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%===============================title page=============================
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% bold faces: no ensuremath for them
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%% calibration and cluster separation
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% hadronic variables
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% beams
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%% observables
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\newcommand{\antikt}{\ensuremath{{\mathrm{anti-}k_{\mathrm{T}}}}\xspace}
\newcommand{\bkt}{\ensuremath{\bm{k}_{\boldsymbol{\mathrm{T}}} }\xspace}
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%%% systematic uncertainties
\newcommand{\dHFS}[2][] {\dabs[RCES]{#2}\xspace}
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%% CS tables
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%%% Cross sections
\newcommand{\sI} {\ensuremath{\sigma_{\rm jet}}\xspace}
\newcommand{\sD} {\ensuremath{\sigma_{\rm dijet}}\xspace}
\newcommand{\sT} {\ensuremath{\sigma_{\rm trijet}}\xspace}
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%\newcommand{\sTN} {\ensuremath{\nicefrac{\sT}{\sNC}}\xspace}



% Journal macro
\def\Journal#1#2#3#4{{#1}~{\bf #2} (#3) #4}
%\def\NCA{\em Nuovo Cimento}
%\def\NIM{\em Nucl. Instrum. Methods}
%\def\NIMA{{\em Nucl. Instrum. Methods} {\bf A}}
%\def\NPB{{\em Nucl. Phys.}   {\bf B}}
%\def\PLB{{\em Phys. Lett.}   {\bf B}}
%\def\PRL{\em Phys. Rev. Lett.}
%\def\PRD{{\em Phys. Rev.}    {\bf D}}
%\def\ZPC{{\em Z. Phys.}      {\bf C}}
%\def\EJC{{\em Eur. Phys. J.} {\bf C}}
%\def\CPC{\em Comp. Phys. Commun.}
%
\def\NPB{Nucl. Phys.~}
\def\PRL{Phys. Rev. Lett.~}
\def\EPJC{Eur. Phys. J.~}
\def\PLB{Phys. Lett.~}
\def\NIM{Nucl. Instrum. Meth.~}
\def\PRD{Phys. Rev.~}
\def\JHEP{JHEP~}
\def\PROC{Conf. Proc.~}
\def\CPC{Comp. Phys. Commun.~}









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% title page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{titlepage}

\noindent
\begin{flushleft}
{\tt Erratum to: DESY 16-200    \hfill    ISSN 0418-9833} \\
{\tt May 2021}                  \\
\end{flushleft}

%\noindent
%Date:      \ \ \ 13.10.2016      \\
%Version:   2.0 (draft for final reading) \\
%Editors:   D. Britzger (daniel.britzger@desy.de) \\
%Referees:  J. Gayler, V. Chekelian \\
%Final reading scheduled for 24.10.2016         \\
%\noindent

\vspace{2cm}
\begin{center}
\begin{Large}

Erratum to \\
{\bf Measurement of Jet Production Cross Sections in Deep-inelastic ep Scattering at HERA}\\
Eur.Phys.J. C77 (2017) 215

\vspace{2cm}

H1 Collaboration

\end{Large}
\end{center}

\vspace{2.5cm}

\begin{abstract}
  \noindent
  The measurement of the jet cross sections by the H1 collaboration had
  been compared to various predictions including the next-to-next-to-leading
  order (NNLO) QCD calculations which are corrected in this erratum for an
  implementation error in one of the components of the NNLO calculations.
  The jet data and the other predictions remain unchanged.
  Eight figures, one table and conclusions are adapted accordingly, exhibiting even better agreement between the corrected NNLO predictions
  and the jet data.
\end{abstract}

\vspace{2.5cm}

\begin{center} Submitted to EPJ C \end{center}

\end{titlepage}

\begin{flushleft}
 % \input{h1autsnew}
%-- H1AUTS Author list by names 
%-- Status: Mon Jul 11 16:22:36 CEST 2016  Number of authors = 139 

