Qualification Theses  
The ErdosStraus conjecture  Diplomarbeit (Master Thesis equivalent)  
Sums of k unit fractions  Ph.D. Thesis (parts appeared in Transactions of the AMS)  
Combinatorial Prime Number Theory  Habilitationsschrift (Postdoc Thesis). (External referees: Brüdern (Stuttgart), HeathBrown (Oxford), A. Sárközy (Budapest). Internal referees: Lucht, Hilgert). 
Title  Abstract  Author(s)  Status  Download  
1  Primzahlen der Form $p=4k+1$ sind Summe von zwei Quadratzahlen  Christian Elsholtz  Mathematik Lehren. (Februar 1994, Heft 62, 5861)  
2  A remark on Hofmann and Wolke's additive decompositions of the set of primes.  Christian Elsholtz  Arch. Math. 76, No.1, 3033 (2001).  Link to publisher Springer Link pdf  
3  Sums of $k$ unit fractions  Christian Elsholtz  Trans. Amer. Math. Soc. 353 (2001), no. 8, 32093227.  Transactions of the AMS dvi ps pdf  
4  The inverse Goldbach problem  Christian Elsholtz  Mathematika 48 (2001), 151158  dvi ps pdf scanned pdf  
5  The distribution of sequences in residue classes  Christian Elsholtz  Proc. Amer. Math. Soc. 130 (2002), no.8, 22472250  Proceedings of the AMS dvi ps pdf  
6  Some remarks on the additive structure of the set of primes  Christian Elsholtz  Number theory for the millennium, I (Urbana, IL, 2000), 419427, (Proceedings of the Millennial Conference on Number Theory (Bennett et.al.), AK Peters, 2002).  ps pdf  
7  Upper bounds for prime $k$tuples of size $\log N$ and oscillations  Christian Elsholtz  Arch. Math. 82 (2004), 3339.  Link to publisher ps pdf  
8  Lower bounds for multidimensional zero sums  Christian Elsholtz  Combinatorica 24, (2004) no. 3, 351  358.  Link to publisher dvi ps pdf  
9  Triples of primes in arithmetic progressions  Christian Elsholtz  Quart. J. Math. 53 (2002), 393395.  Link to publisher pdf ps  
10 
The LiouvilleHeathBrownZagier
proof of the two squares theorem
Kombinatorische Beweise des Zweiquadratesatzes und Verallgemeinerungen 
Christian Elsholtz  The english version contains all details of the proofs. The German version has a more general introduction and is directed towards a general audience and contains another short proof. The German version is published in Mathematische Semesterberichte 50, Heft 1, 7793, 2003.  English preprint.
ps
German version pdf Link to publisher 

11  Zeroes of $L$functions attached to cuspidal representation of congruence subgroups of $GL_n (\mathbb{Z})$  Christian Elsholtz, Jacek Pomykala, Warsaw  submitted, Preprint  
12  On primes of the form $x^2+y^4$  Christian Elsholtz  submitted, Preprint  
13  On cluster primes  Christian Elsholtz  Acta Arithmetica, 110 (2003), 281284.  Link to publisher dvi ps pdf  
14  Von Lücken zwischen Primzahlen zur Optimierung von Gitterpunkten.  Survey for a general audience (in German)  Christian Elsholtz  invited paper by TU Contact, magazine of the TU Clausthal, issue 12, 2003, pages 6164 
1) zum
Heft
2) pdf 3) internet links that are mentioned in the paper 
15  A short survey on upper and lower bounds for multidimensional zero sums  This survey appeared with years of delay. For an update see "Zerosum problems in finite abelian groups and affine caps" below.  Christian Elsholtz  Springer Lecture Notes in Computer Science, LNCS 4123, pages 10791080, 2006. 
pdf
Link to the publisher 
16  On variants of the larger sieve  Ernie Croot, Georgia Tech., Christian Elsholtz  Acta Mathematica Hungarica 103 (3) (2004) 243254  Link to publisher ps pdf  
17  On Thin Sets of Primes Expressible as Sumsets  Ernie Croot, Georgia Tech., Christian Elsholtz  Acta Mathematica Hungarica, 106 (3), 2005, 197  226  Link to publisher ps pdf  
18  A simple proof of Sen's possibility theorem on majority decisions  Christian Elsholtz, C. List, LSE, London  Elemente der Mathematik 60 (2005), 4556.  Link to publisher pdf  
19  Shifted products that are coprime pure powers  Rainer Dietmann, Stuttgart; Christian Elsholtz, Katalin Gyarmati and Miki Simonovits, Budapest  Journal of combinatorial Theory, Series A, 111 (1), 2005, 2436.  Link to publisher ps pdf  
20  Maximal dimension of unit simplices  Christian Elsholtz, Walter Klotz  Discrete Comput. Geom. 34 (2005), no. 1, 167177.  Link to publisher dvi ps pdf  
21  Zerosum problems in finite abelian groups and affine caps  Yves Edel, Christian Elsholtz, Alfred Geroldinger, Silke Kubertin, Laurence Rackham  Quarterly Journal of Mathematics 58 (2007), 159186  Link to publisher pdf  
22  Additive decomposability of multiplicatively defined sets  Christian Elsholtz  This paper was invited for an issue at the occasion of the 75th birthday of Prof. Wirsing. Functiones et Approximatio 35 (2006), 6177. 
