{
"type": "Article",
"authors": [
{
"type": "Person",
"familyNames": [
"Sourmelidis"
],
"givenNames": [
"Athanasios"
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"type": "Person",
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"This note deals with an application of Voronin’s universality theorem for the Riemann zeta-function ",
{
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"text": "ζ",
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"text": "ζ(σ+it)",
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"text": "t",
"meta": {
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" is fixed. In this sense, the values of the zeta-function on\nany such vertical line provide an atlas for plane curves."
]
}
],
"identifiers": [],
"references": [
{
"type": "Article",
"id": "bib-bib1",
"authors": [],
"title": "\nJ. Andersson, Lavrent’ev’s approximation theorem with nonvanishing\npolynomials and universality of zeta-functions. In New directions in\nvalue-distribution theory of zeta and L-functions, pp. 7–10, Shaker Verlag, Aachen (2009) "
},
{
"type": "Article",
"id": "bib-bib2",
"authors": [],
"title": "\nB. Bagchi, The statistical behaviour and universality properties of the\nRiemann zeta-function and other allied Dirichlet series, PhD Thesis,\nCalcutta, Indian Statistical Institute (1981)\n"
},
{
"type": "Article",
"id": "bib-bib3",
"authors": [],
"title": "\nH. Bohr and R. Courant,\nNeue Anwendungen der\nTheorie der Diophantischen Approximationen auf die Riemannsche\nZetafunktion. J. Reine Angew. Math. 144, 249–274 (1914)\n",
"url": "https://dx.doi.org/10.1515/crll.1914.144.249"
},
{
"type": "Article",
"id": "bib-bib4",
"authors": [],
"title": "\nK. Dajani and C. Kraaikamp, Ergodic theory of numbers. Carus\nMathematical Monographs 29, Mathematical Association of America, Washington,\nDC (2002) ",
"url": "https://doi.org/10.5948/upo9781614440277"
},
{
"type": "Article",
"id": "bib-bib5",
"authors": [],
"title": "\nM. P. do Carmo, Differential geometry of curves and surfaces.\nPrentice-Hall, Englewood Cliffs, NJ (1976) "
},
{
"type": "Article",
"id": "bib-bib6",
"authors": [],
"title": "\nS. M. Gonek, Analytic properties of zeta and L-functions.\nPhD Thesis, University of Michigan (1979) "
},
{
"type": "Article",
"id": "bib-bib7",
"authors": [],
"title": "\nS. M. Gonek and H. L. Montgomery, Spirals of the zeta function I. In\nAnalytic number theory, pp. 127–131, Springer, Cham (2015)\n",
"url": "https://doi.org/10.1007/978-3-319-22240-0_9"
},
{
"type": "Article",
"id": "bib-bib8",
"authors": [],
"title": "\nS. N. Mergelyan, Uniform approximations to functions of a complex variable. (In Russian.)\nUspehi Matem. Nauk (N.S.) 7, 31–122 (1952); English translation: Am. Math. Soc. Transl. 101 (1954) "
},
{
"type": "Article",
"id": "bib-bib9",
"authors": [],
"title": "\nH. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I.\nClassical theory. Cambridge Studies in Advanced Mathematics 97, Cambridge\nUniversity Press, Cambridge (2007) ",
"url": "https://doi.org/10.1017/cbo9780511618314"
},
{
"type": "Article",
"id": "bib-bib10",
"authors": [],
"title": "\nH. Sagan,\nSpace-filling\ncurves. Universitext, Springer-Verlag, New York (1994) ",
"url": "https://dx.doi.org/10.1007/978-1-4612-0871-6"
},
{
"type": "Article",
"id": "bib-bib11",
"authors": [],
"title": "\nJ. Steuding, Value-distribution of L-functions. Lecture Notes in\nMathematics 1877, Springer, Berlin (2007) ",
"url": "https://doi.org/10.1007/978-3-540-44822-8"
},
{
"type": "Article",
"id": "bib-bib12",
"authors": [],
"title": "\nS. M. Voronin, Theorem on the “universality” of the Riemann\nzeta-function. (In Russian.) Izv. Akad. Nauk SSSR Ser. Mat. 39, 475–486,\n703 (1975); English translation: Math. USSR, Izv. 9 (1975), 443–453 (1976) ",
"url": "https://doi.org/10.1070/im1975v009n03abeh001485"
}
],
"title": "An atlas for all plane curves",
"meta": {},
"content": [
{
"type": "Heading",
"id": "S1",
"depth": 1,
"content": [
"1 Curves generated by the Riemann zeta-function"
]
},
{
"type": "Paragraph",
"id": "S1.p1",
"content": [
"Curves appear naturally in life, perhaps not as ideal objects, as Euclid defined a line as “a length without breadth”\nin his ",
{
"type": "Emphasis",
"content": [
"Elements"
]
},
", but as orbits of planets, trajectories in physics and technology, or drawings in art. Taking\ninto account their variety, it might be surprising that one can find them all realized in a single curve. Following\nTolkien, we may state this also as a “Lord of the Curves” poem:"
]
},
{
"type": "Figure",
"id": "S1-F1",
"caption": [
{
