Computing 𝜋(𝑁): An elementary approach in 𝑂̃(√𝑁) time

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
 Subject Classification: 97F60, 11A25

Several authors (e.g., see [8]) have considered the Fibonacci numbers F x where the subscript x is an arbitrary real number and showed that these (complex) numbers enjoy most of the properties of the usual Fibonacci numbers F m (m integral). A quite natural extension of the numbers F x leads to the definition of the Fibonacci numbers F z and Lucas numbers L z $$ {F_z} = \left( {{\alpha ^z} - {\beta ^z}} \right)/\sqrt 5 $$ (1.1) $$ {L_z} = {\alpha ^z} + {\beta ^z}, $$ (1.2) where the subscript z is an arbitrary complex number and \( \alpha = - 1/\beta = \left( {1 + \sqrt 5 } \right)/2 \)

It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\), where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation.

The theory of non-selfadjoint differential operators has great importance in several applications. Developing a fruitful method of V. A. II'in [1], in [21 I. Jod described a new method which makes possible to avoid the use of the basic solution in such investigations. Using this method, a general equiconvergence theorem was proved in [9] for the SchrSdinger operator, having generalized the results of the papers [5], [6], [7] and [8]. This theorem concerns the expansions by Riesz bases formed by eigenfunctions of higher order of the SchrSdinger operator (but seems to be new also for the case of orthonormal bases). The existence of such Riesz bases was proved for a wide class of operators (also of higher order) by V. P. Mikhailov [10] and G. M. Keselman [111. Later on, the method of [2] was generalized for the case of differential operators of higher order in [41, [12] and [13]. These generalizations are based on a new formula which extends the well-known mean-value formula of E. C. Titchmarsh for more general operators. Using the method and the results of these papers, we shall prove a general equiconvergence theorem for linear differential operators of higher order. In this theorem the eigenvalues of the Riesz basis may be arbitrary complex numbers. Our result is new also for orthonormal bases.

Diagonalizability of operators occurring in linear transport theory is discussed from a general point of view. In particular, the operator A−1T, where T is the multiplication operator in L2 (−1, 1) and A is given by a formula of the type A f = f − Σ o n aj pj , is investigated. Diagonalization of this operator which is connected with one-group neutron transport is carried out in the general case that the coefficients aj are arbitrary complex numbers. Also, a peculiarity of multi-group theory, where the operator involved has a multiple continuous spectrum, is pointed out. A correct interpretation of the main result of that theory is provided.

Let P(z) be a polynomial of degree n 1 . In this paper we consider a more general problem of investigating the dependence of maximum of ∣∣∣P(Rz) − αP(z) + β {( R + 1 2 )n − |α| } P(z) ∣∣∣, R 1, on the maximum of |P(z)| on |z| = 1 where α , β are arbitrary complex numbers with |α| 1 , |β | 1 and obtain certain sharp compact generalizations of well-known Bernsteintype polynomial inequalities. Mathematics subject classification (2000): 30C10, 30A10.

Let {m0} be a strictly increasing sequence of positive integers. S. Chowla (1965) conjectured that n~~~~~~~~~~ min cos 27rmtkX O being an absolute constant. Let K1, K2, , KKN be the distinct integers ml-Mk, 1?k<l_n; r,, 1_j_N, the number of pairs (k, 1) with 1 ?k<lfn and mnl-mk=Kj; and r(n) = max r,. 1< j<N Lower bounds for Jo Cke 27me2rTm A' dx, Ck, arbitrary complex numbers and f J 1 Yk cos 27r(nlkXl+Xk) I dx, Yk _ O, Xk real, are obtained in terms of n, r(n) and the Ck, and Yk respectively and it has been deduced that in case r(n)=(5, independent of n, then 1 1~~~~~/ min cos 27rn1kX < n_ _ _ _ l_ O<x<l k=1 25/2 ((5 + 1)1/2

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, $$D: = \frac{d}{{dx}}$$ and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre $$\{L_n^{(2)}\}_{n\geq0}$$ and the Jacobi $$\{P_n^{(\alpha-2,2)}\}_{n\geq0}$$

The recently secured mathematical formalism of direct methods is here generalized to the case that the atomic scattering factors are arbitrary complex numbers, thus including the special case that one or more anomalous scatterers are present. Once again the neighborhood concept plays the central role. Final results from the probabilistic theory of the two- and three-phase structure invariants are briefly summarized. In particular, the conditional probability distribution of the three-phase structure invariant, given the six magnitudes |E| in its first neighborhood, is described. The distribution yields an estimate for the three-phase structure invariant which is particularly good in the favorable case that the variance of the distribution happens to be small (the neighborhood principle). Particularly noteworthy is the fact that, in sharp contrast to all earlier work, the estimate is unique in the whole range 0 to 2π. An example shows that the method is capable of yielding unique estimates for tens of thousands of three-phase structure invariants with unprecedented accuracy, even in the macromolecular case. The clear implication is that the fusion of the traditional techniques of direct methods with anomalous dispersion, which is described here, will facilitate the solution of those crystal structures which contain one or more anomalous scatterers.

Suppose throughout that l, a n ( n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

The coupling of sound fields through a finite-sized aperture in a plane rigid baffle where the region (half-space) on one side is unbounded can be described by an integral equation which constitutes a boundary condition for the field on the other side of the aperture. Such a boundary condition, when the pressure and the normal velocity are expanded in basis functions defined over the aperture, can be recast into a matrix form relating the coefficients of the basis functions in the expansions, the principal feature being a matrix of fourfold (double-area) integrals analogous to those encountered in studies of radiation from flexible pistons in rigid baffles. A substantial analytical reduction to sums of single nonsingular integrals is derived for the elements of this radiation impedance matrix when the aperture is rectangular and the basis functions are expressible as a sum of products of exponential functions of the Cartesian coordinates of the aperture plane, with the exponential coefficients being arbitrary complex numbers. The validity of the result is substantiated by its reduction to previously published results for less general cases. Its utility is demonstrated with the example of diffraction by a square hole in a screen.

Suppose throughout that s, an (n=0,1, 2,…) are arbitrary complex numbers, that α&gt;0 and β is real and that N is a non-negative integer such that αN+β≧l. Letwhere z=x+iy is a complex variable and the power zr is assumed to have its principal value.

A class of exponential sums studied by D. H. Lehmer and Emma Lehmer is generalized by replacing a root of unity by an arbitrary complex number x .

Domain of meromorphy of a class of Bateman-like totient zeta functions