V.~Andreev$^{19}$,             %LPI -PD        8/88            Andreev             
A.~Baghdasaryan$^{31}$,        %YERE-PD        09/03           Baghdasaryana       
K.~Begzsuren$^{28}$,           %ULBA-PD        04/06           Begzsuren           
A.~Belousov$^{19}$,            %LPI -HON       12/12           Belousov            
A.~Bolz$^{12}$,                %HDB1-DIP       09/14           Bolz                
V.~Boudry$^{22}$,              %ECPL-PD        1/93            Boudry              
G.~Brandt$^{41}$,              %GOET-PD        03/07           Brandt              
V.~Brisson$^{21}$,             %ORSA-HON       01/12           Brisson             
D.~Britzger$^{10}$,            %DESY-PD        06/13           Britzger            
A.~Buniatyan$^{2}$,            %BIRM-HON       06/14           Buniatyan           
A.~Bylinkin$^{43}$,            %MIPT-PD        12/14           Bylinkin            
L.~Bystritskaya$^{18}$,        %ITEP-PD        05/99           Bystritskaya        
A.J.~Campbell$^{10}$,          %DESY-PD        10/84           Campbella           
K.B.~Cantun~Avila$^{17}$,      %MEX1-PD        05/13           Cantunavila         
K.~Cerny$^{25}$,               %PRG2-PD        11/08           Cernyk              
V.~Chekelian$^{20}$,           %MPIM-PD        01/90           Chekelian           
J.G.~Contreras$^{17}$,         %MEX1-PROF      03/98           Contreras           
J.~Cvach$^{24}$,               %PRAG-HON       11/14           Cvach               
J.B.~Dainton$^{14}$,           %LIVE-PROF      01/94           Dainton             
K.~Daum$^{30}$,                %WUPP-HON       08/15           Daum                
C.~Diaconu$^{16}$,             %MARS-PD        09/96           Diaconu             
M.~Dobre$^{4}$,                %BUCH-PD        12/12           Dobre               
V.~Dodonov$^{10}$,             %DESY-PD        04/98           Dodonov             
G.~Eckerlin$^{10}$,            %DESY-PD        8/88            Eckerlin            
S.~Egli$^{29}$,                %PSI -PD        01/10           Egli                
E.~Elsen$^{10}$,               %DESY-PROF      01/06           Elsen               
L.~Favart$^{3}$,               %BRUX-PROF      10/99           Favart              
A.~Fedotov$^{18}$,             %ITEP-PD        8/88            Fedotov             
J.~Feltesse$^{9}$,             %SACL-HON       11/06           Feltesse            
J.~Ferencei$^{44}$,            %NPIC-PD        08/88           Ferencei            
M.~Fleischer$^{10}$,           %DESY-PD        07/95           Fleischer           
A.~Fomenko$^{19}$,             %LPI -PD        02/89           Fomenko             
E.~Gabathuler$^{14,\dagger{}}$,          %LIVE-HON       01/12           Gabathulere         
J.~Gayler$^{10}$,              %DESY-HON       02/05           Gayler              
S.~Ghazaryan$^{10}$,           %DFLC-PD        09/09           Ghazaryan           
L.~Goerlich$^{6}$,             %CRAC-PD        8/88            Goerlich            
N.~Gogitidze$^{19}$,           %LPI -PD        05/91           Gogitidze           
M.~Gouzevitch$^{35}$,          %IPNL-PD        10/11           Gouzevitch          
C.~Grab$^{33}$,                %ZUTH-PROF      08/07           Grab                
A.~Grebenyuk$^{3}$,            %BRUX-PD        04/12           Grebenyuk           
T.~Greenshaw$^{14}$,           %LIVE-PD        8/88            Greenshaw           
G.~Grindhammer$^{20}$,         %MPIM-HON       09/11           Grindhammer         
D.~Haidt$^{10}$,               %DESY-HON       03/04           Haidt               
R.C.W.~Henderson$^{13}$,       %LANC-PROF      10/04           Henderson           
J.~Hladk\`y$^{24}$,            %PRAG-HON       12/02           Hladky              
D.~Hoffmann$^{16}$,            %MARS-PD        07/00           Hoffmann            
R.~Horisberger$^{29}$,         %PSI -PD        01/10           Horisberger         
T.~Hreus$^{3}$,                %BRUX-PD        10/08           Hreus               
F.~Huber$^{12}$,               %HDB1-PD        01/13           Huberf              
M.~Jacquet$^{21}$,             %ORSA-PD        09/96           Jacquet             
X.~Janssen$^{3}$,              %ANTW-PD        02/03           Janssenx            
H.~Jung$^{10,3}$,              %DESY-PROF      12/99           Jungh               
M.~Kapichine$^{8}$,            %JINR-PD        3/97            Kapichine           
J.~Katzy$^{10}$,               %DESY-PD        01/94           Katzy               
C.~Kiesling$^{20}$,            %MPIM-HON       09/11           Kiesling            
M.~Klein$^{14}$,               %LIVE-PROF      12/06           Klein               
C.~Kleinwort$^{10}$,           %DESY-PD        8/88            Kleinwort           
R.~Kogler$^{11}$,              %HAM2-PD        12/10           Kogler              
P.~Kostka$^{14}$,              %ZEUT-PD        10/87           Kostka              
J.~Kretzschmar$^{14}$,         %LIVE-PD        03/08           Kretzschmar         
D.~Kr\"ucker$^{10}$,           %DESY-PD        01/96           Kruecker            
K.~Kr\"uger$^{10}$,            %DESY-PD        01/04           Kruegerk            
M.P.J.~Landon$^{15}$,          %QMWC-PD        10/83           Landon              
W.~Lange$^{32}$,               %ZEUT-PD        8/88            Lange               
P.~Laycock$^{14}$,             %LIVE-PD        11/03           Laycock             
A.~Lebedev$^{19}$,             %LPI -HON       12/12           Lebedev             
S.~Levonian$^{10}$,            %DESY-PD        08/88           Levonian            
K.~Lipka$^{10}$,               %DESY-PD        06/01           Lipka               
B.~List$^{10}$,                %DESY-PD        11/99           Listb               