"type": "Paragraph",
"content": [
"The values of ",
{
"type": "MathFragment",
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"text": "ζ(3/4+it)",
"meta": {
"altText": "\\zeta(3/4+it)"
}
},
" for ",
{
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"mathLanguage": "mathml",
"text": "0≤t≤35",
"meta": {
"altText": "0\\leq t\\leq 35"
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"; one can already imagine an approximation of a shifted unit circle (in yellow)."
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}
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}
]
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"type": "Paragraph",
"content": [
"Of course, our statement above needs to be clarified. Here and in the sequel we consider only ",
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]
},
{
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"id": "S1.E1",
"label": "(1)",
"mathLanguage": "mathml",
"text": "γ:[0,1]→ℝ2,t↦γ(t)",
"meta": {
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},
{
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"content": [
"such that ",
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"content": [
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"]); this\nincludes line segments, ellipses, and many more curves that easily come to mind. The single curve that realizes all\nthese smooth curves, however, is an artifact and has to be ",
{
"type": "Emphasis",
"content": [
"infinite"
]
},
" for obvious reasons. In this respect, our\ntheorem below has some implications to our understanding of infinity."
]
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"content": [
"This infinite curve originates from the Riemann zeta-function, defined by\n"
]
},
{
"type": "MathBlock",
"id": "S1.E2",
"label": "(2)",
"mathLanguage": "mathml",
"text": "ζ(s)=(21−s−1)−1∑n=1∞(−1)nns,",
"meta": {
"altText": "\\zeta\\mkern 1.0mu(s)=\\bigl(2^{1-s}-1\\bigr)^{-1}\\sum_{n=1}^{\\infty}\\frac{(-1)^{n}}{n^{s}},"
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},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
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"altText": "s=\\sigma+it"
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"type": "MathFragment",
"mathLanguage": "mathml",
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{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "0\" display=\"inline\">σ>0",
"meta": {
"altText": "\\sigma>0"
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},
".\nThe complex-valued function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζ(s)",
"meta": {
"altText": "\\zeta(s)"
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},
" plays a central role in analytic number theory and the distribution of prime numbers in particular (see [",
{
"type": "Cite",
"target": "bib-bib9",
"content": [
"9"
]
},
"]).\nFor our result, we need to allow deviations by a quantity as small as desired.\nThe mathematical language allows for a precise formulation:"
]
},
{
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"id": "S1.Thmtheorem1",
"claimType": "Theorem",
"label": "Theorem 1.1.",
"title": [
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"Theorem 1.1"
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{
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"content": [
"."
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"content": [
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{
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},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "0\" display=\"inline\">ε>0",
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"altText": "\\varepsilon>0"
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},
" be fixed. Then, every plane curve is, up to an error of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
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},
"\nand affine translation, contained in the graph of the curve ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℝ∋t↦ζ(σ+it)∈ℂ",
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}
},
"."
]
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]
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"id": "S1.p5",
"content": [
"Here, of course, we regard any curve on the\nEuclidean plane, via ",
{
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"mathLanguage": "mathml",
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"meta": {
"altText": "\\mathbb{R}^{2}\\simeq\\mathbb{C}"
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", also as a curve on the complex plane."
