Trivial zeros of Riemann auxiliary function
It is proved that $s=-2n$ is a simple zero of $\mathop{\mathcal R}(s)$ for each integer $n\ge1$. Here $\mathop{\mathcal R}(s)$ is the function found by Siegel in Riemann's posthumous papers.
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arXiv subjects
Juan Arias de Reyna
Publications and source records attributed to Juan Arias de Reyna.
On the number of zeros of $\mathop{\mathcal R}(s)$
We prove that the number of zeros $\varrho=β+iγ$ of $\mathop{\mathcal R}(s)$ with $0<γ\le T$ is given by \[N(T)=\frac{T}{4π}\log\frac{T}{2π}-\frac{T}{4π}-\frac12\sqrt{\frac{T}{2π}}+O(T^{2/5}\log^2 T).\] Here $\mathop{\mathcal R}(s)$ is the function that Siegel found in Riemann's papers. Siegel related the zeros of $\mathop{\mathcal R}(s)$ to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel.
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On Siegel results about the zeros of the auxiliary function of Riemann
We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\sqrt{T/2π}P(\sqrt{T/2π})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$. We precise also the connection of the position on the zeros of $\mathop{\mathcal R}(s)$ with the zeros of $ζ(s)$ in the critical line.
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Riemann's Auxiliary Function. Right limit of zeros
Numerical data suggest that the zeros $ρ$ of the auxiliary Riemann function in the upper half-plane satisfy $\mathop{\mathrm{Re}}(ρ)<1$. We show that this is true for those zeros with $\mathop{\mathrm{Im}}(ρ)> 3.9211\dots10^{65}$. We conjecture that this is true for all of them.
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Note on the asymptotic of the auxiliary function
To define an explicit regions without zeros of $\mathop{\mathcal R}(s)$, in a previous paper we obtained an approximation to $\mathop{\mathcal R}(s)$ of type $f(s)(1+U)$ with $|U|< 1$. But this $U$ do not tend to zero when $t\to+\infty$. In the present paper we get an approximation of the form $f(s)(1+o(t))$. We precise here Siegel's result, following his reasoning. This is essential to get the last Theorems in Siegel's paper about $\mathop{\mathcal R}(s)$.
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Asymptotic Expansions of the auxiliary function
Siegel in 1932 published a paper on Riemann's posthumous writings, including a study of the Riemann-Siegel formula. In this paper we explicitly give the asymptotic developments of $\mathop{\mathcal R }(s)$ suggested by Siegel. We extend the range of validity of these asymptotic developments. As a consequence we specify a region in which the function $\mathop{\mathcal R }(s)$ has no zeros. We also give complete proofs of some of Siegel's assertions. We also include a theorem on the asymptotic behaviour of $\mathop{\mathcal R }(\frac12-it)$ for $t \to+\infty$. Although the real part of $e^{-i\vartheta(t)}\mathop{\mathcal R }(\frac12-it)$ is $Z(t)$ the imaginary part grows exponentially, this is why for the study of the zeros of $Z(t)$ it is preferable to consider $\mathop{\mathcal R }(\frac12+it)$ for $t>0$.
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Regions without zeros for the auxiliary function of Riemann
We give explicit and extended versions of some of Siegel's results. We extend the validity of Siegel's asymptotic development in the second quadrant to most of the third quadrant. We also give precise bounds of the error; this allows us to give an explicit region free of zeros, or with only trivial zeros. The left limit of the zeros on the upper half plane is extended from $1-σ\ge a t^{3/7}$ in Siegel to $1-σ\ge A t^{2/5}\log t$. Siegel claims that it can be proved that there are no zeros in the region $1-σ\ge t^\varepsilon$ for any $\varepsilon>0$. We show that Siegel's proof for the exponent $3/7$ does not extend to prove his claim.
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Complexity of natural numbers and arithmetic compact sets
The complexity $\Vert n\Vert$ of a natural number is the least number of $1$ needed to represent $n$ using the 5 symbols $(, ), *, +, 1$. A natural number $n$ is called stable is $\Vert 3^kn\Vert =\Vert n\Vert +3k$. For each natural number $n$, the number $3^an$ is stable for some $a\ge0$, and we define the stable complexity of $n$ as $\Vert n \Vert _{\rm st}=\Vert 3^an\Vert -3a$. We show that the closure of the set of all fractions $n/3^{\lfloor \Vert {n}\Vert _{\rm st}/3\rfloor}$ has remarkable properties; self-similarity $3K'''=K$, well-ordered, and certain arithmetical properties. We pose the question about the unicity of this compact. This question raises some problems about the complexity of natural numbers that we are unable to answer.
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High Precision Computation of Riemann's Zeta Function by the Riemann-Siegel Formula, II
(This is only a first preliminary version, any suggestions about it will be welcome.) In this paper it is shown how to compute Riemann's zeta function $ζ(s)$ (and Riemann-Siegel $Z(t)$) at any point $s\in\mathbf C$ with a prescribed error $\varepsilon$ applying the, Riemann-Siegel formula as described in my paper "High Precision ... I", Math of Comp. 80 (2011) 995--1009. This includes the study of how many terms to compute and to what precision to get the desired result. All possible errors are considered, even those inherent to the use of floating point representation of the numbers. The result has been used to implement the computation. The programs have been included in"mpmath", a public library in Python for the computation of special functions. Hence they are included also in Sage.
