Search ResultsShowing 1-20 of 30
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arXiv:2408.04481 (Published 2024-08-08)
On the $p$-ranks of class groups of certain Galois extensions
Comments: 44 pagesCategories: math.NTLet $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $\zeta_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the class group of $\mathbb{Q}(\zeta_p, N^{1/p})$ using Galois cohomological methods and obtain an exact formula for the $p$-rank in terms of the dimensions of certain Selmer groups. Using our formula, we provide a numerical criterion to establish upper and lower bounds for the $p$-rank, analogous to the numerical criteria provided by F.~Calegari--M.~Emerton and K.~Schaefer--E.~Stubley for the $p$-ranks of the class group of $\mathbb{Q}(N^{1/p})$. In the case $p=3$, we use Redei matrices to provide a numerical criterion to exactly calculate the $3$-rank, and also study the distribution of the $3$-ranks as $N$ varies through primes which are $4,7 \pmod{9}$.
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arXiv:2404.02367 (Published 2024-04-02)
A note on the exact formulas for certain $2$-color partitions
Let $p\leq 23$ be a prime and $a_p(n)$ count the number of partitions of $n$ using two colors where one of the colors only has parts divisible by $p$. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula for $\log a_p(n)$. Our results partially extend the work of Mauth, who proved the asymptotic formula for $\log a_2(n)$ conjectured by Banerjee et al.
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arXiv:2311.07702 (Published 2023-11-13)
On a conjecture of Ghorpade, Datta and Beelen for the number of points of varities over finite fields
Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Ghorpade, Datta and Beelen had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$.
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arXiv:2309.16072 (Published 2023-09-28)
An exact formula for a family of floor function sets
Categories: math.NTLet $X$ and $t$ be real numbers, $X\ge 1$ and $t>1$. The family of sets $$\left\{\left\lfloor \frac{X}{n^t} \right\rfloor ~:~ 1\leq n\leq X\right\}$$ are very sparse sets that satisfy the prime number theorem as $X$ increases. We give an exact formula for the cardinality of these sets.
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arXiv:2309.05233 (Published 2023-09-11)
Uniform bounds for Kloosterman sums of half-integral weight, same-sign case
Categories: math.NTIn the previous paper [Sun23], the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied $\tilde m\tilde n<0$. In this paper, we prove the same uniform bound when $\tilde m\tilde n>0$ for further applications.
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arXiv:2307.10441 (Published 2023-07-19)
Exact formula for 1-lower run overpartitions
We are going to show an exact formula for lower $1$-run overpartitions. The generating function is of mixed mock-modular type with an overall weight $0.$ We will apply an extended version of the classical Circle Method. The approach requires bounding modified Kloosterman sums and Mordell integrals.
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arXiv:2306.07459 (Published 2023-06-12)
Log-concavity for partitions without sequences
We prove log-concavity for the function counting partitions without sequences. We use an exact formula for a mixed-mock modular form of weight zero, explicit estimates on modified Kloosterman sums and the saddle point method.
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arXiv:2305.03396 (Published 2023-05-05)
Exact formula for cubic partitions
Categories: math.NTWe obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng.
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arXiv:2204.04459 (Published 2022-04-09)
The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I
Comments: 30 pagesCategories: math.NTFor $B \in \mathbb{F}_q [T]$ of degree $2n \geq 2$, consider the number of ways of writing $B=E^2 + \gamma F^2$, where $\gamma \in \mathbb{F}_q^*$ is fixed, and $E,F \in \mathbb{F}_q [T]$ with $\mathrm{deg} \hspace{0.25em} E = n$ and $\mathrm{deg} \hspace{0.25em} F = m < n$. We denote this by $S_{\gamma ; m} (B)$. We obtain an exact formula for the variance of $S_{\gamma ; m} (B)$ over intervals in $\mathbb{F}_q [T]$. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing $B=E^2 + T F^2$.
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arXiv:2203.16469 (Published 2022-03-30)
Factorials and Legendre's three-square theorem: II
Comments: 14 pages, 6 figuresCategories: math.NTLet $\bar{S}$ denote the set of integers $n$ such that $n!$ cannot be written as a sum of three squares. Let $\bar{S}(n)$ denote $\bar{S} \cap [1, n]$. We establish an exact formula for $\bar{S}(2^k)$ and show that $\bar{S}(n) = 1/8*n + \mathcal{O}(\sqrt{n})$. We also list the lengths of gaps appearing in $\bar{S}$. We make use of the software package Walnut to establish these results.
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arXiv:2009.08222 (Published 2020-09-17)
On Fibonacci partitions
We prove an exact formula for OEIS A000119, which counts partitions into distinct Fibonacci numbers. We also establish an exact formula for its mean value, and determine the asymptotic behaviour.
