Theory Of Zeta Functions Research Articles

On some additive problems of Goldbach’s type

In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the precise formulafor Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem each zerois solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the precise formulasfrom Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.

Обобщенная предельная теорема для периодической дзета-функции Гурвица

С времен Бора и Йессена (1910-1935) в теории дзета-функций прмменяются вероятностные методы. В 1930 г. они доказали первую теорему для дзета-функции Римана $$\zeta(s), $$ $$s=\sigma+it,$$ которая является прототипом современных предельных теорем, характеризующих поведение дзета-функции при помощи слабой сходимости вероятностных мер. Более точно, они получили, что при $$\sigma>1$$ существует предел $$lim_{T\to\infty} \frac{1}{T} \mathrm{J} \left\{t\in[0,T]: \log\zeta(\sigma+it)\in R\right\},$$ где R - прямоугольник на комплексной плоскости со сторонами, паралельными осям, а $$\mathrm{J}A$$ обозначает меру Жордана множества $$A\subset \mathbb{R}.$$ Два года спустя они распространили приведенный результат на полуплоскость $$\sigma>\frac{1}{2}.$$ Идеи Бора и Йессена были развиты в работах Винтнера, Борщсениуса, Йессена, Сельберга и других известных математиков. Современные версии теорем Бора-Йессена для широкого класса дзета-функций были получены в работах К. Матсумото. В основном теория Бора-Йессена применялась для дзета-функций, имеющих эйлерово произведение по простым числам. В настоящей статье доказывается предельная теорема для дзета-функций, не имеющих эйлерова произведения и являющихся обобщением классичесской дзета-функции Гурвица. Пусть $$\alpha, 0<\alpha \leqslant 1, $$ фиксированный параметр, а $$\mathfrak{a}=\{a_m: m\in \mathbb{N}_0= \mathbb{N}\cup\{0\}\}$$ - периодическая последовательность комплексных чисел. Тогда периодическая дзета-функция Гурвица $$\zeta(s,\alpha; \mathfrak{a})$$ в полуплоскости $$\sigma>1$$ определяется рядом Дирихле $$\zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty frac{a_m}{(m+\alpha)^s}$$ и мероморфно продолжается на всю комплексную плоскость. Пусть $$\mathcal{B}(\mathbb{C})$$ - борелевское $$\sigma$$-поле комплексной плоскости, $$\mathrm{meas}A$$ - мера Лебега измеримого множества $$A\subset \mathbb{R},$$ а функция $$\varphi(t)$$ при $$ t\geqslant T_0$$ имеет монотонную положительную производную $$\varphi'(t), $$ при $$t\to\infty$$ удовлетворяющую оценкам $$(\varphi'(t))^{-1}=o(t)$$ и $$\varphi(2t) \max_{t\leqslant u\leqslant 2t} (\varphi'(u))^{-1}\ll t. $$ Тогда в статье получено, что при $$\sigma>\frac{1}{2}$$ $$ \frac{1}{T} \mathrm{meas}\left\{t\in[0,T]: \zeta(\sigma+i\varphi(t), \alpha; \mathfrak{a})\in A\right\},\quad A\in \mathcal{B}(\mathbb{C}), $$ при $$T\to\infty$$ слабо сходится к некоторой в явном виде заданной вероятностной мере на $$(\mathbb{C}, \mathcal{B}(\mathbb{C})).$$

Non-backtracking walk centrality for directed networks

The theory of zeta functions provides an expression for the generating fu nction of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is determined by the spect rum of a three-by-three block matrix involving the original adjacency matrix. This giv es a means to choose appropriate values of the attenuation parameter. We find that three important a dditional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation para meter. Second, because the radius of convergence of the generating function is invariant under the remov al of certain types of nodes, we can gain computational efficiencies through reducing the dimension of t he resulting eigenvalue problem. Third, the dimension of the linear system defining the centrali ty measures may be reduced in the same manner. We show that the new centrality measure may be interp reted as standard Katz on a modified network, where self loops are added, and where nonreciproca l edges are augmented with negative weights. We also give a multilayer interpretation, wh ere negatively weighted walks between layers compensate for backtracking walks on the only non-emp ty layer. Studying the limit as the attenuation parameter approaches its upper bound allows us to propose an eigenvector-based nonbacktracking centrality measure in this directed network setting. We find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrat e the centrality measure on a synthetic network, where it is shown to eliminate a localization effect p resent in standard Katz centrality. Finally, we give results for real networks.

On the classification of prime-power groups by coclass

By giving a constructive proof of Conjecture P of Newman and O’Brien, we reduce the classification of certain p-groups of coclass r to a finite calculation. For the case p=2 we show that all 2-groups of coclass r can be classified by finitely many parametrised presentations. A non-constructive proof of Conjecture P was given by du Sautoy, using the theory of zeta functions. Our constructive proof uses homological algebra. It yields more precise results and detailed structure theorems for the p-groups under consideration.