J.~List$^{10}$,                %DFLC-PD        10/00           Listj               
B.~Lobodzinski$^{20}$,         %MPIM-LEFT      01/15           Lobodzinski         
E.~Malinovski$^{19}$,          %LPI -PD        01/89           Malinovskie         
H.-U.~Martyn$^{1}$,            %AAC1-HON       01/12           Martyn              
S.J.~Maxfield$^{14}$,          %LIVE-PD        8/88            Maxfield            
A.~Mehta$^{14}$,               %LIVE-PROF      06/00           Mehta               
A.B.~Meyer$^{10}$,             %DESY-PD        10/97           Meyeran             
H.~Meyer$^{30}$,               %WUPP-HON       12/00           Meyerhi             
J.~Meyer$^{10}$,               %DESY-HON       10/14           Meyerj              
S.~Mikocki$^{6}$,              %CRAC-PROF      10/10           Mikocki             
A.~Morozov$^{8}$,              %JINR-PD        06/99           Morozova            
K.~M\"uller$^{34}$,            %ZUER-PD        11/94           Muellerk            
Th.~Naumann$^{32}$,            %ZEUT-PROF      01/05           Naumannt            
P.R.~Newman$^{2}$,             %BIRM-PROF      10/09           Newman              
C.~Niebuhr$^{10}$,             %DESY-PD        3/93            Niebuhr             
G.~Nowak$^{6}$,                %CRAC-PROF      02/12           Nowakg              
J.E.~Olsson$^{10}$,            %DESY-HON       09/10           Olsson              
D.~Ozerov$^{29}$,              %PSI -PD        09/15           Ozerov              
C.~Pascaud$^{21}$,             %ORSA-HON       09/05           Pascaud             
G.D.~Patel$^{14}$,             %LIVE-PD        8/88            Patel               
E.~Perez$^{37}$,               %SACL-PD        10/07           Perez               
A.~Petrukhin$^{35}$,           %IPNL-PD        09/09           Petrukhin           
I.~Picuric$^{23}$,             %PODG-PROF      12/07           Picuric             
H.~Pirumov$^{10}$,             %DESY-PD        04/13           Pirumov             
D.~Pitzl$^{10}$,               %DESY-PD        8/88            Pitzl               
R.~Pla\v{c}akyt\.{e}$^{10}$,   %DESY-PD        10/06           Placakyte           
R.~Polifka$^{25,39}$,          %PRG2-PD        05/11           Polifka             
V.~Radescu$^{45}$,             %OXFU-PD        10/06           Radescu             
N.~Raicevic$^{23}$,            %PODG-PD        06/02           Raicevic            
T.~Ravdandorj$^{28}$,          %ULBA-PROF      09/00           Ravdandorj          
P.~Reimer$^{24}$,              %PRAG-PD        07/90           Reimer              
E.~Rizvi$^{15}$,               %QMWC-PROF      03/04           Rizvi               
P.~Robmann$^{34}$,             %ZUER-PD        8/88            Robmann             
R.~Roosen$^{3}$,               %BRUX-HON       03/12           Roosen              
A.~Rostovtsev$^{42}$,          %IITP-PROF      02/11           Rostovtsev          
M.~Rotaru$^{4}$,               %BUCH-PD        06/11           Rotaru              
D.~\v S\'alek$^{25}$,          %PRG2-PD        10/10           Salek               
D.P.C.~Sankey$^{5}$,           %RAL -PD        01/90           Sankey              
M.~Sauter$^{12}$,              %HDB1-PD        10/09           Sauter              
E.~Sauvan$^{16,40}$,           %MARS-PD        11/1            Sauvan              
S.~Schmitt$^{10}$,             %DESY-PD        09/99           Schmittst           
L.~Schoeffel$^{9}$,            %SACL-PROF      10/10           Schoeffel           
A.~Sch\"oning$^{12}$,          %HDB1-PROF      01/09           Schoening           
F.~Sefkow$^{10}$,              %DFLC-PD        09/99           Sefkow              
S.~Shushkevich$^{36}$,         %SINP-PD        08/11           Shushkevich         
Y.~Soloviev$^{10,19}$,         %LPI -PD        08/89           Soloviev            
P.~Sopicki$^{6}$,              %CRAC-PD        03/14           Sopicki             
D.~South$^{10}$,               %DESY-PD        05/03           South               
V.~Spaskov$^{8}$,              %JINR-PD        12/97           Spaskov             
A.~Specka$^{22}$,              %ECPL-PROF      09/05           Specka              
M.~Steder$^{10}$,              %DESY-PD        09/08           Steder              
B.~Stella$^{26}$,              %ROME-HON       11/10           Stella              
U.~Straumann$^{34}$,           %ZUER-PROF      09/95           Straumann           
T.~Sykora$^{3,25}$,            %ANTW-PD        01/06           Sykora              
P.D.~Thompson$^{2}$,           %BIRM-PD        08/99           Thompsonp           
D.~Traynor$^{15}$,             %QMWC-PD        10/01           Traynor             
P.~Tru\"ol$^{34,\dagger{}}$,             %ZUER-HON       10/06           Truoel              
I.~Tsakov$^{27}$,              %SOFI-PROF      03/12           Tsakov              
B.~Tseepeldorj$^{28,38}$,      %ULBA-PROF      06/06           Tseepeldorj         
A.~Valk\'arov\'a$^{25}$,       %PRG2-PD        8/88            Valkarova           
C.~Vall\'ee$^{16}$,            %MARS-PD        06/86           Vallee              
P.~Van~Mechelen$^{3}$,         %ANTW-PROF      01/06           Vanmechelen         
Y.~Vazdik$^{19}$,              %LPI -PD        8/88            Vazdik              
D.~Wegener$^{7}$,              %DORT-HON       01/15           Wegener             
E.~W\"unsch$^{10}$,            %DESY-PD        8/88            Wuensch             
J.~\v{Z}\'a\v{c}ek$^{25}$,     %PRG2-PROF      10/05           Zacek               
Z.~Zhang$^{21}$,               %ORSA-PD        10/92           Zhang               
R.~\v{Z}leb\v{c}\'{i}k$^{25}$, %PRG2-ST        03/12           Zlebcik             
H.~Zohrabyan$^{31}$,           %YERE-PD        11/02           Zohrabyan           
and
F.~Zomer$^{21}$                %ORSA-PROF      09/06           Zomer          

