## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (192.979 recursos)

Functiones et Approximatio Commentarii Mathematici

1. #### The tail of the singular series for the prime pair and Goldbach problems

Goldston, Daniel A.; Ziegler Hunts, Julian; Ngotiaoco, Timothy
We obtain an asymptotic formula for a weighted sum of the square of the tail in the singular series for the Goldbach and prime-pair problems.

2. #### Simple zeros of Dedekind zeta functions

Louboutin, Stéphane R.
Using Stechkin's lemma we derive explicit regions of the half complex plane $\Re (s)\leq 1$ in which the Dedekind zeta function of a number field $K$ has at most one complex zero, this zero being real if it exists. These regions are Stark-like regions, i.e. given by all $s=\beta +i\gamma$ with $\beta\geq 1-c/\log d_K$ and $\vert\gamma\vert\leq d/\log d_K$ for some absolute positive constants $c$ and $d$. These regions are larger and our proof is simpler than recently published such regions and proofs.

3. #### On the equality between two diametral dimensions

Bastin, Françoise; Demeulenaere, Loïc
The paper gives sufficient conditions to have the equality between two diametral dimensions of metrizable locally convex spaces and examples of Köthe echelon spaces satisfying them. It also provides examples for which the equality does not hold.

4. #### On relations equivalent to the generalized Riemann hypothesis for the Selberg class

Mazhouda, Kamel; Smajlović, Lejla
We prove that the generalized Riemann hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $\lambda_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $\Re(\lambda_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally...

5. #### Family of elliptic curves with good reduction everywhere over number fields of given degree

Takeshi, Nao
We show that for Dirichlet characters $\chi_1,\ldots ,\chi_s$ mod $p^m$ the sum $$\mathop{\sum_{x_1=1}^{p^m} \dots \sum_{x_s=1}^{p^m}}_{ A_1x_1^{k_1}+\dots+ A_sx_s^{k_s}\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_s(x_s),$$ has a simple evaluation when $m$ is sufficiently large.

6. #### Jacobi-type sums with an explicit evaluation modulo prime powers

Alsulmi, Badria; Pigno, Vincent; Pinner, Christopher
We show that for Dirichlet characters $\chi_1,\ldots ,\chi_s$ mod $p^m$ the sum $$\mathop{\sum_{x_1=1}^{p^m} \dots \sum_{x_s=1}^{p^m}}_{ A_1x_1^{k_1}+\dots+ A_sx_s^{k_s}\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_s(x_s),$$ has a simple evaluation when $m$ is sufficiently large.

7. #### Small solutions of diagonal congruences

Cochrane, Todd; Ostergaard, Misty; Spencer, Craig
We prove that for $k \geq 2$, $0 <\varepsilon< \frac 1{k(k-1)}$, $n>\frac {k-1}{\varepsilon }$, prime $p> P(\varepsilon, k)$, and integers $c,a_i$, with $p \nmid a_i$, $1 \le i \le n$, there exists a solution $\underline{x}$ to the congruence $$\sum_{i=1}^n a_ix_i^k \equiv c \mod p$$ in any cube $\mathcal{B}$ of side length $b \ge p^{\frac 1k + \varepsilon}$. Various refinements are given for smaller $n$ and for cubes centered at the origin.

8. #### Elliptic curves with rank $0$ over number fields

Dey, Pallab Kanti
Let $E: y^2 = x^3 + bx$ be an elliptic curve for some nonzero integer $b$. Also consider $K$ be a number field with $4 \nmid [K : \mathbb{Q}]$. Then in this paper, we obtain a necessary and sufficient condition for $E$ having rank $0$ over $K$.

9. #### The $3x+1$ problem: a lower bound hypothesis

Rozier, Olivier
Much work has been done attempting to understand the dynamic behaviour of the so-called $3x+1$'' function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two. In this paper, we formulate a new hypothesis asserting that the first terms of those sequences have a lower bound which depends on the binary entropy of the ones-ratio''. It is in agreement with all computations so far. Furthermore it implies accurate upper bounds for the total stopping time and the maximum excursion of an integer. Theses...

