numbering section lemma theorem etc

Question

A referee suggests that

The style of numbering of results in the paper make them hard to find. I would suggest at least numbering by section and also probably using the same counter for theorems, lemmas, definitions etc.

My code in overleaf is as follows, use the springer template, but I can not figure out how to make it as the referee expected. Very much appreciate the help!

\begin{filecontents*}{example.eps}
%!PS-Adobe-3.0 EPSF-3.0
%%BoundingBox: 19 19 221 221
%%CreationDate: Mon Sep 29 1997
%%Creator: programmed by hand (JK)
%%EndComments
gsave
newpath
  20 20 moveto
  20 220 lineto
  220 220 lineto
  220 20 lineto
closepath
2 setlinewidth
gsave
  .4 setgray fill
grestore
stroke
grestore
\end{filecontents*}
%
\RequirePackage{fix-cm}
%
%\documentclass{svjour3}                     % onecolumn (standard format)
%\documentclass[smallcondensed]{svjour3}     % onecolumn (ditto) 
\documentclass[smallextended]{svjour3}       % onecolumn (second format)
%\documentclass[twocolumn]{svjour3}          % twocolumn
%
\smartqed  % flush right qed marks, e.g. at end of proof
%
\usepackage{graphicx}
%
% \usepackage{mathptmx}      % use Times fonts if available on your TeX system
%
% insert here the call for the packages your document requires
%\usepackage{latexsym}
% etc.
%
% please place your own definitions here and don't use \def but
% \newcommand{}{}
%
% Insert the name of "your journal" with
% \journalname{myjournal}
%
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{bbm}

%Includes "References" in the table of contents
\usepackage[nottoc]{tocbibind}
\usepackage{mathrsfs,amsfonts,amssymb,amsmath}
\let\proof\relax\let\endproof\relax
\usepackage{amsthm}
\usepackage{enumerate}
\usepackage{graphicx,cite}
\usepackage{romannum}
\usepackage{hyperref}
\usepackage{authblk}
\usepackage{graphicx}
\newcommand\sbullet[1][.5]{\mathbin{\vcenter{\hbox{\scalebox{#1}{$\bullet$}}}}}
\allowdisplaybreaks

\hypersetup{colorlinks=true, linkcolor=blue, citecolor=red}

\textwidth=15.0cm \textheight=21.0cm \hoffset=-1.1cm \voffset=-0.5cm

%\textwidth=13.8cm \textheight=21.7cm \topmargin=0.0cm
%\oddsidemargin=1.0cm \evensidemargin=1.0cm

%\usepackage{amsmath, amsthm, amssymb}
%\usepackage{graphicx}
%\usepackage{float}

\iffalse
\setlength{\textwidth}{170mm}
\setlength{\textheight}{215mm}
\setlength{\oddsidemargin}{0mm}
\setlength{\evensidemargin}{0mm}
\setlength{\topmargin}{-10mm}
\setlength{\headheight}{10mm}
\setlength{\headsep}{0mm}
\setlength{\footskip}{10mm}
\setlength{\parindent}{0mm}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\Leb}{Leb}
\DeclareMathOperator{\Cor}{Cor}
\DeclareMathOperator{\interior}{int}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\Emb}{Emb}

\newcommand{\cqfd}{{\nobreak\hfil\penalty50\hskip2em\hbox{}
\nobreak\hfil $\square$\qquad\parfillskip=0pt\finalhyphendemerits=0\par\medskip}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\RNum}[1]{\uppercase\expandafter{\romannumeral #1\relax}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\p}{\partial}
\newcommand{\dt}{\partial_t}
\newcommand{\dx}{\partial_x}
\newcommand{\de}{\delta}
\newcommand{\ds}{\displaystyle}
\newcommand{\eps}{ \varepsilon}
\newcommand{\Lip}{\mbox{\rm Lip}}
\newcommand{\pr}{{\bf \textit{Proof : }}}
\newcommand{\ph}{ \varphi}

\numberwithin{equation}{section}

\def\bR{\mathbf{R}}
\def\bS{\mathbf{S}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{notation}[theorem]{Notation}
\newtheorem{convention}[theorem]{Convention}
\newtheorem{assumption}{Assumption}
% for article


%Import the natbib package and sets a bibliography style
\usepackage[square,numbers]{natbib}
\bibliographystyle{abbrvnat}


\begin{document}
\pagenumbering{arabic}
\setlength{\belowdisplayskip}{0pt}

content

\bibliography{bibtext}


\end{document}

Answer 1

Since you're using a template, it's best to stick to their setup. And for this Springer class (svjour3), it already defines a number of theorem-like structures.

The class provides an option to have all theorem-like structures use the same counter: envcountsame

\documentclass[smallextended,envcountsame]{svjour3}       % onecolumn (second format)

\usepackage{amsmath}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\numberwithin{theorem}{section}

\newtheorem{notation}[theorem]{Notation}
\newtheorem{convention}[theorem]{Convention}
\newtheorem{assumption}[theorem]{Assumption}

\begin{document}

\setcounter{section}{6}

\section{A section}
\begin{theorem} A theorem \end{theorem}
\begin{corollary} A corollary \end{corollary}
\begin{proposition} A proposition \end{proposition}
\begin{lemma} A lemma \end{lemma}
\begin{definition} A definition \end{definition}
\begin{remark} A remark \end{remark}
\begin{notation} A notation \end{notation}
\begin{convention} A convention \end{convention}
\begin{assumption} An assumption \end{assumption}

\end{document}

---

A more crude way to achieve this would be to revoke the counters associated with already-defined theorem-like structures, and re-establish them as aliases of the theorem counter.

\documentclass[smallextended]{svjour3}       % onecolumn (second format)

\smartqed  % flush right qed marks, e.g. at end of proof

\usepackage{amsmath}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\numberwithin{theorem}{section}% Number theorems within sections: <section>.<theorem>
\makeatletter
% Remove existing theorem-like counters
\let\c@corollary\relax% Corollary
\let\c@proposition\relax% Proposition
\let\c@lemma\relax% Lemma
\let\c@definition\relax% Definition
\let\c@remark\relax% Remark
\makeatother

% Slave duplicate counter (https://tex.stackexchange.com/a/65827/5764)
\usepackage{aliascnt}
\newaliascnt{corollary}{theorem}% Corollary numbering: <section>.<theorem>
\newaliascnt{proposition}{theorem}% Proposition numbering: <section>.<theorem>
\newaliascnt{lemma}{theorem}% Lemma numbering: <section>.<theorem>
\newaliascnt{definition}{theorem}% Definition numbering: <section>.<theorem>
\newaliascnt{remark}{theorem}% Remark numbering: <section>.<theorem>
\newtheorem{notation}[theorem]{Notation}% Notation numbering: <section>.<theorem>
\newtheorem{convention}[theorem]{Convention}% Convention numbering: <section>.<theorem>
\newtheorem{assumption}[theorem]{Assumption}% Assumption numbering: <section>.<theorem>

\begin{document}

\setcounter{section}{6}

\section{A section}
\begin{theorem} A theorem \end{theorem}
\begin{corollary} A corollary \end{corollary}
\begin{proposition} A proposition \end{proposition}
\begin{lemma} A lemma \end{lemma}
\begin{definition} A definition \end{definition}
\begin{remark} A remark \end{remark}
\begin{notation} A notation \end{notation}
\begin{convention} A convention \end{convention}
\begin{assumption} An assumption \end{assumption}

\end{document}