Link to publisher 

23  Sudoku im Mathematikunterricht  Christian Elsholtz, A. Mütze  Mathematische Semesterberichte 54 (1), 2007, 6993 
Springerlink: An extended version is posted here. Also the internetlinks can be reached from here. 

24  The number Gamma(k) in Waring's problem  Christian Elsholtz  Acta Arith. 131 (2008), 4349  pdf Acta Arithmetica  
25  Multiplicative decomposability of shifted sets  Christian Elsholtz  Bulletin of the London Mathematical Society 2008, 40: 97107  pdf publisher's site  
26  Sums of two squares and one biquadrate  Rainer Dietmann, Christian Elsholtz  Funct. Approx. Comment. Math. Volume 38, Number 2 (2008), 233234.  pdf publisher's site  
27  Nonsimple abelian varieties in a family: geometric and analytic approaches  J.Ellenberg (Madison, WI), Christian Elsholtz, C.Hall (U o Michigan, Ann Arbor), E.Kowalski (ETH Zürich)  Journal of the LMS 2009 80(1), 135154  Arxiv, Link to Publisher  
28  Independence and interdependence in collective decision making: an agentbased model of nestsite choice by honeybee swarms  Christian Elsholtz, C.List (LSE), T.D.Seeley (Cornell)  Phil. Trans. R. Soc. B Volume 364, Number 1518 / March 27, 2009 
Publisher,
pdf Press coverage: Daily Telegraph, Economist, Economist BBC Radio Wales, Friday 13th February 2009, 4.56pm (Good Evening Wales) Recommended by F1000 biology The paper is top ten in the most downloaded of the journal in 2009. I am mentioned (as a mathematician, albeit not by name) in the cover story of Science News: Swarm Savvy How bees, ants and other animals avoid dumb collective decisions By Susan Milius May 9th, 2009; Vol.175 #10 (p. 16) Science News Cover 

29  A survey on additive and multiplicative decompositions of sumsets and of shifted sets  Christian Elsholtz  Chapter in "Combinatorial Number Theory and Additive Group Theory" Series: Advanced Courses in Mathematics  CRM Barcelona, Geroldinger, Alfred, Ruzsa, Imre Z. 2009, Birkhäuser, (based on the DocCourse on Additive Combinatorics, Barcelona 2008). Pages, 213231. Publisher 
pdf, Publisher 

30  A combinatorial approach to sums of two squares and related problems  Christian Elsholtz  Invited paper, appeared in: Additive Number Theory: Festschrift in Honor of the sixtieth birthday of Melvyn B. Nathanson, Springer Verlag, 2010, pages 115140. 
pdf,
link to publisher 

31  Hilbert cubes in progressionfree sets and in the set of squares  Rainer Dietmann, Christian Elsholtz  Israel Journal of Mathematics, November 2012, Volume 192, Issue 1, pp 5966  link to publisher  
32  Prime divisors of thin sequences  Christian Elsholtz  The American Mathematical Monthly, Vol. 119, No. 4 (April 2012), pp. 331333 
Link to publisher (JSTOR)
A major part of this paper was kindly included in "Proofs from the BOOK", by M. Aigner and G. Ziegler, (in the first chapter: Six proofs of the infinity of primes, with an appendix on infinitely many more proofs) Link to publisher (Springer), needs subscription. That chapter "Six proofs ..." is also included in "Pi und Co.: Kaleidoskop der Mathematik" (editors: Ehrhard Behrends, Peter Gritzmann, Günter M. Ziegler.) 

33  The number of representations of rationals as a sum of unit fractions  Printing took a while (printed in 2013!), but IJM may be one of the few journals, which allows the author to use the final(!) version for private/homepages/print servers.  Tim Browning, Christian Elsholtz  Illinois J of Mathematics 55 (2), 2011, 685696.  