%\newpage

%-- H1 Institutes 

\bigskip{\it
 $ ^{\dagger{}}$ Deceased

\medskip
 $ ^{1}$ I. Physikalisches Institut der RWTH, Aachen, Germany \\
 $ ^{2}$ School of Physics and Astronomy, University of Birmingham,
          Birmingham, UK$^{ b}$ \\
 $ ^{3}$ Inter-University Institute for High Energies ULB-VUB, Brussels and
          Universiteit Antwerpen, Antwerp, Belgium$^{ c}$ \\
 $ ^{4}$ Horia Hulubei National Institute for R\&D in Physics and
          Nuclear Engineering (IFIN-HH) , Bucharest, Romania$^{ i}$ \\
 $ ^{5}$ STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire, UK$^{ b}$ \\
 $ ^{6}$ Institute of Nuclear Physics Polish Academy of Sciences,
          Krakow, Poland$^{ d}$ \\
 $ ^{7}$ Institut f\"ur Physik, TU Dortmund, Dortmund, Germany$^{ a}$ \\
 $ ^{8}$ Joint Institute for Nuclear Research, Dubna, Russia \\
 $ ^{9}$ Irfu/SPP, CE Saclay, GIF-SUR-YVETTE, France \\
 $ ^{10}$ DESY, Hamburg, Germany \\
 $ ^{11}$ Institut f\"ur Experimentalphysik, Universit\"at Hamburg,
          Hamburg, Germany$^{ a}$ \\
 $ ^{12}$ Physikalisches Institut, Universit\"at Heidelberg,
          Heidelberg, Germany$^{ a}$ \\
 $ ^{13}$ Department of Physics, University of Lancaster,
          Lancaster, UK$^{ b}$ \\
 $ ^{14}$ Department of Physics, University of Liverpool,
          Liverpool, UK$^{ b}$ \\
 $ ^{15}$ School of Physics and Astronomy, Queen Mary, University of London,
          London, UK$^{ b}$ \\
 $ ^{16}$ Aix Marseille Universit\'{e}, CNRS/IN2P3, CPPM UMR 7346,
          13288 Marseille, France \\
 $ ^{17}$ Departamento de Fisica Aplicada,
          CINVESTAV, M\'erida, Yucat\'an, M\'exico$^{ g}$ \\
 $ ^{18}$ Institute for Theoretical and Experimental Physics,
          Moscow, Russia$^{ h}$ \\
 $ ^{19}$ Lebedev Physical Institute, Moscow, Russia \\
 $ ^{20}$ Max-Planck-Institut f\"ur Physik, M\"unchen, Germany \\
 $ ^{21}$ LAL, Universit\'e Paris-Sud, CNRS/IN2P3, Orsay, France \\
 $ ^{22}$ LLR, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France \\
 $ ^{23}$ Faculty of Science, University of Montenegro,
          Podgorica, Montenegro$^{ j}$ \\
 $ ^{24}$ Institute of Physics, Academy of Sciences of the Czech Republic,
          Praha, Czech Republic$^{ e}$ \\
 $ ^{25}$ Faculty of Mathematics and Physics, Charles University,
          Praha, Czech Republic$^{ e}$ \\
 $ ^{26}$ Dipartimento di Fisica Universit\`a di Roma Tre
          and INFN Roma~3, Roma, Italy \\
 $ ^{27}$ Institute for Nuclear Research and Nuclear Energy,
          Sofia, Bulgaria \\
 $ ^{28}$ Institute of Physics and Technology of the Mongolian
          Academy of Sciences, Ulaanbaatar, Mongolia \\
 $ ^{29}$ Paul Scherrer Institut,
          Villigen, Switzerland \\
 $ ^{30}$ Fachbereich C, Universit\"at Wuppertal,
          Wuppertal, Germany \\
 $ ^{31}$ Yerevan Physics Institute, Yerevan, Armenia \\
 $ ^{32}$ DESY, Zeuthen, Germany \\
 $ ^{33}$ Institut f\"ur Teilchenphysik, ETH, Z\"urich, Switzerland$^{ f}$ \\
 $ ^{34}$ Physik-Institut der Universit\"at Z\"urich, Z\"urich, Switzerland$^{ f}$ \\

