## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (204.172 recursos)

Functiones et Approximatio Commentarii Mathematici

1. #### Diophantine properties of the sequences of prime numbers

Budarina, Natalia
The solvability over the ring of integers $\mathbb Z$ of some Diophantine equations is connected with the property of integers to form sequences of prime numbers, in particular, with the property of numbers to be twins. The Diophantine description of the sequences of prime numbers is obtained using the deformation method of quadratic matrix equations.

2. #### Rankin-Cohen Brackets on Hilbert Modular forms and Special values of certain Dirichlet series

Kumari, Moni; Sahu, Brundaban
Given a fixed Hilbert modular form, we consider a family of linear maps between the spaces of Hilbert cusp forms by using the Rankin-Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Hilbert cusp forms constructed using this method involve special values of certain Dirichlet series of Rankin-Selberg type associated to Hilbert cusp forms.

3. #### A short account of the values of the zeta function at integers

Huxley, Martin N.
We use methods of real analysis to continue the Riemann zeta function $\zeta(s)$ to all complex $s$, and to express the values at integers in terms of Bernoulli numbers, using only those infinite series for which we could write down an explicit estimate for the remainder after $N$ terms. This paper is self-contained, apart from appeals to the uniqueness theorems for analytic continuation and for real power series, and, verbis in Latinam translatis, would be accessible to Euler.

4. #### On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function

Ramakrishnan, B.; Sahu, Brundaban; Singh, Anup Kumar
In this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + x_3^2+ x_3x_4 + x_4^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G.A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan tau function.

5. #### A note on the congruences with sums of powers of binomial coefficients

Shen, Zhongyan; Cai, Tianxin
Let $p\geq 7$ be a prime, $l\geq 0$ be an integer and $k,~m$ be two positive integers, we obtain the following congruences, $\sum\limits_{s=lp}^{(l+1)p-1}\binom{kp-1}{s}^m\equiv\begin{cases}\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3},&\text{if}~2\nmid m,\\ \binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}},&\text{if}~2\mid m; \end{cases}$ and $\sum\limits_{s=lp}^{(l+1)p-1}(-1)^s\binom{kp-1}{s}^m\equiv\begin{cases}(-1)^l\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3}, &\text{if}~2\mid m,\\ (-1)^l\binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}}, &\text{if}~2\nmid m. \end{cases}$ Let $p$ and $q$ are distinct odd primes and $k$ be a positive integer, we have $\binom{kpq-1}{(pq-1)/2}\equiv \binom{kp-1}{(p-1)/2}\binom{kq-1}{(q-1)/2}\pmod {pq}.$

6. #### Derivation relations and duality for the sum of multiple zeta values

Li, Zhonghua
We show that the duality relation for the sum of multiple zeta values with fixed weight, depth and $k_1$ is deduced from the derivation relations, which was first conjectured by N. Kawasaki and T. Tanaka.

7. #### Gallagherian $PGT$ on $PSL(2,\mathbb{Z})$

Avdispahić, Muharem
Taking the Iwaniec explicit formula as a starting point, we bound the exponent in the error term of the prime geodesic theorem for the modular surface to $2/3$, outside a set of finite logarithmic measure.

8. #### MSTD sets and Freiman isomorphisms

Nathanson, Melvyn B.
An MSTD set is a finite set with more pairwise sums than differences. $(\Upsilon,\Phi)$-ismorphisms are generalizations of Freiman isomorphisms to arbitrary linear forms. These generalized isomorphisms are used to prove that every finite set of real numbers is Freiman isomorphic to a finite set of integers. This implies that there exists no MSTD set $A$ of real numbers with $|A| \leq 7$, and, up to Freiman isomorphism and affine isomorphism, there exists exactly one MSTD set $A$ of real numbers with $|A| = 8$.

9. #### On the arithmetic of translated monomial maps

Hajli, Mounir
Inspired by the work of Silverman on the geometry and the arithmetic of monomial maps and also on the translated maps on Abelian varieties, we generalize his results to the case of the translated monomial maps.

