Trivial Zero Research Articles

Local zeta factors and geometries under

The first part of this note shows that the odd-period polynomial of each Hecke cusp eigenform for the full modular group produces via the Rodriguez-Villegas transform ([1]) a polynomial satisfying the functional equation of zeta type and having non- trivial zeros only in the middle line of its critical strip. The second part discusses the Chebyshev lambda- structure of the polynomial ring as Borger's descent data to and suggests its role in a possible relation of the -factor to `real geometry over ' (cf. [2]).

Functional equations for double series of Euler type with coefficients

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.

Multiple Solutions of Riemann Type of Functional Equations

Linearly independent Dirichlet L-functions satisfying the same Riemann-type of functional equation have been supposed for long time to possess off critical line non trivial zeros. We are taking a closer look into this problem and into its connection with the Generalized Riemann Hypothesis

On Davenport and Heilbronn-Type of Functions

A correction is brought to the opinion expressed in a previous note published in this journal that the off critical line points indicated by some authors as being non trivial zeros of the Davenport and Heilbronn function are affected of approximation errors and illustrations are presented which enforce the conclusion that they are true zeros. It is shown also that linear combinations of L-functions satisfying the same Riemann-type of functional equation do not offer counterexamples to RH, contrary to a largely accepted position.

Zero distribution of Dirichlet L-functions

The series Lχ converges absolutely and uniformly in the region Re(s) ≥ 1 + e, for any e > 0. It therefore represents a holomorphic function on the half-plane Re(s) > 1, which further extends to a meromorphic function in the complex plane C. In particular, for the principal character χ = 1, we get back the Riemann zeta function ζ(s). For a ∈ C, the zeros of L χ − a which we denote by ρa = βa + iγa, are called the a-points of Lχ, and their distribution has long been an object of study (see [17, 18, 20]). The function Lχ has only real zeros in the half plane Re(s) < 0, these zeros are called the trivial zeros. If χ(−1) = 1, the trivial zeros of Lχ are s = −2n for all non-negative integers n. If χ(−1) = −1, the trivial zeros of Lχ are s = −2n − 1 for all non-negative integers n. Beside the trivial zeros of Lχ, there are infinitely many non-trivial zeros lying in the strip 0 < Re(s) < 1. For a 6= 0, it can be shown that there is always a a-point in some neighbourhood of any trivial zero of Lχ with sufficiently large negative real part, and with finitely many exceptions there are no other in the left half-plane, thus the number of these a-points having real part in [−R, 0] is asymptotically 1 2 R. The remaining a-points all lie in a strip 0 < Re(s) < A, where A depends on a, and we call these non-trivial a-points (see [16, 19]). For a positive number T , let N χ(T ) denote the number of non-trivial a-points ρa = βa+ iγa of Lχ with |γa| ≤ T . We have the following formula

A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

Let f be a modular form of weight k and Nebentypus $\psi$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes \xi)$, where $\xi$ is a Dirichlet character. When s=k-1 and $\xi=\psi^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implies a conjecture of Benois.

On Equalities Involving Integrals of the Logarithm of the Riemann ς-Function with Exponential Weight Which Are Equivalent to the Riemann Hypothesis

Integral equalities involving integrals of the logarithm of the Riemann ς-function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(−2πt). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.

Kink excitation spectra in the (1+1)-dimensional φ8 model

We study excitation spectra of BPS-saturated topological solutions — the kinks — of the φ8 scalar field model in (1 + 1) dimensions, for three different choices of the model parameters. We demonstrate that some of these kinks have a vibrational mode, apart from the trivial zero (translational) excitation. One of the considered kinks is shown to have three vibrational modes. We perform a numerical calculation of the kink-kink scattering in one of the considered variants of the φ8 model, and find the critical collision velocity vcr that separates the different collision regimes: inelastic bounce of the kinks at vin ≥ vcr, and capture at vin < vcr. We also observe escape windows at some values of vin < vcr where the kinks escape to infinity after bouncing off each other two or more times. We analyse the features of these windows and discuss their relation to the resonant energy exchange between the translational and the vibrational excitations of the colliding kinks.

Constraints of distance from boundary to skeleton: For the control of length scale in level set based structural topology optimization

Abstract A method is proposed for the control of minimum/maximum length scale in the level set based structural topology optimization. The minimum/maximum length scale of structure is characterized by using the concept of smallest/biggest maximal inscribable ball. In order to prevent trivial zero minimum length scale, the skeleton of structure is utilized and trimmed. The control of length scale is realized by constraining the distance from boundary to skeleton, and the distance is explicitly constructed by using the highly efficient fast marching method. Numerical examples in two dimensions are investigated.