\bigskip
 $ ^{35}$ Now at IPNL, Universit\'e Claude Bernard Lyon 1, CNRS/IN2P3,
          Villeurbanne, France \\
 $ ^{36}$ Now at Lomonosov Moscow State University,
          Skobeltsyn Institute of Nuclear Physics, Moscow, Russia \\
 $ ^{37}$ Now at CERN, Geneva, Switzerland \\
 $ ^{38}$ Also at Ulaanbaatar University, Ulaanbaatar, Mongolia \\
 $ ^{39}$ Also at  Department of Physics, University of Toronto,
          Toronto, Ontario, Canada M5S 1A7 \\
 $ ^{40}$ Also at LAPP, Universit\'e de Savoie, CNRS/IN2P3,
          Annecy-le-Vieux, France \\
 $ ^{41}$ Now at II. Physikalisches Institut, Universit\"at G\"ottingen,
          G\"ottingen, Germany \\
 $ ^{42}$ Now at Institute for Information Transmission Problems RAS,
          Moscow, Russia$^{ k}$ \\
 $ ^{43}$ Now at Moscow Institute of Physics and Technology,
          Dolgoprudny, Moscow Region, Russian Federation$^{ l}$ \\
 $ ^{44}$ Now at Nuclear Physics Institute of the CAS,
          \v{R}e\v{z}, Czech Republic \\
 $ ^{45}$ Now at Department of Physics, Oxford University,
          Oxford, UK \\


\bigskip
 $ ^a$ Supported by the Bundesministerium f\"ur Bildung und Forschung, FRG,
      under contract numbers 05H09GUF, 05H09VHC, 05H09VHF,  05H16PEA \\
 $ ^b$ Supported by the UK Science and Technology Facilities Council,
      and formerly by the UK Particle Physics and
      Astronomy Research Council \\
 $ ^c$ Supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT
      and by Interuniversity Attraction Poles Programme,
      Belgian Science Policy \\
 $ ^d$ Partially Supported by Polish Ministry of Science and Higher
      Education, grant  DPN/N168/DESY/2009 \\
 $ ^e$ Supported by the Ministry of Education of the Czech Republic
      under the project INGO-LG14033 \\
 $ ^f$ Supported by the Swiss National Science Foundation \\
 $ ^g$ Supported by  CONACYT,
      M\'exico, grant 48778-F \\
 $ ^h$ Russian Foundation for Basic Research (RFBR), grant no 1329.2008.2
      and Rosatom \\
 $ ^i$ Supported by the Romanian National Authority for Scientific Research
      under the contract PN 09370101 \\
 $ ^j$ Partially Supported by Ministry of Science of Montenegro,
      no. 05-1/3-3352 \\
 $ ^k$ Russian Foundation for Sciences,
      project no 14-50-00150 \\
 $ ^l$ Ministery of Education and Science of Russian Federation
      contract no 02.A03.21.0003 \\
}
\end{flushleft}
%
% Please not that the author list may need re-formatting.


\clearpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%  Describe the erratum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The measurement of absolute and normalised inclusive jet and dijet
cross sections by the H1 collaboration~\cite{H1_jets} were compared to
next-to-next-to-leading order QCD predictions by the NNLOJET
program~\cite{NNLOJET}.
An implementation error of specific integrated initial-final antenna
functions that has impact on the numerical predictions for jet production
cross sections in DIS was found in this numeric calculation~\cite{NNLO,NNLO17}.
The data, the next-to-leading (NLO) and approximate
next-to-next-to-leading order (aNNLO) predictions remain unchanged, as
well as figures for three-jet cross sections and the results on the strong
coupling constant $\asmz$, which are based on NLO predictions.

In this erratum, we provide corrected figures for the comparison of 
inclusive jet and dijet cross section data, both for absolute and
normalised jet cross sections.
The changes to the NNLO predictions are overall small, such that
differences are only visible in figures of cross section ratios.
The calculated values for $\chi^2/\ndf$ are corrected in
table~\ref{tab:chisq}. The discussion is corrected accordingly.


\setcounter{section}{7}
\section{Cross section measurements}
The differential cross sections are presented for absolute and normalised inclusive jet,
dijet and trijet production at hadron level in Ref.~\cite{H1_jets}.
The agreement of the various predictions with the data is judged by calculating values of $\chi^2/\ndf$~\cite{H1Multijets}.
Here \ndf\ is the number of data points in the calculation.
The values of $\chisq/\ndf$ for the absolute and the normalised jet cross sections are listed in table~\ref{tab:chisq}.
All calculations provide a reasonable value of $\chisq/\ndf$, taking into account
the fact that uncertainties on the theory predictions, such as scale variations
or the PDF uncertainties, are not included.

