Andrés Chirre
Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $ζ'/ζ$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery's pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of $ζ'/ζ$ was considered.
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Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line
本文研究了黎曼ξ函数在临界线上方区域的实部对数导数正负性，并给出了显式下界及假设非平凡零点存在的情况。
A note on the zeros of the derivatives of Hardy's function $Z(t)$
本文利用黎曼 zeta 函数的扭曲四阶矩改进哈代函数导数零点间大间隙的结果，并用光滑化离散二阶矩证明其偶数阶导数与 $Z(t)$ 零点的小距离现象。
A note on log-type GCD sums and derivatives of the Riemann zeta function
本文证明了对数型GCD和的上界，推广了Gál定理，并分别用无条件的谱范数方法和基于黎曼zeta函数导数条件上界的方法给出两种证明。
We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is fin…
We investigate the large values of the derivatives of the Riemann zeta function $ζ(s)$ on the 1-line. We give a larger lower bound for $\max_{t\in[T,2T]}|ζ^{(\ell)}(1+{\rm i} t)|$, which improves the previous result established by Yang.
Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $ζ$-function to be a Lehmer pair in terms of derivatives of Hardy's $Z$-function. Furthermore, we connect Newman's conjecture with stationary points of the $Z$-function, and present some numerical results.
In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $ζ'(ρ)$ where $ζ(s)$ is the Riemann zeta function and $ρ$ is a non-trivial zero of the Riemann zeta function.
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region. Theorem 2 gives an asymptotic estimate for the number of nontrivial zeros to height T. Theorem 3 is a zero density estimate.
In this brief note we connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.
Yıldırım obtained an asymptotic formula of the discrete moment of $|ζ(\frac{1}{2}+it)|$ over the zero of the higher derivatives of Hardy's $Z$-function. We give a generalization of his result on Hardy's $Z$-function.
Mean Values of the Logarithmic Derivative of the zeta Function and the GUE Hypothesis
GUE假设涉及到黎曼ζ函数零点的分布，它被用来计算一些涉及zeta函数对数导数的积分。GUE假说与其他假说之间存在一定的联系。
In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $μ\geq1$ an integer, we compute the mean value of the $μ$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order te…
We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.
Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional equation and mean square formula of $L_f(s)$. In this paper, we shall generalize these formulas for the derivatives of $L_f(s)$.
Assuming the Riemann hypothesis, we prove that $$ N_k(T) = \frac{T}{2π}\log \frac{T}{4πe} + O_k\left(\frac{\log{T}}{\log\log{T}}\right), $$ where $N_k(T)$ is the number of zeros of $ζ^{(k)}(s)$ in the region $0<\Im s\le T$. We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo.
For a number field $K$, let $ζ_{K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $ζ_{K}$ as well as its derivative $ζ_{K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^γ$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].
In unpublished notes, Pila discussed some theory surrounding the modular function $j$ and its derivatives. A focal point of these notes was the statement of two conjectures regarding $j$, $j'$ and $j"$: a Zilber-Pink type statement incorporating $j$, $j'$ and $j"$, which was an extension of an apparently weaker conjecture of Andre-Oort type. In this paper, I first cover some background regarding $j$, $j'$ and $j"$, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila-Zannier strategy to prove a weakened version of Pila's "Modular Andre-Oo…
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, gamma.
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