]
},
{
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"claimType": "Proof",
"label": "Proof.",
"title": [
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"content": [
"The proof relies on Voronin’s celebrated universality theorem [",
{
"type": "Cite",
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"content": [
"12"
]
},
"] from 1975 which states, roughly speaking,\nthat certain shifts of the zeta-function approximate every zero-free analytic function, defined on a sufficiently small\ndisk – a remarkable approximation property!"
]
},
{
"type": "Paragraph",
"content": [
"For our purpose, we recall the universality theorem [",
{
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"content": [
"12"
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},
"] in a stronger form: ",
{
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{
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"mathLanguage": "mathml",
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{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "g(s)",
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"type": "MathFragment",
"mathLanguage": "mathml",
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{
"type": "MathFragment",
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"mathLanguage": "mathml",
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"mathLanguage": "mathml",
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{
"type": "MathBlock",
"id": "S1.E3",
"label": "(3)",
"mathLanguage": "mathml",
"text": "maxs∈𝒦|ζ(s+iτ)−g(s)|<ε",
"meta": {
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"type": "Emphasis",
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},
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{
"type": "Cite",
"target": "bib-bib11",
"content": [
"11"
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},
"]).\nThe main differences from Voronin’s original statement in [",
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"type": "Cite",
"target": "bib-bib12",
"content": [
"12"
]
},
"] are the positive lower density of the set of\nshifts ",
{
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"mathLanguage": "mathml",
"text": "τ",
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{
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"mathLanguage": "mathml",
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{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
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},
"]\nand later in Bagchi’s PhD thesis [",
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{
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"8"
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"] and [",
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"type": "Cite",
"target": "bib-bib11",
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"11"
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", p. 107]). We will also make use of the following observation, due\nto Andersson [",
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"1"
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{
"type": "Cite",
"target": "S1-E1",
"content": [
"1"
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")) is a real-valued function that measures the deviation of the curve from a straight line.\nIt is a well-known fact that a smooth plane curve is determined by its curvature; this follows from the fundamental\ntheorem of the local theory of curves (see [",
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},
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{
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},
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{
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"content": [
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},
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"We note that, in view of the universality theorem, the target function just needs to be continuous if ",
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"content": [
"1"
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") exists (see [",
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"] and Figure ",
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" for the Peano curve as the limit of an iteration). This\nPeano curve maps the unit interval ",
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"Is it possible to extend these results further? To answer this question we recall that, more than a century ago, Bohr\n(the mathematician Harald, younger brother of the physicist Niels) and Courant [",
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{
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")). These limitations also hold for the\napproximation of plane curves, which is obvious for ",
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"], who\nshowed that the curvature of ",
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" as well. The latter result is conditional subject to the truth of the famous, yet unproven, Riemann\nHypothesis that"
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},
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"This open conjecture is one of the seven Millennium Problems in mathematics."
]
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"; see [",
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"A more detailed account of our study with additional results in this context will appear elsewhere."
]
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"content": [
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"\nWith this short note the authors want to express their gratitude to the EMS for the\nfinancial support of the conference “",
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]
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"” at Centre\ninternational de rencontres mathématiques in Luminy in August 2023.\nWe are also grateful to the referee for carefully reading our note.\nThe first author was supported by the Austrian Science Fund (FWF): project M 3246-N."
]
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"content": [
"\nAthanasios\nSourmelidis has graduated from the University of Patras in 2013 and completed his doctoral studies at the University of Würzburg in 2019.\nHe is currently a postdoc researcher in the Insitute of Analysis and Number Theory in the Technical\nUniversity of Graz, being awarded a Lise Meitner fellowship from the Austrian Science Fund (FWF). His research interests lie on the field of analytic number theory with focus on the value-distribution of zeta- and ",
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"-functions.\n",
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"content": [
"sourmelidis@math.tugraz.at"
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},
"\nJörn Steuding studied and received his PhD in Hannover; later he was a postdoc in Frankfurt and Madrid. Since 2006 he\nhas been a professor of mathematics at the University of Würzburg. His research focuses on aspects of analytic number\ntheory and, in particular, the value-distribution of the Riemann zeta-function and related functions.\n",
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