APOLOGY FOR THE PROOF OF THE RIEMANN HYPOTHESIS Louis de Branges Abstract. An apology is an explanation or defense of actions which may otherwise be misunderstood. There are several sources of misunderstanding concerning the proof of the Riemannhypothesis. An obstacle liesinthe narrowperceptionof the Riemannhypothesisas a mechanism for counting prime numbers. The Riemann hypothesis is signi(cid:12)cant because of its signi(cid:12)cance in mathematical analysis. The proof cannot be read as an isolated argument because of its roots in the history of mathematics. Another obstacle lies in the unexpected source of the proof of the Riemann hypothesis. The proof is made possible by events which seem at (cid:12)rst sight to have no relevance to mathematics. Exceptional people and exceptional circumstances prepared the proof of the Riemann hypothesis. Good writing about mathematics is di(cid:14)cult because the expected reader knows either too much or too little. Those with graduate experience are biased by the choice of a specialty. Those without graduate experience exist in a state of ignorance. Expository writing about mathematics needs to present the reader with a view of the subject which is convincing at several levels of knowledge. Readers without graduate experience need to be supplied with information which justi(cid:12)es mathematical research. Readers with graduate experience need to place their speciality within a larger perspective. These objectives are achieved by a history of mathematics as it relates to the Riemann hypothesis. The Riemann hypothesis culminates a renewal of mathematical analysis after a mille- nium in which Greek analysis lay dormant in libraries. The Renaissance is stimulated by the Cartesian philosophy that problems are best solved by prior thought, as opposed to the Roman philosophy that problems are solved by immediate action. Analysis is not exclusive to mathematics since it is little else than the consistent application of thought. A common feature of e(cid:11)ective analysis is the need for hypotheses, without which no conclusion is valid. Although analysis has striking successes, the analysis applied in mathematics sur- passes other applications of analysis in the extent and consistency of its logical structure. Other applications of analysis emulate the application made in mathematics. Mathematical analysis di(cid:11)ers in purpose from other applications of analysis. Serious projects need to exhibit an evident purpose if they expect to receive the means required for their achievement. The value of a proposed contribution is weighed against the cost of its realization. Mathematical analysis does not admit a statement of purpose which is meaningful without preparation. The discovery of purpose is a historical process which perpetually diversi(cid:12)es itself into new channels and persistently returns to a clari(cid:12)cation of original aims. The earliest known applications of mathematical analysis are witnessed by architectural achievements, such as Egyptian pyramids, and by astronomical observations essential to agriculture. Mathematical analysisoriginatesasthegeometry ofspace withnumbers asac- cessories inmeasurement. Numbers are discovered asintegers from which rationalnumbers are constructed. An essentially di(cid:11)erent purpose is discovered for mathematical analysis when geometric objects are constructed which are not measured by rational numbers. 1 2 LOUIS DE BRANGES April 27, 2010 American readers may be pleased to learn how the attraction of irrational numbers has shaped their history. The (cid:12)ve{pronged star which is their cultural heritage has a fascination which cannot be explained by beauty. Since beauty is akin to symmetry, the six{pronged star wins when beauty is the issue. The attraction of the (cid:12)ve{pronged star lies in its dynamic quality which appeals for action because it is less complete. The star originates in the construction of an irrational number disturbing an eye which prefers the restfulness of rational proportions. The distinction between constructions which terminate and those which do not assigns a purpose to mathematical analysis. The Euclidean algorithm marks the discovery of mathematical analysis as applied to in(cid:12)nite constructions. Discoveries of purpose in the Renaissance are illustrated in the lives of Ren(cid:19)e Descartes (1596{1652), Pierre de Fermat (1601{1665), and Blaise Pascal (1623{1662). Cartesian space reinforces the classical conception of space by the introduction of rect- angular coordinates. Cartesian analysis applies the properties of real numbers to obtain the properties of geometrical (cid:12)gures. The success of the method justi(cid:12)es the Cartesian philosophy that problems are solved by thought. The contribution of Descartes to science is the discovery of orderly structure in nature which exceeds previous expectations. Of signi(cid:12)cance for the Riemann hypothesis is the conception of space as structured. The characterization of the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron as regular solids demonstrates Greek awareness of the properties of space. Descartes completes this achievement with the observation that in every case the number of faces minus the number of edges plus the number of vertices is equal to two. The (cid:12)res which ravaged the library of Alexandria are disastrous events in the history of mathematical analysis. Those books salvaged by Muslim scholars leave an incomplete record of Greek achievement. The theorem that every positive integer is the sum of four squares is not found in any surviving book of Diophantus. Yet the conditions stated for the representation as a sum of two squares presume a knowledge of the general representation. The mathematical contributions of Fermat are stimulated by the desire to recover and continue such classical knowledge. His problem of (cid:12)nding positive integers a;b, and c such that an +bn = cn for a positive integer n challenged subsequent generations of analysts. The in(cid:12)nitessimal calculus is however his most original contribution to mathematical analysis. Although the logical skills required for mathematical analysis clearly require a special education, thereisnoagreementaboutwhatitscontentshouldbe. Examplesofasuccessful mathematicaleducationareinstructiveforthosewhodesiretonurturemathematicaltalent in themselves and in others. Pascal received an exceptional education because his mother diedbefore hereached school age. Hisfather personallytaught himthereading andwriting skills of a traditional curriculum aimed at an understanding of current political, social, and religious structures in a historical perspective. When he was twelve, he learned about the nature and purpose of a discipline called geometry. Curiosity stimulated him to attempt his own implementation of that purpose. Only then did his father supply him with Euclid’s Elements. At that time there already existed in Paris learned societies for the presentation of scienti(cid:12)c work. The contributions of Pascal were well received initially because they contain new arguments in support of known results and eventually because the results themselves are new. Memorable contributions are combinatorial principles which underlie the binomial theorem and the calculus of (cid:12)nite di(cid:11)erences. The education of Blaise Pascal is described with loving care by his sister Jacqueline in the preface to his Pens(cid:19)ees. An APOLOGY 3 illuminating portrait of her by the court artist Philippe de Champaigne is preserved in the museum on the site of the ancient Abbaye de Port{Royal{des{Champs. It becomes clear that her dedication as a Jansenist nun was a major ingredient in the success of her brother’s education. The revival of mathematical analysis in the Renaissance relies on foundations which were discovered in ancient times and which are formalized in modern times. The geometric concept of a line is implemented by an algebraic structure which is now called a (cid:12)eld. The elements of a (cid:12)eld can be added and multiplied to produce elements of the (cid:12)eld. If a and b are elements of a (cid:12)eld, a unique element c = a+b is de(cid:12)ned as the sum of a and b. Addition satis(cid:12)es the commutative law a+b = b+a and the associative law (a+b)+c = a+(b+c): If a and b are elements of a (cid:12)eld, the equation c+a = b admits a unique solution c in the (cid:12)eld. The origin is a unique element 0 which satis(cid:12)es the identity 0+c = c for every element c of the (cid:12)eld. If a and b are elements of a (cid:12)eld, a unique element c = ab is de(cid:12)ned as the product of a and b. Multiplication satis(cid:12)es distributive laws c(a+b) = ca+cb and (a+b)c = ac+bc: Multiplication satis(cid:12)es the commutative law ab = ab and the associative law (ab)c = a(bc): If a and b are elements of a (cid:12)eld with b nonzero, the equation cb = a admits a unique solution c = a=b 4 LOUIS DE BRANGES April 27, 2010 in the (cid:12)eld. The unit is the unique element 1 of the (cid:12)eld which satis(cid:12)es the identity 1c = c for every element c. The rational numbers, which are ratios a and b with a and b integers of which b is nonzero, are a (cid:12)eld with the generally accepted de(cid:12)nitions of addition and multiplication. The concept of a (cid:12)eld is useful in explaining that there are related real numbers having the (cid:12)eld properties and that they are essential to the description of points on a line. This information was not new in the Renaissance but received its (cid:12)rst major applications then. A dynamical contribution of Isaac Newton (1642{1729) to mathematical analysis is to treat the origin of Cartesian coordinates as a center surrounded by the trajectories of mov- ingparticles. Momentum isintroduced asa concept which resembles positionsinceitliesin a space isomorphic to Cartesian space. Momentum is observable by its action on position. The motion of a particle is formulated as a voyage in time through a phase space which is composed of Cartesian space and momentum space. Implicit are mappings of phase space into itself which are de(cid:12)ned by the motion of particles in time. An evolution of the in(cid:12)nitessimal calculus is required for a solution of the equations of motion. Newton applies a limiting case of the calculus of (cid:12)nite di(cid:11)erences. The application to planetary motion owes its success to the understanding of Cartesian space obtained from the equations of motion. Applications of the in(cid:12)nitessimal calculus are typical of subsequent research results submitted to national scienti(cid:12)c academics. A fundamental treatment of the propagation of light was presented by Christian Huygens (1629{1695) to the Acad(cid:19)emie des Sciences. An in(cid:12)nite product of rational numbers converging to the area (cid:25) enclosed by a unit circle was presented by John Wallis (1616{1703) to the Royal Academy. An in(cid:12)nite sum of rational numbers converging to (cid:25) was discovered by Wilhelm Leibnitz (1646{1716), a member of both academies who founded a predecessor of the Preussische Akademie der Wissenschaften. In(cid:12)nite series whose sums are products representing (cid:25) were discovered by Jakob Bernoulli (1654{1705). Applications of the in(cid:12)nitessimal calculus which underlie computations of (cid:25) were explored by Johann Bernoulli (1667{1748). Complexanalysisoriginatesinthediscovery ofAbraham deMoivre(1667-1754)thatthe complex plane is a (cid:12)eld acceptable as domain of de(cid:12)nition for polynomials and functions represented by power series. The exponential function of a complex variable combines the sine and cosine functions of a real variable with the exponential function of a real variable to parametrize the complex plane in polar coordinates. TheNewtoninterpolationpolynomialsinthecalculus of(cid:12)nitedi(cid:11)erences aremotivating special cases of hypergeometric functions discovered by Leonard Euler (1707{1783). The gamma function appears in 1729 as an in(cid:12)nite limit of Newton polynomials. The classical zeta function is discovered in 1737 by analogy of its Euler product to the in(cid:12)nite product for the gamma function. The functional identity for the zeta function is obtained in 1761 by a calculation with hypergeometric series. Mathematical analysis was subsidized during the Enlightenment by absolute rulers who appliedtheresources ofemergingnationstotheperceivedneeds ofthegoverned. Catherine the Great in Petersburg and Frederick the Great in Potsdam followed the example of Louis XIV in Versailles by maintaining courts as centers of cultural, artistic, and scienti(cid:12)c activity. Leonard Euler was one of many contributors to scienti(cid:12)c knowledge who bene(cid:12)ted from this support. A derivation of the Newtonian equations of motion by minimizing an integral of the action of momentum on position was made by Jean le Rond d’Alembert (1717{1783). APOLOGY 5 The nonexistence of solutions of the Fermat equation for exponent three was proved by Comte Louis de Lagrange (1736{1813). The result is an application of the Euclidean algorithmfor the (cid:12)eld obtained by adjoining a square rootof three to the rationalnumbers. The algorithm determines solutions of the equation a3 +b3 = rc3 in positive integers a;b, and c when r is a given positive integer. TheFrench RevolutiondelayedappreciationofLagrange’smost signi(cid:12)cant contribution. Atheorem whichisattributedonindirect evidencetoDiophantus statesthatevery positive integer is the sum of four squares of integers. If he left a proof in the library of Alexandria, it was one of the many losses caused by (cid:12)re. The (cid:12)rst known proof was obtained by Lagrange. The algebra of the proof is clari(cid:12)ed by the quaternions of Rowan Hamilton (1805{1865). Students of the in(cid:12)nitessimal calculus learn about quaternions as vectors. A vector space is de(cid:12)ned over a coe(cid:14)cient (cid:12)eld. Vectors can be added to vectors to produce vectors. Vectors can be multiplied by elements of the (cid:12)eld to produce vectors. If a and b are vectors, a unique vector c = a+b is de(cid:12)ned as the sum of a and b. Addition satis(cid:12)es the commutative law a+b = b+a and the associative law (a+b)+c = a+(b+c): If a and b are vectors, the equation c+a = b admits a unique vector solution c. The origin is the unique vector 0 which satis(cid:12)es the identity 0+c = c for every vector c. If a is a vector and if b is an element of the (cid:12)eld or if a is an element of the (cid:12)eld and if b is a vector, a unique vector c = ab is de(cid:12)ned as the product of a and b. Multiplication satis(cid:12)es the distributive laws c(a+b) = ca+cb and (a+b)c = ac+bc whenever the products are meaningful. Multiplication satis(cid:12)es the commutative law ab = ba and the associative law (ab)c = a(bc) 6 LOUIS DE BRANGES April 27, 2010 whenever the products are meaningful. If a is a vector and if b is a nonzero element of the (cid:12)eld, the equation cb = a admits a unique vector solution c = a=b: The unit 1 of the (cid:12)eld satis(cid:12)es the identity 1c = c for every vector c. A skew{(cid:12)eld is constructed when a (cid:12)eld admits no representation 0 = a2 +b2 +c2 +d2 of zero with elements a;b;c, and d which are not all zero. The skew{(cid:12)eld is a vector space of dimension four over the (cid:12)eld which is spanned by elements i;j;k, and 1. Products are de(cid:12)ned by the multiplication table ij = k; jk = i; ki = j; ji = −k; kj = −i; ik = −j; ii = −1; jj = −1; kk = −1: If a and b are elements of the skew{(cid:12)eld, a unique element c = ab is de(cid:12)ned as the product of a and b. Multiplication satis(cid:12)es the distributive laws c(a+b) = ca+cb and (a+b)c = ac+bc: Multiplication satis(cid:12)es the associative law (ab)c = a(bc): If a and b are elements of the skew{(cid:12)eld with b nonzero, the equation cb = a admits a unique solution c = a=b in the skew{(cid:12)eld. The unit is the unique element 1 of the skew{(cid:12)eld which satis(cid:12)es the identity 1c = c for every element c. APOLOGY 7 The noncommutative nature of multiplication is compensated by conjugation, an anti{ automorphism c into c− which takes i into −i;j into −j, k into −k, and 1 into 1. The identity − − − (ab) = b a holdsfor allelements aand bof theskew{(cid:12)eld. A self{conjugate element coftheskew{(cid:12)eld is an element of the (cid:12)eld since it satis(cid:12)es the identity − c = c: A skew{conjugate element c of the skew{(cid:12)eld satis(cid:12)es the identity c− = −c: An element of the skew{(cid:12)eld is the unique sum of a self{conjugate element and a skew{ conjugate element. Multiplication as taught in the calculus decomposes a product into self{conjugate and skew{conjugate components. The Euclidean algorithm applied by Lagrange in the proof of the Diophantus theorem is clari(cid:12)ed by Adolf Hurwitz (1859{1919). A skew{(cid:12)eld is constructed from the (cid:12)eld of rational numbers. An element (cid:24) = d+ia+jb+kc of the skew{(cid:12)eld is de(cid:12)ned as integral if the coordinates a;b;c and d are all integers or if they are all halves of odd integers. A positive integer n is a sum of four squares of integers if, and only if, it admits a representation − n = (cid:24) (cid:24) with (cid:24) an integral element of the skew{(cid:12)eld. Sums and products of integral elements are integral. The conjugate of an integral element is integral. The product of a nonzero integral element with its conjugate is a positiveinteger. TheEuclideanalgorithmisasearchforanintegralelement whichsucceeds because a nonempty set of positive integers contains a least element. The search is made in the nonzero elements of a