Zeta distributions associated to a representation of a Jordan algebra

For a regular representation of a Euclidean Jordan algebra, we introduce multi-parameter zeta distributions with harmonic polynomial coefficient. Bernstein-Sato type identities are obtained and used to prove a functional equation. Examples are discussed in relation with Sato's general theory of zeta functions.

Calculation of the free energy of fullerenes in the Gross-Neveu model

Methods of field theory on Riemannian manifolds were used to study the Gross–Neveu model (incorporating the Hubbard model as a special case) at T ≠ 0. The generating functional of the Gross–Neveu model on a torus, S2r ⊗ S1β, was obtained by the functional integration method. The model was regularized using the theory of zeta functions. Double sums were calculated using recurrent formulas. For the zeta function of the Dirac operator in the limit of s = 1, we obtained a polar singularity of the form 1/(s – 1), characteristic of the local limit of Green's function. The free energy density was computed as a function of the radius of the sphere r and inverse temperature β using the Maple 6 pack. The results show no anomalies, indicating that there are no phase transitions to the principal order in 1/N. However, taking into account the kink–antikink configurations of the scalar field A(x) in calculations without the 1/N expansion may drastically change the phase structure of the model.

Zeta functions of algebraic cycles over finite fields

In this paper, we generalize the concept of a zeta function from zero cycles to higher dimensional cycles and investigate its meromorphic continuation and order of pole at t = 1 by means of a Riemann-Roch approach. We begin with an example which motivated the present work. A classical problem about finite fields is to find a formula for the number N~(d, n) of irreducible polynomials over Fq of degree d in n variables as d varies. This question goes back to Gauss (see the notes of Chapter 3 in [21]), who first found a formula in the case when there is only one variable and the finite field is a prime field Fp. Dickson generalized Gauss's formula from a prime finite field to a general finite field Fq. Since then, at tempts have been made to find a formula in the case of several variables. In 1936, Carlitz ([2]) found an analogous formula for the number of irreducible factorizable polynomials of degree d, where a polynomial over Fq is factorizable if it factors as a product of linear factors over the algebraic closure of Fq. This generalizes Gauss's formula in some sense. Later, Carlitz ([3-4]) established the asymptotic behavior of g'(d,n) (as d --, c~). Some of Carlitz's results were generalized by Cohen ([6-7I). Both Carlitz [3-4] and Cohen [6-7] said that no explicit formula for N~(d, n) is known in the several variable case. Using ideas from the theory of zeta functions, we found an explicit but complicated combinatorial formula for Nr n). The proof actually shows that in the several variable case, there is no combinatorial formula of the Gauss type; rather, there is a nice p-adic formula generalizing Gauss's formula. In some sense, this answers the question of Carlitz and Cohen. To understand this, we consider homogeneous polynomials. Let Nq (d, n) be the number of homogeneous irreducible polynomials over Fq of degree d in n

The significance of conformal inversion in quantum field theory

The 2-point functions of Euclidean conformal invariant quantum field theory are looked at as intertwining kernels of the conformal group. In this analysis a fundamental role is played by a two-element groupW, whose non-identity element ℛ=R·I consists of the conformal inversionR multiplied by a space-time reflectionI. The propagators of conformal invariant quantum field theory are determined by the requirement of ℛ-covariance. The importance of the ℛ-inversion in the theory of Zeta-functions is mentioned.

Zeta Functions and Green's Functions for Linear Partial Differential Operators of Elliptic Type with Constant Coefficients

in which T(nj) is an arbitrary polynomial of elliptic (will be defined) type of any even degree, and Q, is any further polynomial and we will demonstrate that these series really have sums, for any real variables xj and all complex s; they they are Cx and even analytic in the xj, and entire functions in s; except that for (xj) $ (0)-and of course for congruent points-there are some simple poles in the s-plane of the kind familiar from the theory of Zeta functions, Epstein Zeta functions, and such like expansions. There are no equations present however, at least not in an analytically fruitful manner, unless one chooses to interpret Theorem 11 as embodying a functional equation incipiently; and some remarks tending to substantiate such an interpretation for Theorem 11 have been presented in a separate note, see [1}. Actually, in our theorem proper, T(nj) and also Q(nj) will not be polynomials at all, but rather more general functions of non-oscillatory behavior reminiscent of such, and it will be syllogistically advisable to have these generalizations undertaken in the body of the statements, but we will not attempt to assess the gains from such generalizations in terms of concretely novel consequences thus resulting. If with a linear operator with constant coefficients h ,~~~~~~~~rl+ * +rkf (3) Af = (1) E a,, ...rk 1 . a~k O<rl+. +rk < 2h OX1 * an