10. #### Convolutions with probability distributions, zeros of $L$-functions, and the least quadratic nonresidue

Banks, William D.
Let ${\tt d}$ be the density of a probability distribution that is compactly supported in the positive semi-axis. Under certain mild conditions we show that $$\lim_{x\to\infty}x\sum_{n=1}^\infty \frac{{\tt d}^{*n}(x)}{n}=1,\qquad\text{where}\quad {\tt d}^{*n}:=\underbrace{{\tt d} *{\tt d}*\cdots*{\tt d}}_{n\text{~times}}.$$ We also show that if $c>0$ is a given constant for which the function $f(k):=\widehat{\tt d}(k)-1$ does not vanish on the line $\{k\in\mathbb{C}:\Im k=-c\}$, where $\widehat{\tt d}$ is the Fourier transform of ${\tt d}$, then one has the asymptotic expansion $$\sum_{n=1}^\infty\frac{{\tt d}^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty),$$ where the sum is taken over those zeros $k$ of $f$ that lie in the strip $\{k\in\mathbb{C}:-c<\Im k<0\}$, $m(k)$ is the multiplicity of any such zero, and the implied constant depends only on $c$....

11. #### Functional equation for the Mordell-Tornheim multiple zeta-function

Okamoto, Takuya; Onozuka, Tomokazu
We show a relation between the Mordell-Tornheim multiple zeta-function and the confluent hypergeometric function, and using it, we give the functional equation for the Mordell-Tornheim multiple zeta-function. In the double case, the functional equation includes the known functional equation for the Euler-Zagier double zeta-function.

12. #### Resolving Grosswald's conjecture on GRH

McGown, Kevin; Treviño, Enrique; Trudgian, Tim
In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.

13. #### Counting lattice points in certain rational polytopes and generalized Dedekind sums

Kozuka, Kazuhito
Let ${\mathcal P} \subset {\mathbf R}^n$ be a rational convex polytope with vertices at the origin and on each positive coordinate axes. On the basis of the study on counting lattice points in $t{\mathcal P}$ with positive integer $t$, which is deeply connected with reciprocity laws for generalized Dedekind sums, we study the number of lattice points in the shifted polytope of $t{\cal P}$ by a fixed rational point. Certain generalized multiple Dedekind sums appear naturally in the main result.

14. #### An explicit result for primes between cubes

We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.

15. #### $abc$ triples

Martin, Greg; Miao, Winnie
The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we...

16. #### Polynomials with polar derivatives

Mir, Abdullah; Wani, Ajaz
In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

17. #### Small fractional parts of polynomials

Baker, Roger
Let $k \ge 6$. Using the recent result of Bourgain, Demeter, and Guth \cite{1586:bdg} on the Vinogradov mean value, we obtain new bounds for small fractional parts of polynomials $\alpha_kn^k + \cdots + \alpha_1n$ and additive forms $\beta_1n_1^k + \cdots + \beta_sn_s^k$. Our results improve earlier theorems of Danicic (1957), Cook (1972), Baker (1982, 2000), Vaughan and Wooley (2000), and Wooley (2013).

18. #### On extended Eulerian numbers

Bayad, Abdelmejid; Hernane, Mohand Ouamar; Togbé, Alain
In this paper, we will study some properties of the extended Eulerian numbers $H(n,\lambda)$, with a parameter $\lambda$. In fact, for any integer $n$, we investigate the asymptotic behavior, find lower and upper bounds for $H(n,\lambda)$. As application, for a champion number $N$, we obtain asymptotic formulas, lower and upper bounds of the arithmetic functions $\omega(N)$ and $\Omega(N)$.

19. #### The Picard groups of the stacks $\mathscr{Y}_0(2)$ and $\mathscr{Y}_0(3)$

Niles, Andrew
We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which $6$ is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a~wide variety of base schemes.

20. #### Exact divisors of polynomials with prime variable

Scourfield, Eira J.
In 1952 Paul Erdős obtained upper and lower bounds of the same order of magnitude for the number $N(x)$ of divisors of an irreducible polynomial $f(n)$ with integer coefficients for $n$ up to $x$; an asymptotic formula for $N(x)$ when $f$ has degree at least $3$ has not yet been established. However progress has been made in the corresponding problem when the divisors of $f(n)$ are restricted in some way and $f$ is not necessarily irreducible. In this paper we consider a~polynomial $f$ with integer coefficients that may not be irreducible or squarefree. Our aim is to obtain an asymptotic formula for the number of exact divisors...

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