right ideal of integral elements. A right ideal is a set of integral elements which contains the origin, which contains the sum of any two elements, and which contains the product ab of an element a with every integral element b of the skew{(cid:12)eld. − If a nonzero element a of the ideal minimizes a a, then every element of the ideal is a product ab with an integral element b of the skew{(cid:12)eld. An estimate of the number of primes which are less than a given positive number was made by Adrien Marie Legendre (1752{1833). The Riemann hypothesis is a conjecture which treats the accuracy of the estimate. The Enlightenment is notable not only for the advancement of science but also for the dissemination of information. An Encyclop(cid:19)edie des Sciences, des Arts, et des M(cid:19)etiers supplied the needs of critical readers. The Encyclopedia Britannica was created as an equivalent in the English language. When publication of the original encyclopedia ceased intheFrenchRevolution, itssuccessor continued withinformativearticlesonmathematical analysis. Fourier analysis in the decomposition of a function which is subject to symmetries into elementary functions which exhibit these symmetries. The techniques of Joseph Fourier (1768{1830) implement this purpose by a relaxation of the accepted concept of function. 8 LOUIS DE BRANGES April 27, 2010 A function treated by Fourier need not be de(cid:12)ned prior to analysis. It is reconstructed indirectly by a determination of its symmetric components. Fourier analysis introduces a new perception of orbital motion. The Newtonian equa- tions of motion determine the orbits of an isolated particle. In Fourier analysis all orbits of the particle are treated with no preconception as to which is occupied. Some spacial variable is chosen to measure the probability that the particle has a given position at a given time. Fourier illustrates the method in his treatment of heat flow. The mechanism for trans- porting energy is irrelevant to his analysis. Temperature is observed as a function of position and time. A di(cid:11)erential equation is proposed which is generally accepted as cor- rect. Heat flow is distinct from previous treatments of motion since energy is lost in the process. A conflict with the law of conservation of energy is circumvented by declaring that not all energy is observed. The di(cid:11)erential equation for the flow of heat admits a solution which was previously introduced in celestial mechanics by Pierre Simon Marquis de Laplace (1749{1827). In the present application the Laplace transformation supplies a spectral analysis of the in(cid:12)nites- simal generator of heat flow. The in(cid:12)nitessimal generator is a di(cid:11)erential operator which is converted by the Laplace transformation into a multiplication operator. The Laplace transformation is fundamental to Fourier analysis because of its intimate relationship to the Fourier transformation. In applications to orbital motion the Fourier transformation acts on functions of position to produce functions of momentum. The Laplace transform of a function and the Laplace transform of its Fourier transform are easily computed from each other. This procedure is commonly employed in computations of Fourier transforms. The applications of the Laplace transformation in Fourier analysis demonstrate the fun- damental nature of heat flow. An insight into the nature of the Laplace transformation is supplied by Johann Radon (1887{1956). The Radon transformation formally factors the Fourier transformation for a plane as a composition with the Fourier transformation for a line. The Laplace transformation supplies a spectral analysis of the Radon transformation. Since the Laplace transformation supplies a spectral analysis for the in(cid:12)nitessimal gener- ator for the flow of heat, a relationship is found between the Radon transformation and the in(cid:12)nitessimal generator for the flow of heat. The Radon transformation is formally the inverse of the in(cid:12)nitessimal generator in the flow of heat. Anappreciationofthefundamental natureofheat flow isgainedthroughitsrelationship to the Radon transformation. If a space has the additive and topological properties which permit Fourier analysis, then the Cartesian product of the space with itself also has these properties. A Radon transformation relates Fourier analysis on the space with Fourier analysis on the Cartesian