way, we improve the asymptotic constants using recent results of Hinrichs and Markhasin. Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.
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arXiv:1906.09718 (Published 2019-06-24)
Lower bound for class number and a proof of Chowla and Yokoi's conjecture
Comments: 21 pages. Comments are welcomeCategories: math.NTLet $d$ be a square-free positive integer and $h(d)$ the class number of the real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ In this paper we give an explicit lower bound for $h(n^2+r)$, where $r=1,4$, and also establish an equivalent criteria to attain this lower bound in terms of special value of Dedekind zeta function. Our bounds enables us to provide a new proof of a well-known conjecture of Chowla and that of Yokoi. Also applying our results, we obtain some criteria for class group of prime power order to be cyclic.
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arXiv:1807.11065 (Published 2018-07-29)
On growth of the set $A(A+1)$ in arbitrary finite fields
Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a prime. For a set $A \subset \mathbb{F}_q$, under certain structural restrictions, we prove a new explicit lower bound on the size of the product set $A(A + 1)$. Our result improves on the previous best known bound due to Zhelezov and holds under more relaxed restrictions.
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arXiv:1805.03850 (Published 2018-05-10)
Asymptotic lower bound of class numbers along a Galois representation
Comments: 14 pagesCategories: math.NTLet T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K_n, which is the Galois extension field of K fixed by the stabilizer of T/p^n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p^\infty])/K for a given abelian variety A with certain conditions in terms of the Mordell-Weil group. We also prove another asymptotic inequality for the cases when A is a Hilbert--Blumenthal or CM abelian variety.
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arXiv:1802.00085 (Published 2018-01-31)
Explicit bounds for primes in arithmetic progressions
Comments: 66 pages. Results of computations, and the code used for those computations, can be found at: http://www.nt.math.ubc.ca/BeMaObRe/Categories: math.NTWe derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{3600} \frac q{\phi(q)} \frac{x}{\log x}, $$ for all $x \geq 7.94 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod q$ when $q\le4500$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.
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arXiv:1712.04214 (Published 2017-12-12)
Explicit Small Heights in Infinite Non-Abelian Extensions
Comments: 23 pages, comments welcomeCategories: math.NTLet $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower bound for the height of non-zero elements in $\mathbb{Q}(E_{\text{tor}})$ that are not a root of unity, only depending on the conductor of the elliptic curve. As a side result we will give an explicit bound for a small supersingular prime for an elliptic curve.
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arXiv:1602.01307 (Published 2016-02-03)
On an explicit lower bound for the star discrepancy in three dimensions
Comments: 12 pagesCategories: math.NTFollowing a result of D.~Bylik and M.T.~Lacey from 2008 it is known that there exists an absolute constant $\eta>0$ such that the (unnormalized) $L^{\infty}$-norm of the three-dimensional discrepancy function, i.e, the (unnormalized) star discrepancy $D^{\ast}_N$, is bounded from below by $D_{N}^{\ast}\geq c (\log N)^{1+\eta}$, for infinitely many $N\in\mathbb{N}$, where $c>0$ is some constant independent of $N$. This paper builds upon their methods to verify that the above result holds with $\eta<1/(32+4\sqrt{41})\approx 0.017357\ldots$