代码如下：
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\begin{document}
\frontmatter % only in book class (roman page #s)
\title{The ternary Goldbach problem}
\author{Harald Andr\'es Helfgott}
\makehalftitle % Print title page.
\makepuptitle
\begin{bookepigraph}
\begin{greektext}
á¼Î³Î³á½ºÏ‚ δ᾽ ἦν τέλεος· ὃ δὲ Ï„á½€ Ï„Ïίτον ἧκε χ[αμᾶζε·\\
σὺν τῶι δ᾽ á¼Î¾á½³Ï†Ï…γεν θάνατον καὶ κῆ[Ïα μέλαιναν
\end{greektext}
\epigraphsource{Hesiod (?), {\em Ehoiai}, fr. 76.21--2 Merkelbach and West}
%apud Ioannis Ziogas, Ovid and Hesiod: The Metamorphosis of the Catalogue of Women
\end{bookepigraph}
\tableofcontents % Print table of contents
\begin{thepreface}
\begin{mysage}
cei=(lambda x,d : N((((RBF(10^d)*RBF(x)).upper()).ceil())/10^d))
flo=(lambda x,d : N((((RBF(10^d)*RBF(x)).lower()).floor())/10^d))
import re
def pf(x):
return sage.misc.latex.LatexExpr(re.sub(r'(\d)\.?0+($|\ )',r'\1\2',latex(RR(x))).replace(r'\times',r'\cdot'))
\end{mysage}
% def pf(x):
% return x.str(no_sci=2, skip_zeroes=True, truncate=True)
\begin{mysage}
import numpy as np
\end{mysage}
% def pf(x):
% exponent = floor(abs(x).log10())
% if exponent<=4 and exponent>=-4:
% return x.str(skip_zeroes=True, truncate=True)
% fracstring = (x*10^(-exponent)).str(skip_zeroes=True, truncate=True)
% expstring = "\cdot 10^{" + exponent.str() + "}"
% return fracstring + expstring
The ternary Goldbach conjecture (or {\em three-prime conjecture})
states that every odd number $n$ greater than $5$ can be written as
the sum of three primes. The main
purpose of this book is to give the first full proof of this conjecture.
The text is intended as a definitive,
peer-reviewed form of the proof, which
first appeared in 2012 and 2013 as a series of preprints
\cite{Helf}, \cite{HelfMaj}, \cite{HelfTern}. It is also meant to be
an accessible account.
The proof builds on the great advances made in the early 20th century by
Hardy and Littlewood (1922) and Vinogradov (1937). Progress since then has
been gradual. In some ways, it proved necessary to proceed independently
of more recent developments and work using only the main existing ideas towards the problem, together with techniques that were originally developed for many other purposes.
Part of the aim has been to keep the exposition as accessible as possible, with
an emphasis on qualitative improvements and technical ideas that should
be of use elsewhere. The guiding idea was to give an analytic approach
that is efficient, relatively clean, and, as it must be for this problem,
explicit; the focus does not lie in optimizing explicit constants, or in
performing calculations, necessary as these tasks are.
The minimal background required of the reader is that typically possessed
by a graduate student who has taken a course in
analytic number theory and feels comfortable with basic
real and complex analysis.
For instance, when we define $L$-functions and the Fourier transform, we really do
it so as to set the terminology once and for all; one would hope all
readers are already acquainted with them, and need at most
a quick refresher of their basic properties. However, it
is not assumed that all readers
are conversant with the circle method, exponential-sum estimates, the large
sieve, etc.
It will actually be convenient to develop some of these
matters from first principles.
It should thus be feasible to use the text to teach (or teach oneself)
the circle method -- at least considered as a way to address additive problems
involving primes, as opposed to Diophantine problems. For that matter, it is
not hard to see how a proper subset of
the book could be used as the basis for a more general
advanced course in analytic number theory, or as a reference
for foundational material in explicit analytic number theory.
%Some care has
%been taken to explain certain standard techniques, such as the large
%sieve or the saddle-point method.
%Naturally, in a course, one would actually cover a proper subset
%of the book.
%The guides provided in the introduction should help readers with
%different purposes in mind decide what to omit.
Besides discussing the actual path we follow,
we will sometimes mention roads not taken.
We will also attempt to distinguish what is traditional and what is new -- or
relatively new, in the sense of being recent work done in part
by people other than the author. The intent is to orient students and save
the time of specialist readers.