34  Monochromatic solutions of linear equations.  Christian Elsholtz, Frank Weber  submitted  
35  Patterns and complexity of multiplicative functions  Yvonne Buttkewitz, Christian Elsholtz  Journal of the London Mathematical Society 84 (3) 2011, 578594.  Publisher  
36  A problem of Ramanujan, Erdos and Katai on the iterated divisor function  Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, JanChristoph SchlagePuchta  IMRN 2012, issue 17, 40514061. 
arxiv, free final version, with permission of OUP 

37  Egyptian fractions with restrictions  YongGao Chen, Christian Elsholtz, and LiLi Jiang (Nanjing, China)  Acta Arith. 154 (2012), 109123  arxiv, publisher,  
38  The central limit theorem for subsequences in probabilistic number theory  Christoph Aistleitner, Christian Elsholtz  Canad. J. Math. 64(2012), 12011221  publisher,  
39  Counting the number of solutions to the ErdosStraus equation on unit fractions  Christian Elsholtz, Terence Tao  J. Aust. Math. Soc. 94 (2013), 50–105  arxiv Link to publisher  
40  The number of Huffman codes, compact trees, and sums of unit fractions  Christian Elsholtz, Clemens Heuberger, Helmut Prodinger  IEEE Transactions of Information Theory 59 (2) 2013, 10651075.  arxiv IEEE (final version free)  
41  On Gaps Between Primitive Roots in the Hamming Metric  Rainer Dietmann, Christian Elsholtz, Igor Shparlinski  Quarterly J. Math. 64 (no 4), 2013, 10431055  arxiv, publisher's link publisher's link 2  
42  An Alternative Proof on FourDimensional ZeroSums  Christian Elsholtz  Papers in Number Theory, RMSLecture Notes Series No. 23, 2016, pp. 29–36.  
43  On Romanov's constant  Christian Elsholtz, JC SchlagePuchta  Mathematische Zeitschrift (2018) 288:713724  publisher , publisher ,  
44  Sumsets being squares  Andrej Dujella, Christian Elsholtz  Acta Math. Hungar. 141 (2013), no. 4, 353–357.  publisher , final publisher pdf version, with kind permission of the publisher  
45  On Gaps Between Quadratic Nonresidues in the Euclidean and Hamming Metrics  Rainer Dietmann, Christian Elsholtz, Igor Shparlinski  Indag. Math. (N.S.) 24 (2013), no. 4, 930–938.  publisher  
46  On Diophantine quintuples and D(1)quadruples  Christian Elsholtz, Alan Filipin, Yasutsugu Fujita  Monatsh. Math. 175 (2014), 227239.  publisher pdf  
47  Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations  Christian Elsholtz, David S. Gunderson  Funct. Approx. Comment. Math.Volume 52, Number 2 (2015), 263281.  publisher ,  
48  Almost all primes have a multiple of small Hamming weight  Christian Elsholtz  Bull. Australian Math. Soc. 94 (2) 2016, 224235.  publisher, arxiv  
49  Additive decompositions of sets with restricted prime factors  Christian Elsholtz, Adam J. Harper  Trans. Amer. Math. Soc. 367 (2015) 74037427.  pdf arxiv  
50  Hilbert cubes in arithmetic sets  Rainer Dietmann, Christian Elsholtz  Revista Matemática Iberoamericana 31 (4), 2015, 1477–1498.  arxiv publisher  
51  On conjectures of T. Ordowski and Z.W. Sun concerning primes and quadratic forms  Christian Elsholtz, Glyn Harman  In: Springer (eds. C. Pomerance and M. Rassias): Analytic Number Theory (In Honor of Helmut Maier’s 60th Birthday), (2015), 6581  Springer link pdf  
52  Sums of two squares and a power  Rainer Dietmann, Christian Elsholtz  in book: From Arithmetic to ZetaFunctions  Number Theory in Memory of Wolfgang Schwarz, Editors: Sander, Jürgen, Steuding, Jörn, Steuding, Rasa, Springer, 2016, pages 103108.  Springer link, arxiv  
53  Prescribing the binary digits of squarefree numbers and quadratic residues  Rainer Dietmann, Christian Elsholtz, Igor Shparlinski  Trans. Amer. Math. Soc. 369 (2017), 83698388  arxiv publisher  
54  The curvature of sequences moderately dense in the primes  Jörg Brüdern, Christian Elsholtz  in book: Number Theory – Diophantine Problems, Uniform Distribution and Applications, Festschrift in Honour of Robert F. Tichy’s 60th Birthday, C. Elsholtz, P. Grabner (editors), Springer 2017. pages 193210.  arxiv Springer  