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Computation of the secondary zeta function
The secondary zeta function $Z(s)=\sum_{n=1}^\inftyα_n^{-s}$, where $ρ_n=\frac12+iα_n$ are the zeros of zeta with $\Im(ρ)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis the numbers $α_n=γ_n$, but we do not assume the RH. We give an algorithm to compute the analytic prolongation of the Dirichlet series $Z(s)=\sum_{n=1}^\infty α_n^{-s}$, for all values of $s$ and to a given precision.
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Tuples of polynomials over finite fields with pairwise coprimality conditions
Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v$ that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with finite norms.
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Arithmetic of the Fabius function
I solve here a question of Vladimir Reshetnikov in Mathoverflow (question 261649) about the values of Fabius function. Namely, I prove that the numbers $R_n:=2^{-\binom{n-1}{2}}(2n)! F(2^{-n})\prod_{m=1}^{\lfloor n/2\rfloor}(2^{2m}-1)$ are integers. We show also some other arithmetical properties of the values of Fabius function at dyadic points. The Fabius function was defined in 1935 by Jessen and Wintner and has been independently defined at least six times since. We attempt to unify notations related to the Fabius function.
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An infinitely differentiable function with compact support: Definition and properties
This is the English translation of my old paper 'Definición y estudio de una función indefinidamente diferenciable de soporte compacto', Rev. Real Acad. Ciencias 76 (1982) 21-38. In it a function (essentially Fabius function) is defined and given its main properties, including: unicity, interpretation as a probability, partition of unity with its translates, formulas for its $n$-th derivates, rationality of its values at dyadic points, formulas for the effective computation of these values, and some arithmetical properties of these values. Since I need it now for a reference, I have translated it.
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Cancellations in power series of sine type
We present a method to study the behavior of a power series of type $$f(x):=\sum_{n=0}^\infty (-1)^n c_n\frac{x^{2n+1}}{(2n+1)!}$$ when $x\to\infty$. We apply our method to study the function $$f(t):=\int_0^t\frac{dx}{x}\int_0^x\frac{dy}{y}\int_0^y\frac{dz}{z}\bigl\{ \sin x+\sin(x-y)-\sin(x-z)-\sin(x-y+z)\bigr\}.$$ We will derive various different representations of $f(t)$ by means of which it will be shown that $\lim_{t\to+\infty}f(t)=0$, disproving a conjecture by Z. Silagadze, claiming that this limit equals $-π^3/12$.
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A note on the real part of the Riemann zeta-function
We consider the real part $\Re(ζ(s))$ of the Riemann zeta-function $ζ(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $σ_0 = 1.19\ldots$ which is defined to be the supremum of $σ$ such that $\Re(ζ(σ+it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re(ζ(1+it)) \le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals.
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On the distribution (mod 1) of the normalized zeros of the Riemann Zeta-function
We consider the problem whether the ordinates of the non-trivial zeros of $ζ(s)$ are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros $(x_n)$ are uniformly distributed modulo 1. Odlyzko conjectured this to be true. This is far from being proved, even assuming the Riemann hypothesis (RH, for short). Applying the Piatetski-Shapiro $11/12$ Theorem we are able to show that, for $0<κ<6/5$, the mean value $\frac1N\sum_{n\le N}\exp(2πi κx_n)$ tends to zero. The case $κ=1$ is especially interesting. In this case the Prime Number Theorem is sufficient to prove that the mean value is $0$, but the rate of convergence is slower than for other values of $κ$. Also the case $κ=1$ seems to contradict the behavior of the first two million zeros of $ζ(s)$. We make an effort not to use the RH. So our Theorems are absolute. We also put forward the interesting question: will the uniform distribution of the normalized zeros be compatible with the GUE hypothesis? Let $ρ=\frac12+iα$ run through the complex zeros of zeta. We do not assume the RH so that $α$ may be complex. For $0<κ<\frac65$ we prove that \[\lim_{T\to\infty}\frac{1}{N(T)}\sum_{0<\Reα\le T}e^{2iκ\vartheta(α)}=0\] where $\vartheta(t)$ is the phase of $ζ(\frac12+it)=e^{-i\vartheta(t)}Z(t)$.
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Counting tuples restricted by coprimality conditions
Given a set $A=\{(i_1,j_1),\ldots,(i_m,j_m)\}$ we say that $(a_1,\ldots,a_v)$ exhibits pairwise coprimality if $\gcd(a_i,a_j) = 1$ for all $(i,j)\in A$. For a given positive $x$ we give an asymptotic formula for the number of $(a_1,\ldots,a_v)$ with $1 \le a_1,\ldots,a_v \le x$ that exhibit pairwise coprimality. Our error term is better than that of Hu.
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Computation of a Definite Integral
As an application of Cauchy's Theorem we prove that $\int_0^1\arctan\left({\arctanh x-\arctan x\over π+\arctanh x-\arctan x}\right) {dx\over x}= {π\over 8}\log{π^2\over 8}$ answering a question first posted in Mathematics Stack Exchange and then in MathOverflow.
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