%On the whole, our aim will be not just to establish the main theorem, but
%-- in so far as space constraints and a necessary focus allow -- to provide clear expositions of tools that may be useful for similar problems, and also
%give a sense of how and why these tools are needed.
%That part of the
%text makes no
%pretensions to completeness; only material that is useful towards our main
%goal will be treated.
My perspective is that of an analytic number theorist
writing for others with an interest in number theory as a whole, and not
solely in its computational or numerical aspects. There are doubtlessly
achievements of importance in explicit analytic number theory
that we will not discuss, or barely mention,
simply because the bounds they give, while remarkable,
are not useful here. We will have to answer the following questions, natural
to an analytic number theorist: which
basic tools in the field can be made explicit at present, in a way that is
strong enough to be generally useful? Which standard paths need to be avoided
until further progress is made? How do we proceed so that our
reasoning is transparent, while our results are set out precisely and in full?
\textbf{Organization.}
In the introduction, after a summary of the history of the problem, we will
go over a detailed outline of the proof, giving, for each part, first the
general strategy and main ideas, followed by a guide to the text.
Part \ref{part:ground}
sets out the foundations we need for all that follows.
We devote a chapter to general preliminaries -- notation and
basic material from analysis and analytic number theory -- and another one
to computational issues, with an emphasis on rigor as well as efficiency. The last chapter of
Part \ref{part:ground} is dedicated to basic estimates on sums of
arithmetic functions. These estimates require a mixture of analytic
and computational work.
The rest of the book is divided into four parts, structured so that they
can be read independently.
Part \ref{part:sieves} offers an explicit treatment of a natural quadratic sieve
(Ch.~\ref{ch:sieve}), and material on the large sieve that
is in part new (Ch.~\ref{ch:largesieve}--\ref{ch:intri}).
%At the same time, Ch.~\ref{ch:sieve} can be used as a black
%box, as only its main result will be used later.
The proof of the ternary Goldbach conjecture may be said to begin properly
with Part \ref{part:min}.
%Parts \ref{part:min} and \ref{part:maj} (on minor and major arcs)
%are independent from each other. Part \ref{part:min} will use the results on si%eves
%in Part \ref{part:sieves}, but Part \ref{part:maj} will not.
%Part \ref{part:concl} uses only the main results (clearly marked) of
%Parts \ref{part:min} and \ref{part:maj}.
As is the case in most proofs involving the circle method, the problem
is quickly reduced to showing that a certain integral over the ``circle''
$\mathbb{R}/\mathbb{Z}$ is non-zero.
The circle is divided into major arcs
and minor arcs.
In Part \ref{part:min} -- in some ways the technical heart of the proof --
we will see how to give upper bounds on the integrand when $\alpha$ is in the minor arcs.
Part \ref{part:maj} will provide rather
precise estimates for the integrand when the variable $\alpha$ is in the major
arcs. Lastly, Part \ref{part:concl} shows how to use these inputs efficiently to estimate the integral.
Each chapter starts with a general discussion of the strategy
and the main ideas involved. Some of the more
technical bounds and computations are relegated to the appendices.
\vskip 20pt
G\"ottingen and Paris, December 2019
\hskip 100pt
\textsc{H. A. Helfgott}
%\vskip 20pt
%\hskip 160pt
%\noindent
%Georg-August Universit\"at G\"ottingen
%\vskip 20pt
%\hskip 160pt
%\noindent%CNRS/Universit\'e de Paris VI/VII
\end{thepreface}
\vfill
\text{}
\section*{Leitfaden} \vfill
%should be:
% 1 2 3 4
% 5 7 11 14
% 6 8 12 15
% 9 13 16
% 10 A 17
% B? 18
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% \fill (4) +(-1,-1) rectangle
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\chapter{Acknowledgements}
D. Platt worked in close coordination with me to
provide GRH verifications in the necessary ranges, and
also helped me with the usage of interval arithmetic.
My postdocs P. Srivastav and F. Aryan gave the work a thorough, careful
proofreading, double-checked calcuations,
and made numerous suggestions on the exposition.
In reply to my requests, O. Ramar\'e
prepared and sent for publication several auxiliary results, and otherwise
provided much-needed feedback. I am deeply grateful to all of them.
I am also much
indebted to A. Booker, B. Green, R. Heath-Brown,
H. Kadiri, T. Tao and M. Watkins
for many discussions on Goldbach's problem and
related issues. The assistance and feedback
given by several of my students and postdocs --
J. Bajpai, A. Commaret, D. Dona, A. Sedunova and S. Z\'u\~niga Alterman,
besides those already mentioned -- was invaluable.