\setcounter{table}{2}
\begin{table}[hbp!]
  \footnotesize
  \begin{center}
    \begin{tabular}{lccccccc}
      \hline
      \hline
              &  ~~\ndf~~ &\multicolumn{6}{c}{Value of $\chi^2/\ndf$} \\
              &   & NLO & aNNLO  & NNLO & NLO & aNNLO  & NNLO \\
       \hline
                     &&\multicolumn{3}{c} {Absolute jet cross sections} & \multicolumn{3}{c} {Normalised jet cross sections} \\
      Inclusive jet at low-\Qsq           & 48 &1.7&2.1&0.7&1.9&1.6&1.0\\ % ERRATA 2020
      Inclusive jet at low- and high-\Qsq & 78 &1.7&2.0&1.1&1.9&2.2&1.5\\ % ERRATA 2020
      Dijet  at low-\Qsq                  & 48 &1.4&1.9&0.4&1.6&1.7&0.6\\ % ERRATA 2020
      Trijet at low-\Qsq                  & 32 &0.6&   &   &0.6&   & \\
      \hline
      \hline
    \end{tabular}
    \caption{Summary of values of $\chi^2/n_{\rm dof}$ for absolute and normalised jet cross sections for the NLO, aNNLO and NNLO predictions, whenever those are available. }
    \label{tab:chisq}
  \end{center}
\end{table}


\setcounter{section}{8}
\setcounter{subsection}{1}
\subsection{Inclusive jet cross section}
The measured double-differential inclusive jet cross sections as
function of \ptjet\  and \Qsq\ for low and high values of \Qsq\ 
are compared to different theoretical predictions in figure~\ref{figInclJet}.
Ratios of the data and of the predictions in aNNLO and full NNLO to
the NLO predictions are provided in figure~\ref{figInclJetRatio}. 

\begin{boldmath}
\subsubsection{Inclusive jet cross sections at low-$\Qsq$ ($\Qsq<80\,\GeVsq$)}
\end{boldmath}
The conclusions drawn on the agreement between the NNLO predictions
and the data remain largely unchanged with respect to
Ref.~\cite{H1_jets}.
The NNLO predictions give a good description of the \ptjet-distributions
following the excellent value of $\chisq/\ndf$ (table~\ref{tab:chisq}).

The NNLO corrections to the cross section predictions, which are defined as
ratios of NNLO to NLO predictions and are displayed in figure~\ref{figInclJetRatio},
are particularly large at
low values of \ptjet\ or at low values of \Qsq, equivalent to low values of
the renormalisation and factorisation scales \mur\ and \muf.
The NNLO predictions themselves have significantly smaller scale
uncertainties than the NLO predictions.
At low values of \ptjet\ and small \Qsq, where the data are most precise, the
uncertainties from scale variations 
of all predictions, however, are significantly larger than the experimental uncertainties.
At higher values of \ptjet\ the relative theoretical uncertainties are becoming smaller,
but the data uncertainties, both statistical and systematic, increase
and overshoot the uncertainties from scale variations. 


% [...]
%The central values of the  NLO and the aNNLO predictions are lower than the data in most bins, while the NNLO predictions 
%have a tendency to lie above the data.  {\color{green} We think, this is still true!}
% [...]
%The NNLO predictions give a good description of the \ptjet-distributions
%explaining the excellent value of $\chisq/\ndf$
% (table~\ref{tab:chisq}).{\color{blue} Yes. that's still true!}
% [...]

%The NNLO corrections to the cross section predictions, which are defined as  
%ratios of NNLO to NLO predictions and are displayed in figure~\ref{figInclJetRatio}, 
%are particularly large at low values of \ptjet\ or at low values of \Qsq, equivalent to low values of
%the renormalisation and factorisation scales \mur\ and \muf. {\color{blue} Yes. that's still true!}

%  The NNLO predictions themselves have significantly smaller scale uncertainties than the NLO predictions{\color{red}\sout{, 
%  in particular at high values of the renormalisation
%  scale}}.
%  % {\color{red} Not quite: also at low-\ptjet\ the scale
%  %  uncertainties are now reduced significantly.}
%  
%  At low{\color{red}\sout{er}} values of \ptjet\ {\color{blue}and small \Qsq},
%  where the data are most precise, the uncertainties from scale
%  variations  of all predictions, however, are significantly larger than the
%  experimental uncertainties.
%  % {\color{red} not quite: at medium Q2, already the NNLO scale
%  % uncertainties are competitive with data uncertainties.}

%At higher values of \ptjet\ the relative theoretical uncertainties are becoming smaller, 
%but the data uncertainties, both statistical and systematic, increase and overshoot the uncertainties from scale variations.


%--------------------------------------------------------------------
\begin{boldmath}
\subsubsection{Measurement of inclusive jets at high-$\Qsq$ ($\Qsq>150\,\GeVsq$)}
\end{boldmath}

The phase space of additional inclusive jet cross sections at high values of \Qsq\ is extended to the region $\ptjet<7\,\GeV$ by adding an extra bin at low \ptjet~\cite{H1_jets}.
These additional cross section points as a function of \Qsq\ for
inclusive jet production in the range
$5<\ptjet<7\,\GeV$
are shown in figures~\ref{figInclJet} and ~\ref{figInclJetRatio}.

