Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡ _N (z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡ _N (z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.     

On a problem connected with zeros of ζ(s) on the critical line

The existence of zeros ofZ ^(k)(t) in short intervals of the type [T, T+H] is established, whereHT ^a(k)logT, . Hitherto the sharpest bounds for the constanta(k) are obtained by employing a certain exponential averaging technique and the estimation of the relevant exponential sums. Bounds for are also derived, under the assumption that orZ(t) does not vanish in certain short intervals.     

On the Relationship between the Multidimensional Dirichlet Divisor Problem and the Boundary of Zeros of ζ(s)F

Mathematical Notes -     

A short remark on Fibonacci-type sequences,Möbius strips and the ψ-function

A base for linear recursive sequences, such as the sequence of Fibonacci numbers, is defined within the framework of the sum of the digits of a number. Examples of bases of a number of such sequences are then outlined, and a Möbius strip is also used to illustrate the effects diagrammatically.     

The Perturbation ϕ2, 1 of the M(p,p + 1) Models of Conformal Field Theory and Related Polynomial-Character Identities

Using q-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p + 1) models of conformal field theory perturbed by the operator _{2, 1}. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime 1⁺ of A _p–1 dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for all Virasoro characters of unitary minimal series.     

Jacob’s Ladders,Interactions between ζ-Oscillating Systems,and a ζ-Analogue of an Elementary Trigonometric Identity

In our previous papers, within the theory of the Riemann zeta-function we have introduced the following notions: Jacob’s ladders, oscillating systems, ζ-factorization, metamorphoses, etc. In this paper we obtain a ζ-analogue of an elementary trigonometric identity and other interactions between oscillating systems.     

The zeros of the Weierstrass $$\wp$$–function and hypergeometric series

We express the zeros of the Weierstrass -function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian ₄ F ₃–evaluation. Research of W. Duke was supported in part by NSF Grant DMS-0355564. He wishes to acknowledge and thank the Forschungsinstitut für Mathematik of ETH Zürich for its hospitality and support.     

Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.     

RESEARCH ANNOUNCEMENTS On the Least Pair of Quadratic Non-Residues (mod p)

For each odd prime p,let a(p) denote the least positive residue class n(mod p),so that both n and n+1 are quadratic non-residues.The previously best knownestimate for a(p) obtained by Elliott is as follows     

On the compatibility between cup products,the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture,II

We give a different proof of the famous result on compatibility between cup product (Kontsevich, 2003, [3, Section 8]) in cohomology of degree 0, for a finite-dimensional Lie algebra, from which we deduce an alternative way of re-writing Kontsevich?s star product by means of the Alekseev–Torossian connection (Alekseev and Torossian, 2010, [1]).     

A Binary Additive Problem of Erdös and the Order of 2 mod p2

We show that the problem of representing every odd positive integer as the sum of a squarefree number and a power of 2, is strongly related to the problem of showing that p² divides 2^{p-1}-1 for few primes p.     

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

Distribution of zeros of the Hermite-Padé polynomials for a system of three functions,and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\Re _3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3$ that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ?? \ B of the open (possibly, disconnected) set B ? ?? introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.     

Benford’s law and distribution functions of sequences in (0, 1)

Applying the theory of distribution functions of sequences x _n ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence x _n and show that the satisfaction of this functional equation for a sequence x _n is equivalent to the fact that the sequence x _n to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.