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Number Theory

New submissions

[ total of 20 entries: 1-20 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07319 [pdf, ps, other]
Title: Non-trivial Integer Solutions of $x^r+y^r=Dz^p$
Comments: 12 pages, 1 table
Subjects: Number Theory (math.NT)

In this paper, we use the modular method together with some standard conjectures to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r>5$ is a fixed prime, whenever $p$ is large enough.

[2]  arXiv:2404.07550 [pdf, ps, other]
Title: On the Borisov-Gunnells relations for products of Eisenstein series
Comments: 7 pages, comments welcome!
Subjects: Number Theory (math.NT)

Borisov and Gunnells have proved that certain linear combinations of products of Eisenstein series are Eisenstein series themselves, in analogy with the Manin 3-term relations for modular symbols. We devise a new method to determine and prove such relations, by differentiating with respect to the parameters of the Eisenstein series.

[3]  arXiv:2404.07688 [pdf, ps, other]
Title: Algebraic identities among $q$- analogue of Euler double zeta values
Subjects: Number Theory (math.NT)

In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant. Using a $q$-analogue of the Nielsen Reflexion Formula for $q>1$, we investigate identities involving different versions of $q$-analogues of the Riemann zeta function and the double-zeta function. Additionally, we analyze the behavior of $\zeta_q(s_1, s_2)$ as $s_1$ and $s_2$ approach to $0$ and compare these limits to those of the classical double-zeta function. Finally, we discuss the $q$-analogue of the Mordell-Tornheim $r$-ple zeta function and its relation with the $q$-double zeta function.

[4]  arXiv:2404.07690 [pdf, ps, other]
Title: Transcendental nature of $p$-adic digamma values
Subjects: Number Theory (math.NT)

For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $\psi_p(r/p)+ \gamma_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $\psi_p(r/p^n)+ \gamma_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $\psi_p(r/pq)+ \gamma_p$, where $p, q$ are distinct primes.

[5]  arXiv:2404.07832 [pdf, other]
Title: Zeros of $L$-Functions in Low-Lying Intervals and de Branges spaces
Comments: 26 pages, 1 figure
Subjects: Number Theory (math.NT); Classical Analysis and ODEs (math.CA)

We consider a variant of a problem first introduced by Hughes and Rudnick (2003) and generalized by Bernard (2015) concerning conditional bounds for small first zeros in a family of $L$-functions. Here we seek to estimate the size of the smallest intervals centered at a low-lying height for which we can guarantee the existence of a zero in a family of $L$-functions. This leads us to consider an extremal problem in analysis which we address by applying the framework of de Branges spaces, introduced in this context by Carneiro, Chirre, and Milinovich (2022).

Cross-lists for Fri, 12 Apr 24

[6]  arXiv:2404.07342 (cross-list from math.RT) [pdf, ps, other]
Title: The global Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on unitary groups
Subjects: Representation Theory (math.RT); Number Theory (math.NT)

We prove the Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on unitary groups and an Ichino-Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. We develop the full coarse spectral and geometric expansions of the relative trace formulae, and compute relevant spectral terms via zeta integrals and truncated periods. We compare all geometric terms and characterize the local geometric comparison in terms of spectral data.

[7]  arXiv:2404.07380 (cross-list from math.CO) [pdf, ps, other]
Title: Strong bounds for skew corner-free sets
Comments: 23 pages, comments welcome
Subjects: Combinatorics (math.CO); Number Theory (math.NT)

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew corner-free set has size at most $\exp(-\Omega(\log^{1/12} n))\cdot n^2$, nearly matching the best known lower bound of $\exp(-O(\sqrt{\log n}))\cdot n^2$ by Beker (arXiv'24). Our techniques generalize those of Kelley and Meka's recent breakthrough on three-term arithmetic progression (FOCS'23), answering a question of Beker (arXiv'24).

