Aleksandar Ivić
Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|ζ(1/2+it)|$ on the critical line, and zeros of a class of convolution functions. The first two topics are classical, and the remaining ones are connected with the author's research.
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We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert ζ\vert$. The main ingredient in our argument is an identity which links the zeros of a function $f$ defined on the interval $[-a,a]$ and the values of its derivatives of odd order at $\pm a$.
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a derivative function. It is proved that these are equivalent, and that, if the Riemann hypothesis holds, then all zeros of the zeta function on the critical line are simple. The Riemann hypothesis is then shown to imply the third proposition holds, this being a new necessary condition for the Riemann hypothesis. The third proposition is shown to be equivalent to the…
The main aim of this article is to develop, in a fully detailed fashion, a {\bf unified} theory of the spectral theory of mean values of individual automorphic L-functions which is a natural extension of the fourth moment of the Riemann zeta-function but does not admit any analogous argument and requires a genuinely new method. Thus we first develop a relatively self-contained account of the theory of automorphic representations, especially highlighting the Kirillov model, with which we resolve the problem on the mean value of those L-functions. As another reward, we gain a geometrical underst…
Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros
本文基于对相关猜想，证明在不依赖黎曼假设的情况下，零点对相关猜想可推导出零点的简单性和临界线分布。
Currently there are no reasons to doubt the Riemann Hypothesis: The zeta function beyond the realm of computation
本文考察了质疑黎曼假设的已发表论点，结合解析数论、随机矩阵理论定理及示例解释其不成立原因，探讨ζ函数大高度图像与L函数-随机矩阵特征多项式关系，指出计算证据具误导性。
Mean Values of the auxiliary function
本文研究与Riemann zeta函数相关的Siegel辅助函数的均值，采用更标准的推导路径得到两类均值的主项，给出Siegel相关结果的完整证明，并阐释了理解Siegel论证的难点。
According to two remarkable theorems of Nyman and Báez-Duarte, the Riemann hypothesis is equivalent to a simply-stated criterion concerning least-squares approximation. In carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. In this article we describe the background leading to this observation, and make a conjecture.
Variants on Andrica's conjecture with and without the Riemann hypothesis
本文研究Andrica猜想的变体，在黎曼假设下证明对数修正和m次根形式的显式界，并给出无条件结果及验证区域的更新。
Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta
本文推广了Landau-Gonek定理，研究了zeta函数零点处的和，并通过近似函数方程计算离散矩，进而证明了Shanks猜想。
Bias in cubic Gauss sums: Patterson's conjecture
本研究在广义黎曼假设下证明了关于三次高斯和的渐近公式，证实了Patterson1978年的猜想，解释了Kummer观测到的数值偏差，还得到显式Voronoi求和公式，证明三次大筛法是紧的。
We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter i…
There are no square $L^2$-flat sequences of polynomials of the type $$\frac{1}{\sqrt q}( ε_0 + ε_1z + ε_2z^2 + \cdots + ε_{q-2}z^{q-2} +ε_q z^{q-1}),$$ where for each $j,~~ 0 \leq j\leq q-1,~ε_j = \pm 1$. It follows that Erdös's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only finitely many Barker sequences. We further get that the spectrum of dynamical systems arising from continuous Morse sequences is singular. This settles an old question due to M. Keane. Applying our reasoning to the Liouville function we obtain that the po…
We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima and positive maxima. Intuition for the result, involving illuminating pictures, is described in detail. The generalization of Fourier's theorem to certain entire functions of order one (which is conjectural) suggests that the official description of the Riemann Hypothesis Millennium Problem incorrectly describes an equivalence to the Riemann Hypothesis. The…
In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\Re (s) > \tfrac{1}{2}$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We apply these results to the study of the argument of the zeta function. In another application, we de…
The eñe product over a commutative ring
定义交换环上多项式与形式幂级数的eñe积，研究其代数性质及与对称函数、张量积、Hecke算子的关系，并应用于Riemann zeta函数零点统计和Riemann假设。
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