We bound the total variation distance in the Kubilius model for sequences with positive level of distribution. We obtain a result that we expect is qualitatively optimal. As a special case, it recovers a recent result of Ford on shifted primes, with a slightly simplified proof. In the classical case considered by Kubilius, our theorem gives a simple proof of the optimal bound discovered by Tenenbaum, up to factors of x^o(1) and u^o(u) .
We study an analogy between classical modular forms with Nebentypus characters and degree 2 L -functions in the Selberg class. We prove that every normalized degree 2 L -function with integer conductor, admitting a degree 2 Euler product and stable under twists by Dirichlet characters, possesses a Dirichlet character ψ that plays the role of a Nebentypus character. This result provides evidence for the general conjecture that the elements of integer degree in the Selberg class correspond to automorphic representations of arithmetic groups. Moreover, our method extends to functions in the extended Selberg class 𝒮^♯ that may lack a global Euler product but exhibit multiplicative behavior at selected primes, yielding a characterization of their local Euler factors in terms of an as- sociated Dirichlet character.
We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for L -functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of L -functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the Möbius function, but we also outline the utility of this method in general.
The Hardy-Littlewood-Sobolev inequalities are a distinguished class of functional inequalities, with applications in the theory of nonlinear PDEs and mathematical physics. Those inequalities are known in sharp form, after the work of the existence of optimisers proved by Lieb in 1983. More recently, reverse Hardy-Littlewood-Sobolev inequalities have been established by Dou and Zhu in 2015, in sharp form, together with the existence of optimisers. In this paper, we investigate analogous statements in bounded domains, more specifically, balls and annuli (for different regimes of the parameters involved in the inequalities). The estimates from above and below for the sharp constants of the inequalities are obtained. These bounds of the best constant are helpful to well understand the graviton self-energy and the Coulomb energy in the Thomas-Fermi model describing electron gas.
In Guy’s monograph on unsolved problems in number theory, a remarkable series for Apéry’s constant ζ(3) of convergence rate 1/16 discovered by Gosper was highlighted in the context of series expansions associated with the Riemann zeta function. We introduce the first known Wilf–Zeilberger proof of the Gosper formula for ζ(3) , and we apply a variant of the required WZ pair to accelerate the rate of convergence of a family of _3F_2(1) -series from 1 to 1/16 . Our evaluations build upon the work of Chu on series of convergence rate 1/16 obtained via an extended version of the Gould–Hsu inverse series relations, and our methods have the advantage of producing series of the given convergence rate with linear and quadratic polynomial factors within the summand, by analogy with Ramanujan’s series for 1/π and Gosper’s series for ζ(3) and Guillera’s series for 1/π^2 .
In this paper, we establish estimates for the expectation and variance of the restricted decorrelation between two Hecke eigenforms along vertical geodesic segments and closed horocycle segments on the modular surface.
We study a special case of the Markov-Nikol’skii inequality for the class of algebraic polynomials with complex coefficients that do not vanish in the disc. This inequality estimates the L^q -mean of the derivative of order k of a polynomial on the interval [-1,1] from above by the L^0 -mean (geometric mean) of the polynomial itself on the same interval. We obtain the exact constant in this inequality for q≥2 and arbitrary k and characterize all extremal polynomials. Additionally, for the class of polynomials with zeros in the closed disk, we study the extremal case of Turán's inequality when the L^r -mean on [-1,1] of the derivative of order k=n (the degree of the polynomial) is estimated from below by the L^q -mean of the polynomial itself on the interval. We obtain the exact constant in this inequality for all 0 ≤ r, q ≤∞ and describe the set of extremal polynomials.
The L^1 -norm of kernel functions plays a central role in studying the convergence properties of the Walsh-Fourier system. Although when viewed as a sequence, in many cases only their boundedness matters, the precise values of the individual terms can also be of interest. It is known that for all N ∈ℕ , the L^1 -norms of the 2^N -th and (2^N+1 - 1) -th kernel functions are exactly 1. In this paper, we prove that these are the only such cases: for every other n ∈ℙ , the strict inequality K_n_1 > 1 holds. Finally, we present additional closed-form expressions for selected subsequences of K_n_1 as illustrative examples, using the recursive formula of Toledo [8].
In the present paper, we construct theta functions with two parameters a, b ∈ℝ that satisfy Jacobi's modular relation. Furthermore, we define zeta functions, also depending on two real parameters a, b ∈ℝ and which are derived from these theta functions, which satisfy Riemann's functional equation. To the best of our knowledge, these zeta functions are the first known examples that simultaneously satisfy Riemann's functional equation and involve two independent parameters.
We obtain a Picard-type theorem for holomorphic curves. By introducing a derivative that extends the spherical derivative of meromorphic functions, we obtain an extension of Marty's Theorem and the Zalcman-Pang Lemma to a family of holomorphic curves. Finally, as an application of these results, we obtain a normality criterion for a family of holomorphic curves whose derived curves satisfy certain conditions.
In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note [9] we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery’s proof that on assuming all the zeros are in a narrow vertical box between height T and 2T of width b/log T and centered on the critical line, then, if b=b(T)→ 0 as T→∞ , we have asymptotically at least 2/3 of the zeros are simple and on the critical line.
We study the behavior of the smallest possible constant d(a,b) in Hardy inequality ∫_a^b(1/x∫_a^xf(t) dt )^p dx≤ d(a,b)∫_a^b [f(x)]^p dx. The exact rate of convergence of d(a,b) is established and the “almost extremal” function is found.
We prove that the best constant in the weak type (1, 1) inequality for the ρ -variation operator associated with the Poisson semigroup grows at most like O(n^3/2) . Moreover, we provide a limit result for the weak type (1, 1) bound.
In this paper, we introduce a generalization and refinement of the Cauchy-Schwarz inequality within the framework of positive functionals. By employing elementary techniques, we sharpen the Cauchy-Schwarz inequality for real numbers, specifically for a particular class of unital positive functionals. Additionally, our results yield a variety of related inequalities, demonstrating the broader applicability of this approach.
Our aim in this paper is to derive some Turán type inequalities for modified big q -Bessel functions. Furthermore, some Turán type inequalities of sections for series of aforementioned functions are established. The method is based on proving monotonicity for the special ratio of sections for series of such functions.
We introduce various global smallness conditions for Riemann sums within the vector-valued Riemann-measurable function class. In terms of these conditions, we clarify to some extent the nature of the differences between the absolute Birkhoff, McShane and Henstock integrals from one side and the A-Riemann, A- and Q-integrals from the other side. In particular, we prove that the McShane and A-Riemann integrals do not contradict one another.
