Vol 296, No 1 (2017)

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.

We prove estimates for complete rational arithmetic sums of Bernoulli polynomials whose arguments are formed by the fractional parts of values of a polynomial with rational coefficients. The results are applied to the problem of finding the convergence exponent for the mean values of the sums under consideration.

Let N₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ iT]. It is proved that N₀(T) ≫ T (ln T)^{1/2+1/16−ε}, where ε is an arbitrarily small positive number.

This is primarily an overview article on some results and problems involving the classical Hardy function Z(t):= ζ(1/2 + it)(χ(1/2 + it))^−1/2, ζ(s) = χ(s)ζ(1 − s). In particular, we discuss the first and third moments of Z(t) (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.

It is proved that the denominators of finite continued fractions all of whose partial quotients belong to an arbitrary finite alphabet A with parameter δ > 0.7807... (i.e., such that the set of infinite continued fractions with partial quotients from this alphabet is of Hausdorff dimension δ with δ > 0.7807... ) contain a positive proportion of positive integers. Earlier, a similar theorem has been obtained only for alphabets with somewhat greater values of δ. Namely, the first result of this kind for an arbitrary finite alphabet with δ > 0.9839... is due to Bourgain and Kontorovich (2011). Then, in 2013, D.A. Frolenkov and the present author proved such a theorem for an arbitrary finite alphabet with δ > 0.8333.... The preceding result of 2015 of the present author concerned an arbitrary finite alphabet with δ > 0.7862....

Let Δ(x) and E(x) denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for ζ(s) on the critical line. We consider some general mean values for Δ(x) and E(x) and discover interesting differences between these two functions. In particular, this yields evidence that E(x) is more negative than Δ(x).

In 2007, H. Mishou obtained a joint universality theorem for the Riemann zetafunction ζ(s) and the Hurwitz zeta-function ζ(s, α) with transcendental parameter α. The theorem states that a pair of analytic functions can be simultaneously approximated by the shifts ζ(s + iτ ) and ζ(s + iτ, α), τ ∈ R. In 2015, E. Buivydas and the author established a version of this theorem in which the approximation is performed by the discrete shifts ζ(s + ikh) and ζ(s + ikh, α), h > 0, k = 0, 1, 2.... In the present study, we prove joint universality for the functions ζ(s) and ζ(s, α) in the sense of approximation of a pair of analytic functions by the shifts ζ(s + ik^βh) and ζ(s + ik^βh, α) with fixed 0 < β < 1.

An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Tur´an proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham’s theorem and its conversion by Tur´an. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term.

An asymptotic formula is obtained in an additive problem with the coefficients of Hecke L-functions. The formula is uniform with respect to the parameters of the problem.

The purpose of this note is to prove two results on the quotient sets A/A of finite sets A ⊂ [1, n] of positive integers. They complement the results from the paper by J. Cilleruelo, D.S. Ramana, and O. Ramaré.