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arXiv:1911.11246 (Published 2019-11-25)
The distribution of the $L_4$ norm of Littlewood polynomials
Comments: 23 pages, 4 tablesClassical conjectures due to Littlewood, Erd\H{o}s and Golay concern the asymptotic growth of the $L_p$ norm of a Littlewood polynomial (having all coefficients in $\{1, -1\}$) as its degree increases, for various values of $p$. Attempts over more than fifty years to settle these conjectures have identified certain classes of the Littlewood polynomials as particularly important: skew-symmetric polynomials, reciprocal polynomials, and negative reciprocal polynomials. Using only elementary methods, we find an exact formula for the mean and variance of the $L_4$ norm of polynomials in each of these classes, and in the class of all Littlewood polynomials. A consequence is that, for each of the four classes, the normalized $L_4$ norm of a polynomial drawn uniformly at random from the class converges in probability to a constant as the degree increases.
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arXiv:1907.03026 (Published 2019-07-05)
Exact Formulae for the Fractional Partition Functions
The partition function $p(n)$ has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method'' to estimate the size of $p(n)$, which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined by $\sum_{n = 0}^\infty p_{\alpha}(n)x^n := \prod_{k=1}^\infty (1-x^k)^{-\alpha}$. In this paper we use the Rademacher circle method to find an exact formula for $p_\alpha(n)$ and study its implications, including log-concavity and the higher-order generalizations (i.e., the Tur\'an inequalities) that $p_\alpha(n)$ satisfies.
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arXiv:1808.08455 (Published 2018-08-25)
Additive Volume of Sets Contained in Few Arithmetic Progressions
Comments: 16 pagesA conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$ and for families of structured sets called chains. In this paper we extend the formula to sets of every dimension and prove it for sets composed of three segments, giving structural results for the extremal case. A weaker extension to sets composed of a bounded number of segments is also discussed.
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An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on $\mathbb{P}^2$
Comments: 23 pages; v2: 25 pages, the discussion of asymptotic expansion extended, to appear in Transactions of the AMSTopologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth.
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arXiv:1710.01560 (Published 2017-10-04)
Discrepancy results for the Van der Corput sequence
Comments: 9 pagesCategories: math.NTLet $d_N=ND_N(\omega)$ be the discrepancy of the Van der Corput sequence in base $2$. We improve on the known bounds for the number of indices $N$ such that $d_N\leq \log N/100$. Moreover, we show that the summatory function of $d_N$ satisfies an exact formula involving a $1$-periodic, continuous function. Finally, we show that $d_N$ is invariant under digit reversal in base $2$.
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arXiv:1708.01351 (Published 2017-08-04)
Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields
Comments: 20 pagesCategories: math.NTConsider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.
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arXiv:1612.04539 (Published 2016-12-14)
Exceptional units in the residue class rings of global fields
Categories: math.NTLet $R$ be a commutative ring with $1\in R$ and $R^{*}$ its group of units. A unit $u\in R^*$ is called exceptional if $1-u\in R^{*}$. In the case $R=\mathbb{Z}/n\mathbb{Z}$ of residue classes modulo $n$, recently Sander has determined the number of representations of an arbitrary element $c\in \mathbb{Z}/n\mathbb{Z}$ as the sum of two exceptional units. This result has been immediately generalized by Yang and Zhao, which gives an exact formula for the number of ways to represent each element of $\mathbb{Z}/n\mathbb{Z}$ as the sum of $k \ge 2$ exceptional units. In this paper, we show that the method of Yang and Zhao in fact leads to a generalization in any residue class ring of global fields. Moreover, we completely determine the additive structure of such exceptional units.
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arXiv:1608.06928 (Published 2016-08-24)
The Formula for the Distribution of the 3-Smooth Numbers
Categories: math.NTIn this paper we present a rapidly convergent formula for the arithmetic function $N_{a,b}(x)$, which counts the number of positive integers of the form $a^{p}b^{q}$ less than or equal to $x$. As an application, we give an exact formula for the distribution of the $3$-smooth numbers, which are the numbers of the form $2^{p}3^{q}$. The analog formula for the counting function of the $5$-smooth numbers is given at the end of the paper.
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arXiv:1509.06272 (Published 2015-09-21)
On the sumsets of exceptional units in $\mathbb{Z}_n$
Comments: 7 pages. This is a preliminary draftCategories: math.NTLet $R$ be a commutative ring with $1\in R$ and $R^{\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\ast}$. Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$. In this paper, given an integer $k\ge 2$, we obtain an exact formula for the number of ways to represent each element of $ \mathbb{Z}_n$ as the sum of $k$ exceptional units. This generalizes a recent result of J. W. Sander for the case $k=2$.