We prove an extension of the Landau-Gonek formula. As an application, unconditionally, we recover an important consequence of Montgomery's work on the pair correlation that previously was known under ...the Riemann Hypothesis. We proceed to show that at least two-thirds of the zeros of the zeta function are simple under a milder assumption than the Riemann Hypothesis. We will also prove a corollary on the value-distribution of Dirichlet polynomials.
We establish a new decoupling inequality for curves in the spirit of earlier work of C. Demeter and the author which implies a new mean value theorem for certain exponential sums crucial to the ...Bombieri-Iwaniec method as developed further in the work of Huxley. In particular, this leads to an improved bound \vert\zeta (\frac {1}{2} + it)\vert \ll t^{13/84 + \varepsilon } for the zeta function on the critical line.
In this article, we extend the result of Conrey 4, Theorem 2 to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order ...derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery 13, Ki and Lee 11 on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.
The paper concerns the problem on algebraic difference independence of two functions which come from two different sets G and F, respectively. The first set contains the gamma function and the other ...set contains Riemann zeta function. Meanwhile, a result is obtained which generalizes a theorem of Li-Tahir-Gao in 22 from the gamma function to the function in G.
We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on σ=12+(logT)−θ and t∈T,2T, where 0<θ<12 is a constant.
In this work we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if (an)n=0∞ is a greedy s-energy sequence on the unit circle, the Riesz ...potential UN,s(x):=∑k=0N−1|ak−x|−s, s>0, generated by the first N points of the sequence attains its minimum value at the point aN, for every N≥1. In the case s=0 we minimize instead the logarithmic potential UN,0(x):=−∑k=0N−1log|ak−x|. We analyze the asymptotic properties of these extremal values UN,s(aN), studying separately the cases s=0, 0<s<1, s=1, and s>1. We obtain second-order asymptotic formulas for UN,s(aN) in the cases s=0, 0<s<1, and s=1 (the corresponding first-order formulas are well known). A first-order result for s>1 is proved, and it is shown that the normalized sequence UN,s(aN)/Ns is bounded and divergent in this case. We also consider, briefly, greedy energy sequences in which the minimization condition is required starting from the point ap+1 (instead of the point a1 as previously stated), for some p≥1. For this more general class of greedy sequences, we prove a first-order asymptotic result for 0≤s<1.
In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two non-zero neighbouring Bernoulli numbers and analyses the ...approximating accuracy of the double inequality.
Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of S(t), S1(t), and ζ(1/2+it) while comparing them with recently proven unconditional ones. As a corollary we obtain a ...conditional explicit bound on gaps between consecutive zeros of the Riemann zeta-function.
In recent years, studying degenerate versions of some special polynomials, which was initiated by Carlitz in an investigation of the degenerate Bernoulli and Euler polynomials, regained lively ...interest of many mathematicians. In this paper, as a degenerate version of polyexponential functions introduced by Hardy, we study degenerate polyexponential functions and derive various properties of them. Also, we introduce new type degenerate Bell polynomials, which are different from the previously studied partially degenerate Bell polynomials and arise naturally in the recent study of degenerate zero-truncated Poisson random variables, and deduce some of their properties. Furthermore, we derive some identities connecting the polyexponential functions and the new type degenerate Bell polynomials.
Inspired by the work of Balasubramanian, Conrey and Heath-Brown 1, we obtain an asymptotic expression for the mean of the product of any two finite order derivatives of Hardy's Z-function times ...Dirichlet polynomials in short intervals.
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