55  Golomb's conjecture on prime gaps  Christian Elsholtz  The American Mathematical Monthly 124. no 4, 2017, 365368  arxiv jstor  
56  Title (Schur)  Christian Elsholtz, Laurence Rackham  submitted  
57  On Erdős and Sárközy's sequences with Property P  Christian Elsholtz, Stefan Planitzer  Monatshefte für Mathematik 182, Issue 3, 2017, 565–575  Springer link (free download)  
58  Egyptian fractions with odd denominators  Christian Elsholtz  Q. J. Math. 67, No. 3, 425430 (2016).  arxiv Link to publisher  
59  Maximal sumfree sets of integer lattice grids  Christian Elsholtz, Laurence Rackham  J London Mathematical Society 95(2), 2017, 353–372.  Link to publisher  
60  On the regularity of primes in arithmetic progressions  Christian Elsholtz, Niclas Technau, Robert Tichy  Int. J. Number Theory 13 (2017), no. 5, 1349–1361.  arxiv Link to publisher  
61  Romanov type problems  Christian Elsholtz, Florian Luca, Stefan Planitzer  The Ramanujan journal 47 (2018), Issue 2, 267–289.  Link to publisher (open access) Link to publisher  
62  Number Theory – Diophantine Problems, Uniform Distribution and Applications Festschrift in Honour of Robert F. Tichy’s 60th Birthday,  Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas around R.F. Tichy's interests, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics.  (Editors:) Christian Elsholtz, Peter Grabner  Number Theory – Diophantine Problems, Uniform Distribution and Applications Festschrift in Honour of Robert F. Tichy’s 60th Birthday, Editors: Christian Elsholtz, Peter Grabner, 2017, Springer.  Springer 
63  Sums of two numbers having only prime factors congruent to one modulo four  This asymptotically solves an old problem of Euler.  Rainer Dietmann, Christian Elsholtz 
Banach Center Publications 118 (2019), 177179 Number theory week 2017, Banach centre publications, vol 118, Institute of mathematics, Polish academy of Sciences, 2019, editors: Łukasz Pankowski and Maciej Radziejewski  pdf publisher 
64  The maximal order of iterated multiplicative functions  Christian Elsholtz, Marc Technau, Niclas Technau  Mathematika, 65 (4) 2019, 9901009  publisher  
65  The number of solutions of the ErdősStraus Equation and sums of k unit fractions  Christian Elsholtz, Stefan Planitzer  Proceedings of the Royal Society of Edinburgh Section A, 150(3) (2020), 14011427.  arxiv publisher  
66  Unconditional Primerepresenting Functions, Following Mills  Christian Elsholtz  American Mathematical Monthly 127(7) (2020), 639642.  arxiv, publisher  
67  The density of integers representable as the sum of four prime cubes  Christian Elsholtz, JanChristoph SchlagePuchta  Acta Arithmetica 192 (2020), 363369  arxiv  
68  Longer gaps between values of binary quadratic forms  Rainer Dietmann, Christian Elsholtz  submitted  arxiv  
69  Arithmetic progressions in binary quadratic forms and norm forms  Christian Elsholtz, Christopher Frei  Bulletin London Math. Soc. 51(4), 2019, 595602.  arxiv, publisher (open access)  
70  Algorithmic counting of nonequivalent compact Huffman codes  Christian Elsholtz, Clemens Heuberger, Daniel Krenn  submitted  arxiv  
71  Caps and progressionfree sets in Z_m^n  Christian Elsholtz, Peter Pal Pach  Designs, Codes and Cryptopgraphy  journal (html, open access), journal (pdf, open access) arxiv  
72  Fermat's Last Theorem impies Euclid's infinity of the primes  Christian Elsholtz  to appear in American Mathematical Monthly. Many thanks to Timothy Gowers for advertising this paper, part 1 part 2 
arxiv  
73  unit fractions  Christian Elsholtz, Stefan Planitzer  submitted  
74  exponential  Christian Elsholtz, Gabriel Lipnik  submitted 
Papers on prime numbers: 1,2,4,6,7,9,10,12,13,14,17,22,25,29,30,32,39 42,43,48,51,54,55,60,61,63,64,65,66, 67, 68,72
Papers on sumsets: 2,4,6,8,12,15,17,21,22,24,26,29,31,42,43,44,47,48,49,50,51,52,53,57,59,61,71