product. Heat flow is therefore a general phenomenon in Fourier analysis which competes in importance with the Fourier transformation. This observation underlies the proof of the Riemann hypothesis. An application of the Fourier transformation which discovers unexpected properties of the real line is due to Denis Poisson (1781{1840). The Poisson formula states that the sum of the values of an integrable function at the integers is equal to the sum of the values of its Fourier transform at the integers when the Fourier transform is integrable and Fourier inversion applies. The Poisson formula implies the functional identity for the Euler zeta function. Linear analysis is an aspect of mathematical analysis with applications in Fourier analy- sis. The functions treated by Fourier analysis belong to vector spaces and are subjected to linear transformations of which the Fourier transformation is a fundamental example. The APOLOGY 9 treatment of linear transformations is di(cid:14)cult even in spaces of (cid:12)nite dimension without a determination of invariant subspaces. A major contribution of Carl Friederich Gauss (1777{1855) is a construction of invariant subspaces for linear transformations of a vector space of (cid:12)nite dimension over the complex numbers into the same space. An invariant subspace is constructed in every dimension which admits a subspace. The invariant sub- spaces obtained are nested. If r is a positive integer, the integers modulo r inherit from the integers the additive structure permitting Fourier analysis. The topology of the (cid:12)nite set is discrete. The canonical measure assigns to every subset the number of its elements. The functions with complex values which are de(cid:12)ned on the integers modulo r form a vector space of dimension r which is mapped linearily into itself by the Fourier transformation. The Fourier transformation for the integers modulo r is more elementary than the Fourier transformation for a line since it is de(cid:12)ned by a (cid:12)nite sum. The Laplace transfor- mation for the integers modulo r is applied by Gauss to compute Fourier transforms. The Radon transformation decomposes the space of square integrable functions into orthogonal eigenfunctions with positive eigenvalues. An application given by Gauss is a proof of the Legendre law of quadratic reciprocity. The polynomials of degree less than r form a vector space of dimension r to which another application of invariant subspaces is made. Gaussian quadrature evaluates a non- negative linear functional on polynomials, which is de(cid:12)ned by integration on the real axis, as a sum over a (cid:12)nite set of real numbers determined as the zeros of a polynomial of degree r. Polynomials of degree r suitable for Gaussian quadrature are constructed from the hy- pergeometric series, a generalizationdue to Euler of the Newton interpolationpolynomials. Gaussian quadrature competes with prior results of Legendre as does the Gauss estimate for the number of primes with a given bound. The representation of functions by power series is so useful as to serve e(cid:11)ectively as a de(cid:12)nition of a function in complex analysis. An analytic function is de(cid:12)ned as one which is locally represented by power series. A fundamental theorem of complex analysis is due to Augustin Cauchy (1789{1859). A function f(z) of z in a plane region is analytic if, and only if, the function [f(z)−f(w)]=(z−w) of z is continuous in the region for every element w of the region when suitably de(cid:12)ned at w. The value at w de(cid:12)nes the derivative at w in the sense of complex analysis. The Cauchy formula, on which the characterization depends, states that the integral of a di(cid:11)erentiable function over a closed curve of (cid:12)nite length is equal to zero. The clari(cid:12)cation of hypotheses for the Cauchy formula assigns a purpose to complex analysis. A theorem of Camille Jordan (1838{1921) states that a simple closed curve divides the complex plane into a bounded region and an unbounded region. A necessary condition for the validity of the Cauchy formula is di(cid:11)erentiability in the bounded region. A su(cid:14)cient condition is observed by Bernhard Riemann (1826{1866). A Riemann mapping function is a function which is analytic in the unit disk and which de(cid:12)nes an injective mapping of the disk. The Cauchy formula is valid when the bounded region is the image of the unit