The low-\ptjet\ inclusive jet cross sections at high-\Qsq are underestimated by the
NLO and aNNLO predictions, while the NNLO predictions give a good description of these data points.
In the high-\Qsq domain
the NNLO predictions have significantly smaller scale uncertainties
than the NLO calculations, and the NNLO scale uncertainties typically are smaller than the experimental uncertainties.
Figure~\ref{figInclJetRatio} and
the values of $\chisq/\ndf$ in table~\ref{tab:chisq} indicate that the aNNLO predictions
have difficulties describing the previously published
high-\Qsq\ inclusive jet data \cite{H1Multijets} accurately.
The NNLO predictions provide a good description of both, the low- and high-\Qsq\ inclusive jet data.


%[...]
%The new low-\ptjet\ inclusive jet cross sections at high-\Qsq are underestimated by the 
%NLO and aNNLO predictions, while the NNLO predictions give a good
%description of these new data points. {\color{blue} Yes. that's still true!}
%
%[...]
%In the high-\Qsq domain 
%the NNLO predictions have significantly smaller scale uncertainties 
%than the NLO calculations, and the NNLO scale uncertainties typically
%are smaller than the experimental uncertainties. {\color{blue}
%  Yes. that's even more true now!}

%  Figure ~\ref{figInclJetRatio} and
%  the values of $\chisq/\ndf$ in table~\ref{tab:chisq} indicate that the aNNLO {\color{red}\sout{and NNLO}} predictions
%  have difficulties describing the previously published
%  high-\Qsq\ inclusive jet data \cite{H1Multijets} accurately.
%  {\color{blue}The NNLO predictions provide a good
%    description of both, the low- and high-\Qsq\ inclusive jet data}.
%  % {\color{red}
%  %  No. that's not true. NNLO provides a good description!}
%  

\subsection{Normalised inclusive jet cross section}
\label{sec:NormIjets}
%% NC DIS
In order to obtain the normalised jet cross sections, cross sections for inclusive NC DIS
are measured for $0.2<y<0.6$ in the \Qsq\ bins in the range $5.5<\Qsq<80\,\GeVsq$.
%% normalised jets
The normalised inclusive jet cross sections, derived using the inclusive NC DIS and the
absolute inclusive jet cross sections, are displayed together with theoretical predictions
in figure~\ref{figNormInclJet}.
The ratio of normalised inclusive jet cross sections to NLO predictions and the predictions in aNNLO and full NNLO to the NLO predictions is shown in figure~\ref{figNormInclJetRatio}.
The dominating systematic uncertainties do not cancel in the normalisation, and the systematic uncertainty is significantly reduced only in bins where the overall systematic error is small, typically at low \ptjet.
%The normalised jet cross sections hence do not lead to stronger conclusions
%when confronted with theoretical predictions than the absolute cross sections.
The normalised jet cross sections confronted with theoretical predictions
confirm the observations obtained using the absolute cross sections. 



%--------------------------------------------------------------------- 
\setcounter{subsection}{3}
\subsection{Dijet cross sections}
\label{sec:2jets}
The double-differential dijet cross sections as function of \meanptdi\ and \Qsq\
are displayed in figure~\ref{figDijet} and compared to theoretical
predictions in NLO, aNNLO and NNLO.
A comparison of the ratio of data to NLO predictions is provided in figure~\ref{figDijetRatio}
together with the ratio of NNLO to NLO.

The aNNLO and NNLO predictions provide a better description of the
shapes, while the NNLO predictions provide an
  overall accurate description of the normalisation of the dijet data.
The uncertainty from scale variations of the NLO predictions is larger than 
the experimental uncertainty for  $\meanptdi<35\,\GeV$,
while the scale uncertainty of the NNLO calculations is reduced compared to the 
NLO predictions and is larger than the experimental uncertainties only
at lower \Qsq\ values and for $\meanptdi<11\,\GeV$.


The normalised dijet cross sections are displayed together with theoretical predictions
in figure~\ref{figNormDijet}, and the ratio to NLO predictions is shown in figure~\ref{figNormDijetRatio}.
When comparing normalised dijet cross sections to theory predictions, the features
observed with the absolute dijet cross sections are confirmed.


%  % [...]     
%  The aNNLO and NNLO predictions provide a better description of the
%  shapes{\color{blue}, while the NNLO predictions further provide an
%    overall accurate description of the normalisation of the dijet data}.
%  
%  The uncertainty from scale variations of the NLO predictions is larger than 
%  the experimental uncertainty for  $\meanptdi<35\,\GeV$,
%  while the scale uncertainty of the NNLO calculations is reduced compared to the 
%  NLO predictions and is larger than the experimental uncertainties only
%  {\color{red}\sout{for $\meanptdi<25\,\GeV$}}
%  {\color{blue}at lower \Qsq\ values and for $\meanptdi<11\,\GeV$}.
%  % {\color{red}
%  %  No. The last part is not correct any more.}
%  
%  


%-----------------------------------------------------------------------
\setcounter{section}{9}
\section{Summary}
\label{sect:Conclusion}
% [...]
An error in the implementation of the next-to-next-to-leading order (NNLO) predictions is corrected.
While the data, the NLO and the aNNLO predictions remain unchanged, some conclusions drawn from the
comparison of the data to the NNLO predictions are revisited.
%
The predictions in  next-to-next-to-leading order in perturbative QCD  improve the descriptions of the inclusive jet
and dijet cross sections compared to NLO predictions, and give an
overall good description of the new data at low and high $Q^2$.
% {\color{red}\sout{At high $\ptjet$, in particular when extending the comparisons to previously
%published H1 data at high $Q^2$, discrepancies to the data emerge.}}

% {\color{green}
%  yes/no. The NNLO predictions do quite a bit better.}



 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements} % new for errata
We thank T.~Gehrmann, J.~Niehues and A.~Huss for providing the corrected
NNLO predictions.

