Differential operators and entire functions with simple real zeros

Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz²?₁(z) and f(z)=e−βz²f₁(z), where ?₁ and f₁ have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation.     

Fourier transforms having only real zeros

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the Fourier transform has only real zeros.

     

Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan     

A generalization of Laguerre's theorems on zeros of entire functions

We prove some results generalizing the classical Laguerre theorems about the multiplicity and the number of zeros of the function

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

On the zeros ofζ (1)(s) — a (on the zeros of a class of a generalized Dirichlet series — XVII)

Some very precise results (see Theorems 4 and 5) are proved about thea-values of thelth derivative of a class of generalized Dirichlet series, forl≥l _o =l _o(a) (l _o being a large constant). In particular for the precise results on the zeros ofζ ⁽¹⁾ (s) —a (a any complex constant andl≥l _o) see Theorems 1 and 2 of the introduction.     

Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance between adjacent zeros of cylinder functions are given; and are such that ; stands for the th positive zero of the cylinder (Bessel) function , , .

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions and , give rise to forward ( ) and backward ( ) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions for any real and inside an interval , 0$">, is solved in a simple way.

     

A global Newton method for the zeros of cylinder functions

The zeros of cylinder functions C _u (x)=cos α, J _u (x) - sin α, Y _u(x) coincide with those of the ratios H _u (x)=C _u (x)/C _u-1 (x) except, perhaps, at x = 0. We show monotonicity properties of H u(x) and f _u (x) = x ^2v-1 H u(x) and their derivatives for x > 0. We then build a Newton-Raphson iterative method based on the monotonic function f u(x) which is shown to be convergent, for any real values of u and α and any starting value x ₀ > 0, to an sth positive root c ,s of C _u (x) = 0, s being such that c _,s and x0 belong to the same interval (c _u-1 ,s', c _{u -1} ,s'+1]. We also show applications of the method. In particular, taking advantage of the fact that the ratio H _u (x) for first kind Bessel functions J u(x) can be evaluated by using a continued fraction, a very simple algorithm is built; it becomes especially efficient for low values of u and s and it allows the evaluation of the real zeros for arbitrary orders u, positive or negative. This revised version was published online in August 2006 with corrections to the Cover Date.     

The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .

     

Trigonometric polynomials with many real zeros and a Littlewood-type problem

We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

     

Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers

Let {K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial f _m (x) = x ³ − mx ² − (m + 1)x − 1, where m is an integer with m ≥ 4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to K _m , and compute the values of the Dedekind zeta function of K _m . This work was supported by grant No.R01-2006-000-11176-0 from the Basic Research Program of KOSEF.     

Multicomplexes and polynomials with real zeros

We show that each polynomial a(z)=1+a₁z+?+a_dz^d in N[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen-Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of a(z) belong to the real interval [-1,0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form.     

Asymptotics of zeros of basic sine and cosine functions

We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given.     

Localization of the first zero of the Dedekind zeta function

Using Weil's explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.