One-level density of families of elliptic curves and the Ratios Conjecture

Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This is a new phenomenon for a family of curves with odd rank: the trivial zero at the central point accounts for the Dirac distribution, and also affects the remaining part of the scaling density which is then (maybe surprisingly) the even orthogonal distribution. The one-level density for this family was studied in the past for test functions with Fourier transforms of limited support, but since the Fourier transforms of the even orthogonal and odd orthogonal distributions are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the Ratios Conjecture, and it sheds more light on “independent” and “non-independent” zeroes, and the repulsion phenomenon.

Semifluxons in 0–π–0 infinitely long Josephson junctions

We investigate both analytically and numerically an infinitely long Josephson junction with two π-discontinuities in the phase characterized by a jump of π in the phase difference of the junction, i.e. a 0–π–0 long Josephson junction on an infinite domain. The dynamics of the system is described by a modified one-dimensional perturbed sine-Gordon equation. We investigate an instability region of the trivial zero solution in which semifluxons are spontaneously generated. A perturbation technique is used to show that the existence of static semifluxons depends on the length of the junction, the facet length, and the applied bias current. Numerical simulations are presented accompanying our analytical results.

Note on the Zeros of a Dirichlet Function

The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Contrary to what has been presumed for a long time, we prove that such zeros cannot exist.

On Pellarin’s $L$-series

Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of $L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We determine the exact degree of the special polynomials associated to Pellarin’s $L$-series. The theory of Carlitz polynomial approximations is developed further for both additive and $\mathbb {F}_q$-linear functions. Using Carlitz’s theory we give a generating series for the power sums occurring as the coefficients of the special polynomials associated to Pellarin’s series, and a connection is made between the Wagner representation for $\chi _t$ and the value of Pellarin’s $L$-series at 1.

An exact degree for multivariate special polynomials

We introduce certain special polynomials in an arbitrary number of indeterminates over a finite field. These polynomials generalize the special polynomials associated to the Goss zeta function and Goss–Dirichlet L-functions over the ring of polynomials in one indeterminate over a finite field and also capture the special values at non-positive integers of L-series associated to Drinfeld modules over Tate algebras defined over the same ring. We compute the exact degree in t0 of these special polynomials and show that this degree is an invariant for a natural action of Goss' group of digit permutations. Finally, we characterize the vanishing of these multivariate special polynomials at t0=1. This gives rise to a notion of trivial zeros for our polynomials generalizing that of the Goss zeta function mentioned above.

Shintani cocycles and the order of vanishing of $$p$$ p -adic Hecke $$L$$ L -series at $$s=0$$ s = 0

Let \(\chi \) be a Hecke character of finite order of a totally real number field \(F\). By using Hill’s Shintani cocycle we provide a cohomological construction of the \(p\)-adic \(L\)-series \(L_p(\chi , s)\) associated to \(\chi \). This is used to show that \(L_p(\chi , s)\) has a trivial zero at \(s=0\) of order at least equal to the number of places of \(F\) above \(p\) where the local component of \(\chi \) is trivial.

Trivial zeros of -adic -functions at near-central points

AbstractUsing the$\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of$p$-adic$L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

Analytic continuation of the Lucas zeta and [formula omitted]-functions

We study the Lucas zeta function defined by using the Lucas sequence which is a generalization of the Fibonacci sequence. This zeta function can be meromorphically continued to the whole complex plane, and in a special case, it has “trivial zeros” like the Riemann zeta function. Analogues of Dirichlet’s L-functions are also investigated.

Dérivée en [formula omitted] de la fonction L p-adique du carré symétrique dʼune courbe elliptique sur un corps totalement réel

We prove a formula for the derivative of the p-adic L-function associated with the symmetric square representation of an elliptic curve over a totally real field in which p is inert, under certain assumptions on the conductor. In particular, this proves a conjecture of Greenberg on trivial zeros. The method is to generalize unpublished calculations of Greenberg and Tilouine.

The Exceptional Zero Conjecture for Symmetric Powers of CM Modular Forms: the Ordinary Case

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.

On the connected stability of the large-scale systems of dynamical equations on the time scale in critical cases

The connected stability of a large-scale system of dynamical equations on the time scale is investigated. The case where the trivial zero solution of the system is non-asymptotically stable in the linear approximation is considered. The sufficient conditions for the connected asymptotic stability of the zero solution of the system are obtained on the basis of the Lyapunov direct method.