[8]  arXiv:2404.07463 (cross-list from math.RT) [pdf, ps, other]
Title: Generic representations, open parameters and ABV-packets for $p$-adic groups
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Number Theory (math.NT)

If $\pi$ is a representation of a $p$-adic group $G(F)$, and $\phi$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $\phi$ that will detect when $\pi$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $\phi$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $\phi$ for a $p$-adic group $G(F)$, the ABV-packet $\Pi^{\mathrm{ABV}}_\phi(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,\phi,\mathop{\text{Ad}})$ is regular at $s=1$, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $\phi$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.

[9]  arXiv:2404.07485 (cross-list from math.CO) [pdf, ps, other]
Title: Hook length biases in ordinary and $t$-regular partitions
Comments: 15 pages
Subjects: Combinatorics (math.CO); Number Theory (math.NT)

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular partitions. For a positive integer $k$, let $p_{(k)}(n)$ denote the number of hooks of length $k$ in all the partitions of $n$. We prove that $p_{(k)}(n)\geq p_{(k+1)}(n)$ for all $n\geq0$ and $n\ne k+1$; and $p_{(k)}(k+1)- p_{(k+1)}(k+1)=-1$ for $k\geq 2$. For integers $t\geq2$ and $k\geq1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. We find generating functions of $b_{t,k}(n)$ for certain values of $t$ and $k$. Exploring hook length biases for $b_{t,k}(n)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}(n)\geq b_{2,1}(n)$ for all $n>4$, whereas $b_{2,2}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. Our proofs are both combinatorial and analytic in nature. We also propose some conjectures on biases among $b_{t,k}(n)$.

[10]  arXiv:2404.07752 (cross-list from math.DS) [pdf, ps, other]
Title: Singular linear forms over global function fields
Comments: 32 pages
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)

In this paper, we consider singular linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular linear forms by constructing an appropriate Margulis function over global function fields.

Replacements for Fri, 12 Apr 24

[11]  arXiv:2211.15137 (replaced) [pdf, ps, other]
Title: Primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three
Authors: Hiroshi Onuki
Subjects: Number Theory (math.NT)
[12]  arXiv:2305.09885 (replaced) [pdf, ps, other]
Title: On asymptotically automatic sequences
Authors: Jakub Konieczny
Comments: 32 pages; comments welcome
Subjects: Number Theory (math.NT); Formal Languages and Automata Theory (cs.FL); Combinatorics (math.CO)
[13]  arXiv:2307.16069 (replaced) [pdf, ps, other]
Title: Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes
Comments: 9 pages. Accompanied by a Maple package and numerous output files from <A HREF="this https URL">this url</A> This version corrects minor typos
Journal-ref: INTEGERS v. 23 (2023) #A95
Subjects: Number Theory (math.NT); Combinatorics (math.CO)
[14]  arXiv:2402.11884 (replaced) [pdf, ps, other]
Title: Large prime factors of well-distributed sequences
Comments: Corrects Lemma 3 of the previous version and updates references. 14 pages
Subjects: Number Theory (math.NT)
[15]  arXiv:2403.19453 (replaced) [pdf, ps, other]
Title: The submodularity of the covolume function in global function fields
Authors: Gukyeong Bang
Comments: 19 pages
Subjects: Number Theory (math.NT); Dynamical Systems (math.DS)
[16]  arXiv:2404.01075 (replaced) [pdf, ps, other]
Title: Drinfeld singular moduli, hyperbolas, units
Subjects: Number Theory (math.NT)
[17]  arXiv:2404.05313 (replaced) [pdf, ps, other]
Title: On certain kernel functions and shifted convolution sums of Hecke eigenvalues
Authors: Youjun Wang
Comments: 14 pages
Subjects: Number Theory (math.NT)
[18]  arXiv:2404.05487 (replaced) [pdf, ps, other]
Title: Monogenic Quartic Polynomials and Their Galois Groups
Subjects: Number Theory (math.NT)
[19]  arXiv:2309.10300 (replaced) [pdf, ps, other]
Title: Vojta's conjecture on weighted projective varieties
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
[20]  arXiv:2404.05104 (replaced) [pdf, other]
Title: Uniform approximation of Betti numbers
Comments: Fixed slight inaccuracies and typos
Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); K-Theory and Homology (math.KT); Number Theory (math.NT)
[ total of 20 entries: 1-20 ]
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