Papers using sieve methods: 2,3,4,5,6,7,13,16,17,22,25,27,29,49,50,54,61
Papers on arithmetic functions: 35, 36, 39,47,57,63,64,65,66,70,73
Papers on zero sums: 8,15,21,42,48,71
Papers on multidimensional problems: 8,15,20, 21,42,48,71,74
Papers on arithmetic progressions (in the sense of additive combinatorics): 9,31,34,35,(37),50,53,54,69,71,72,74
Papers on diophantine equations, including Waring's problem: 3,24,26,33, 34, 37, 39,40, 46,52,58,63,65,70,72
Papers on sums of unit fractions (Egyptian fractions): 3, 33,37,39,40,58,65,70,73
Papers on Waring type problems: 24,26,48, 52,57,63, 67
Papers on quadratic nonresidues etc: 31,41,45,48,53
Papers in combinatorics: 8,9,14,15,18,19,20,21,31,34,35,47,50,56,59,70,71,72,74
Papers in algebra: 20,21,27,69
Papers in probabilistic number theory: 38
Papers on mathematical modelling: 28
Papers for general audience (e.g. teachers): 1,10,14,18,23,28,30,32,55,66,72
Papers in 2001: 2,3,4
Papers in 2002: 5,6,9
Papers in 2003: 10,13,14
Papers in 2004: 7,8,16
Papers in 2005: 17,18,19,20
Papers in 2006: 15,22
Papers in 2007: 21,23
Papers in 2008: 24,25,26
Papers in 2009: 27,28,29
Papers in 2010: 30
Papers in 2011: 33,35
Papers in 2012: 31,32,36,37,38
Papers in 2013: 39,40,41,44,45
Papers in 2014: 46
Papers in 2015: 47,49,50,51
Papers in 2016: 42, 48, 52, 58
Papers in 2017: 53,54,55,57,59,60, Book: 62
Papers in 2018: 43,61
Papers in 2019: 63,64,69
Papers in 2020: 65,66,67, ??
Papers in 2020+: 68,70,71,72,73,74
Key publications include:
3. Sums of $k$ unit fractions. Trans. Amer. Math. Soc. 353 (2001), no. 8, 32093227.2)
4. The inverse Goldbach problem. Mathematika 48 (2001), 151158.
8. Lower bounds for multidimensional zero sums. Combinatorica 24 (2004), no. 3, 351 358.
21. Zerosum problems in finite abelian groups and affine caps, (with Y. Edel (Heidelberg), A.
Geroldinger (Graz), S. Kubertin, L. Rackham), Quarterly J. Math. (Oxford) 58 (2007),
159186.
25. The multiplicative decomposability of shifted sets, Bulletin of the London Mathematical
Society 40 (2008), 97107.
27. Nonsimple abelian varieties in a family: geometric and analytic approaches, with J.
Ellenberg (Wisconsin, Madison), C. Hall (U of Michigan), E. Kowalski (ETH Zurich). J. London
Math. Soc. (2) 80 (2009), 135154.
28. Independence and interdependence in collective decisionmaking: an agentbased model of
nestsite choice by honey bee swarms, with C. List (LSE), T.D. Seeley (Cornell), Phil. Trans.
R. Soc. B Volume 364, Number 1518 / March 2009.
31. Hilbert cubes in progressonfree sets and in the set of squares, with Rainer Dietmann,
Israel Journal of Mathematics 192 (2012), 192 (1), pp 5966
35. Patterns and complexity of multiplicative functions,
with Yvonne Buttkewitz, Journal of the London Mathematical Society 84 (3) 2011, 578594.
36. A problem of Ramanujan, Erdos and Katai on the iterated
divisor function, with Yvonne Buttkewitz, Kevin Ford, JanChristoph SchlagePuchta,
IMRN 2012, issue 17, 40514061.
39. Counting the number of solutions to the ErdosStraus equation on unit fractions,
with Terence Tao
40. The number of Huffman codes, compact trees, and sums of unit fractions,
with Clemens Heuberger, Helmut Prodinger
41. On Gaps Between Primitive Roots in the Hamming Metric,
with Rainer Dietmann, Igor Shparlinski, Quarterly J. Math.
43. On Romanov's constant, with JC SchlagePuchta.
49. Additive decompositions of sets with restricted prime factors, with Adam Harper.
50. Hilbert cubes in arithmetic sets, with Rainer Dietmann
53. Prescribing the binary digits of squarefree numbers and quadratic residues,
with Rainer Dietmann, Igor Shparlinski
54. The curvature of sequences moderately dense in the primes, with J Brudern
59. Maximal sumfree sets of integer lattice grids, with L. Rackham
65. The number of solutions of the ErdősStraus Equation and sums of k unit fractions, with S. Planitzer
71. Caps and progressionfree sets in Z_m^n, with P.P. Pach
74. with G. Lipnik