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



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


%%======================= References ==========================%%
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%\bibliographystyle{myutcaps}
%\bibliographystyle{hep}
%\bibliography{MyBibl.bbl}

\begin{flushleft}
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\end{thebibliography}
\end{flushleft}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TABLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

%%%%%%%%%%%%%%%%%%%%%% RESULTS %%%%%%%%%%%%%%%%%%%%%%

%% ---  Inclusive jets
\setcounter{figure}{7}
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.95\textwidth]{d16-200f8}
%  \includegraphics[width=0.95\textwidth]{hInclJetGrand}
  \caption{
    Cross sections for inclusive jet production in NC DIS as a function of \ptjet\ for different \Qsq\ ranges.
    The new data are shown as full circles whereas full triangles indicate previously published data.
    The error bars indicate statistical uncertainties.
    The hatched area indicates all other experimental uncertainties added in quadrature.
    The NLO and NNLO QCD predictions corrected for hadronisation effects together with their uncertainties from scale variations are shown by the shaded and hatched band, respectively.
    The aNNLO calculations are shown as full red line.
    The cross sections in each bin are divided by the bin-size in \ptjet\ and \Qsq.
  }
  \label{figInclJet}
\end{figure}

\setcounter{figure}{8}
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.95\textwidth]{d16-200f9}
%  \includegraphics[width=0.95\textwidth]{hInclJetRatioGrand}
  \caption{
    Ratio of inclusive jet cross sections to the NLO predictions and ratio of aNNLO 
    and NNLO to NLO predictions as function of \Qsq\ and \ptjet.
    More details are given in the caption of figure~\ref{figInclJet}.
  }
  \label{figInclJetRatio}
\end{figure}



% norm. incl jets
\setcounter{figure}{10}
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.95\textwidth]{d16-200f11}
%  \includegraphics[width=0.95\textwidth]{hNormInclJetGrand}
  \caption{
    Normalised inclusive jet cross sections compared to NLO, aNNLO and NNLO predictions as a function of \Qsq\ and \ptjet.
    The cross sections are divided in each bin by the bin size in $\ptjet$.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figNormInclJet}
\end{figure}


\setcounter{figure}{11}
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.95\textwidth]{d16-200f12}
%  \includegraphics[width=0.95\textwidth]{hNormInclJetRatioGrand}
  \caption{
    Ratio of normalised inclusive jet cross sections to NLO predictions and ratio of the NNLO and aNNLO to the NLO predictions as a function of \Qsq\ and \ptjet.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figNormInclJetRatio}
\end{figure}


\newpage
%% ---  Dijets
\setcounter{figure}{12}
\begin{figure}
  \centering
  \includegraphics[width=0.85\textwidth]{d16-200f13}
%  \includegraphics[width=0.85\textwidth]{hDijet}
  \caption{
    Dijet cross sections compared to NLO, aNNLO and  NNLO predictions as a function of \Qsq\ and \meanptdi.
    The cross sections in each bin are divided by the bin-size in
    \meanptdi{} and \Qsq{}.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figDijet}
\end{figure}

\setcounter{figure}{13}
\begin{figure}
  \centering
  \includegraphics[width=0.85\textwidth]{d16-200f14}
%  \includegraphics[width=0.85\textwidth]{hDijetRatio}
 \caption{
  Ratio of dijet cross sections to NLO predictions and ratio of the aNNLO and NNLO to the NLO predictions as a function of \Qsq\ and \meanptdi.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figDijetRatio}
\end{figure}


\setcounter{figure}{14}
\begin{figure}
  \centering
  \includegraphics[width=0.85\textwidth]{d16-200f15}
%  \includegraphics[width=0.85\textwidth]{hNormDijet}
  \caption{
    Normalised dijet cross sections compared to NLO, aNNLO and NNLO predictions as a function of \Qsq\ and \meanptdi.
    The cross sections are divided in each bin by the bin size in \meanptdi.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figNormDijet}
\end{figure}

\setcounter{figure}{15}
\begin{figure}
  \centering
  \includegraphics[width=0.85\textwidth]{d16-200f16}
%  \includegraphics[width=0.85\textwidth]{hNormDijetRatio}
  \caption{
    Ratio of normalised dijet cross sections to NLO predictions and ratio of the aNNLO 
    and NNLO to the NLO predictions as a function of \Qsq\ and \meanptdi.
    Further details can be found in the caption of figure~\ref{figInclJet}.
  }
  \label{figNormDijetRatio}
\end{figure}


\end{document}