disk under a Riemann mapping function. Complex analysis explores the complex plane by methods applied by Newton to Carte- sian space. The simple closed curves of complex analysis aretreated as the paths of moving particles. Theplaneregionsgeneratedminimizeanintegraldiscoveredby Lejeune Dirichlet (1805{1859). The Dirichlet principle is an analogue for the plane of the Alembert principle for Cartesian space. The properties of Dirichlet integrals led Riemann to state on inadequate proof that 10 LOUIS DE BRANGES April 27, 2010 every plane region bounded by a simple closed curve is the image of the unit disk under a Riemann mapping function. The proof given by Hermann Schwarz (1843{1921) applies an estimation theory for functions analytic and bounded by one in the unit disk. The interpolation theory of functions analytic and bounded by one in the unit disk obtained by L(cid:19)eopold Fej(cid:19)er (1880{1959) and Fr(cid:19)ed(cid:19)eric Riesz (1880{1956) is a systematic application of the initial Schwarz lemma which is given a de(cid:12)nitive formulation by Issai Schur (1875{ 1942). A related estimation theory for Riemann mapping functions originates with Ludwig Bieberbach (1886{1982). The Bieberbach conjecture states that the coe(cid:14)cients of a Rie- mann mapping function c +c z +c z2 +::: 0 1 2 satisfy the inequality jc j (cid:20) n n for every nonnegative integer n if the inequality is satis(cid:12)ed when n is zero and when n is one. The elementary proof given by Bieberbach for the second coe(cid:14)cient permits a simpli(cid:12)ed proof of the Schwarz theorem. The proof of the Bieberbach conjecture for the third coe(cid:14)cient by Karl Lo¨wner (1893{1968) parametrizes Riemann mapping functions generated by the paths of moving particles issuing from the origin. The proof of the Bieberbach conjecture for all coe(cid:14)cients obtained in 1984 by the author of the apology applies the L¨owner parametrization in conjunction with a variant of the Schur theory due to Helmut Grunsky (1904{1986). The Euler zeta function is constructed in Fourier analysis on the real line by Carl Jacobi (1804{1851). The construction is an application of the Laplace transformation for the line as it appears in the treatment of heat flow in the plane by Fourier. A compacti(cid:12)cation of the line is implicit since the flow of heat is con(cid:12)ned to a horizontal strip of width one. The theta function is a sum of translates of the Laplace kernel for the line which produces a function periodic of period one. The theta function permits a treatment of heat flow in the strip which adapts the treatment of Fourier in the plane. The Poisson summation formula implies a functional identity for the theta function which has no analogue for the Laplace kernel in the plane. The Euler zeta function and its functional identity are derived from the Jacobi theta function and its functional identity by the Mellin transformation, obtained by change of variables from the Fourier transformation for the line. A new interpretation of hypergeometric series is required for the properties of the theta function. Hypergeometric series are treated by Gauss as formal power series which are so- lutions of di(cid:11)erential equations of second order with quadratic coe(cid:14)cients. The coe(cid:14)cients of hypergeometric series satisfy recurrence relations familiar from the binomial formula. Special functions appearing in Fourier analysis on the real line or the complex plane are expressible in hypergeometric series. In applications made by Jacobi the hypergeometric series is treated by the represented function. Hypergeometric functions have ambiguous values since analytic continuation is applied for their de(cid:12)nition. Dirichlet zeta functions are constructed from Fourier analysis on the complex plane as a generalization of the construction made by Jacobi for the Euler zeta function. The Gauss determination of invariant subspaces for the Fourier transformation on the integers modulo r prepares the concept of a character modulo r. A Dirichlet theta function is a sum of translates of the Laplace kernel for the line which produces a function periodic of period r determined by a character modulo r. The methods of Fourier are applied to the flow of heat in a horizontal strip of width r. The Poisson summation formula implies a functional identity for a Dirichlet theta function. Dirichlet zeta functions are obtained by the Mellin