10. #### Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays

In this paper, we use the contraction mapping principle to obtain mean square asymptotic stability results of a nonlinear stochastic neutral Volterra-Levin equation with Poisson jumps and variable delays. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some previous results due to Burton [5], Becker and Burton [4] and Jin and Luo [10], Ardjouni and Djoudi [1]. Finally, an illustrative example is given.

11. #### Commensurability in Mordell-Weil groups of abelian varieties and tori

We investigate local to global properties for commensurability in Mordell-Weil groups of abelian varieties and tori via reduction maps.

12. #### The minimal number of monochromatic Schur tuples in a cyclic group

Taczała, Katarzyna
We discuss a question of Datskovsky [1] about the minimal number of solutions to Schur-type equation $x_1+\ldots+x_{2n-1}=x_{2n}$ in a cyclic group $\mathbb{Z}_N$. We provide lower and upper bounds for this quantity.

13. #### Points of order $13$ on elliptic curves

Kamienny, Sheldon; Newman, Burton
We study elliptically parametrized families of elliptic curves with a point of order $13$ that do not arise from rational parametrizations. We also show that no elliptic curve over $\mathbb{Q}(\zeta_{13})^+$ can possess a rational point of order $13$.

14. #### A short remark on consecutive coincidences of a certain multiplicative function

Kohnen, Winfried
We study integral solutions $n$ of the equation $A(n+k)=A(n)$, where $A$ is a certain multiplicative function related to Jordan's totient function.

15. #### Euclidean proofs for function fields

Lachmann, Thomas
Schur proved the infinitude of primes in arithmetic progressions of the form $\equiv l\mod m$, such that $l^{2}\equiv1\mod m$, with non-analytic methods by ideas inspired from the famous proof Euclid gave for the infinitude of primes. Ram Murty showed that Schur's method has its limits given by the assumption Schur made. We will discuss analogous for the primes in the ring $\mathbb{F}_{q}[T]$.

16. #### Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

Kritzer, Peter; Laimer, Helene; Pillichshammer, Friedrich
We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

17. #### Representation of a rational number as a sum of ninth or higher odd powers

Reynya, M.A
In the present paper, we substantially generalize one of the results obtained in our earlier paper [RM]. We present a solution of a problem of Waring type: if $F(x_1, \dots ,x_N)$ is a~symmetric form of odd degree $n\ge 9$ in $N=16\cdot 2^{n-9}$ variables, then for any $q\in \mathbb{Q}$, $q\neq 0$, the equation $F(x_i)=q$ has rational parametric solutions, that depend on $n-8$ parameters.

18. #### An extension theorem for generating new families of non-congruent numbers

Reinholz, Lindsey; Spearman, Blair K.; Yang, Qiduan
A technique for generating new families of non-congruent numbers by appending a tail of primes to extend known families of non-congruent numbers is presented. These new non-congruent numbers are comprised of arbitrarily many prime factors belonging to two or three odd congruence classes modulo 8.

19. #### On the value distribution of two Dirichlet $\boldsymbol L$-functions

Laaksonen, Niko; Petridis, Yiannis N.
Let $\rho$ denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of $L(\rho+\sigma,\chi_{1})$ and $L(\rho+\sigma,\chi_{2})$, where $\sigma\in[0,1/2)$ is fixed and $\chi_{1}$, $\chi_{2}$ are two fixed Dirichlet characters to distinct prime moduli. For $\sigma>0$ we prove that a positive proportion of these pairs of values are linearly independent over $\mathbb{R}$, which implies that the arguments of the values are different. For $\sigma=0$ we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive constant $c$.

20. #### On the Diophantine equation $y^{p} = f(x_{1}, x_{2}, ..., x_{r})$

Srikanth, Raghavendran; Subburam, Sivanarayanapandian
In this paper, we study the diophantine equation $$y^{p} = f(x_{1}, x_{2}, ..., x_{r}),$$ where $f(x_{1}, x_{2}, ..., x_{r})$ is a real polynomial in variables $x_{1}, x_{2}, ..., x_{r}$ in $R$, a group of real numbers under the usual addition $+$, having the least element property.

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