Several questions, mathematical and historical, became clearer due to the
help of J. Brandes, B. Bukh, K. Conrad, N. Elkies,
K. Gong, R. Heath-Brown, P. Humphries, G. Kuperberg, A. Mellit,
A. P. Mullhaupt,
K. O'Bryant,
F. Petrov, W. Sawin, Z. Silagadze, L. Silberman, K. Soundararajan,
N. Temme, R. Vaughan and T. Wooley.
F. Johansson and W. Stein were generous with their time
when giving assistance and advice on how to use ARB and SageMath, respectively.
K. Conrad provided me with several helpful examples of pictures in TikZ.
Additional
references were graciously provided by A. Aizenbud,
R. Bryant, E. Carneiro, S. Huntsman, B. Murphy, S. Ramana,
and I. Rezvyakova.
Thanks are also due to
A. Granville and P. Sarnak for their valuable advice, and, of course,
to anonymous referees, for their remarks.
The introduction shares some material with my
article for the Proceedings of the 2014 ICM \cite{HelfICM}.
That article, in turn, is based in part on the informal note \cite{Helblog},
which was published in Spanish translation
\cite{Gaceta}
%, translated by M. A. Morales and the author,
%and revised with the help of J. Cilleruelo and M. Helfgott)
and in a revised French version
\cite{MR3201598}.
%, translated by M. Bilu and revised by H. A. Helfgott).
Beyond the introduction, most of the book descends from the preprints
\cite{Helf}, \cite{HelfMaj}, \cite{HelfTern}, though much has changed
significantly since then.
Travel and other expenses were funded in part by
the Adams Prize, the Philip Leverhulme Prize and the Humboldt Foundation.
My work on the problem started at the
Universit\'e de Montr\'eal (CRM) in 2006. One of the most intensely
productive periods was that spent later at the \'Ecole Normale
Sup\'erieure, to which I am grateful for providing a pleasant working
environment.
%he is grateful to
%Universit\'e de Montr\'eal,
During part of that period, travel was partly
covered by ANR Project Caesar No.\ ANR-12-BS01-0011. Work on the book as such
continued when I moved to Universit\'e de
Paris VI/VII and then when I
became an Alexander von Humboldt professor at G\"ottingen.
The present work would most likely not have been possible without free and
publicly available
software: SageMath, PARI, Maxima, Sympy, gnuplot, ARB, GMP, MPFI,
VNODE-LP, PROFIL / BIAS, and, of course, \LaTeX, Emacs,
the gcc compiler and GNU/Linux in general. Some exploratory work was done
in SageMath and Mathematica; SageMath was also used as an interface to Maxima and
ARB, and to do minor calculations within the TeX files (via SageTeX).
Floating-point calculations were always conducted rigorously by means
of interval arithmetic. Most large interval-arithmetic computations
used D. Platt's \verb|int_double|\index{intdouble (Platt)@int-double (Platt)|boldindex}
%\texttt{int{\\_}double}
package, based in part on Crlibm. Some other ones used ARB/FLINT,\index{ARB|boldindex}
whether as a C library or, in the case of minor\index{ARB|boldindex}
\index{ARB/FLINT|see{ARB}}
calculations and bookkeeping, via SageMath.\index{SageMath|boldindex}
In the earlier stages of the
project, a few interval-arithmetic
computations used MPFI, and,
for numerical integration, VNODE-LP, which uses
PROFIL / BIAS for interval arithmetic.\index{VNODE-LP|boldindex}
Some computations carried out by
D. Platt \cite{Plattfresh}
used a significant amount of resources, kindly donated to D. Platt and me by several institutions. This crucial help was provided by MesoPSL
(affiliated with the Observatoire de Paris and Paris Sciences et Lettres),
Universit\'e de Paris VI/VII (UPMC - DSI - P\^{o}le Calcul), University of Warwick (thanks
to W. Hart), University of Bristol, France Grilles (French National Grid
Infrastructure, DIRAC national instance),
Universit\'e de Lyon 1 and Universit\'e de Bordeaux 1. I would like to thank the donating organizations, their technical
staff, and all those who helped to make these resources available to them.
\mainmatter
% only in book class (arabic page #s)
\end{document}
Class file和支持文件可从以下网址获取：https://press.princeton.edu/resources/manuscript-preparation
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