Analytic number theory

7 articles

Prime number theorem Riemann zeta function Dirichlet series L function Sieve method Distribution of primes Abstract analytic number theory Aperys constant Arithmetic zeta function Artin conjecture l functions Artin l function Barnes g function Basel problem Bateman horn conjecture Bombieri norm Bost connes system Brauer siegel theorem Bruns constant Buchstab function Burgess inequality Character sum Cramers conjecture Critical line theorem De bruijn newman constant Determinant method Deuring heilbronn phenomenon Dickman function Dirichlet character Dirichlet density Dirichlet l function Dirichlet series inversion Dirichlets theorem on arithmetic progressions Elliott halberstam conjecture Erdos arcsine law Erdos kac theorem Erdos tenenbaum ford constant Euler product Exponential sum Functional equation l function Generalized riemann hypothesis Gilbreath conjecture Glaisher kinkelin constant Grosswald schnitzer theorem Hardy littlewood circle method Heilbronn set Hilbert polya conjecture Huas lemma Igusa zeta function Jacobi sum Kloosterman sum Kubilius model Lambert series Landau ramanujan constant Legendres constant Lehmer pair Lindelof hypothesis Linniks theorem Local zeta function Maiers matrix method Meissel mertens constant Mertens conjecture Montgomerys pair correlation conjecture Multiplicative number theory Natural density Newmans conjecture Nivens constant Odlyzko schonhage algorithm Offset logarithmic integral Polya vinogradov inequality Porters constant Poussin proof Prime counting function Probabilistic number theory Proof that the sum of the reciprocals of the primes diverges Redheffer matrix Riemann hypothesis Schinzels hypothesis h Second hardy littlewood conjecture Selberg trace formula Siegel zero Skewess number Smooth number Stephens constant Stirlings approximation Turans method Van der corputs method Voronoi formula Weyls inequality number theory 15 and 290 theorems Abel elliptic functions Abels binomial theorem Additive basis Airy zeta function Alpha beta conjecture Alternating factorial Ap rys constant Ap rys theorem Arakawa kaneko zeta function Arccn Arcdn Arcsn Arithmetic combinatorics Arithmetic geometric mean Artin conductor Artin mazur zeta function Artins conjecture on primitive roots Automorphic l function Balanced polygamma function Barnes zeta function Bernoulli umbra Bernoullis triangle Beta distribution Beta function Beta negative binomial distribution Beurling zeta function Bhargava factorial Binomial Binomial approximation Binomial coefficient Binomial distribution Binomial regression Binomial series Binomial theorem Binomial transform Binomial type Birch and swinnerton dyer conjecture Birchs theorem Bohr mollerup theorem Bombieri vinogradov theorem Brocards problem Brownian motion and riemann zeta function Brumer bound Brumer stark conjecture Brun sieve Brunn minkowski theorem Bruns theorem Cameron erd's conjecture Carlson symmetric form Carlsons theorem Catalan number Centered polygonal number theorem Central binomial coefficient Chebotarev density theorem Chebyshev integral Chowla mordell theorem Chowla selberg formula Classical modular curve Clausen function Combinatorial number system Complex multiplication Conductor of an elliptic curve Congruent number Constant problem Counting points on elliptic curves Cram rs conjecture Curve25519 Curve448 Davenport constant De numeris triangularibus et inde de progressionibus arithmeticis magisteria magna Dedekind eta function Dedekind zeta function Degens eight square identity Difference polynomials Digamma function Dirichlet beta function Dirichlet eta function Divisor function Dixon elliptic functions Dixons identity Double factorial Doubling oriented doche icart kohel curve Doubly triangular number Dwork conjecture Dyson conjecture Dysons transform E function Edwards curve Egorychev method Eichler shimura congruence relation Eisenstein series Elliptic cohomology Elliptic curve point multiplication Elliptic function Elliptic gamma function Elliptic gauss sum Elliptic integral Elliptic rational functions Equianharmonic Equivariant l function Erd's szemer di theorem Erd's tetali theorem Erd's tur n conjecture on additive bases Euler integral Eulerian number Explicit formulae for l functions Exponential factorial Extended negative binomial distribution Fa di brunos formula Factorial moment Factorial moment generating function Factorial number system Factorial prime Factoriangular number Falling and rising factorials Fekete polynomial Feller tornier constant Fermat polygonal number theorem Fermats theorem on sums of two squares Fibonomial coefficient Finite difference Folkmans theorem Fox wright function Frans n robinson constant Freimans theorem Frey curve Friedlander iwaniec theorem Functional equation Fundamenta nova theoriae functionum ellipticarum Fundamental lemma of sieve theory Fundamental pair of periods Fuss catalan number Gamma distribution Gamma function Gan gross prasad conjecture Gauss lemniscate function Gaussian binomial coefficient Gautschis inequality Generalized gamma distribution Generalized hypergeometric function Generalized integer gamma distribution Generalized pochhammer symbol Genocchi number Goldbach conjecture Goldbachs weak conjecture Goldston pintz y ld r m sieve Goss zeta function Goulds sequence Grand riemann hypothesis Green tao theorem Grothendieck ogg shafarevich formula H lders theorem Hadamards gamma function Hadjicostass formula Half period ratio Hardy littlewood zeta function conjectures Hardy ramanujan littlewood circle method Hasse weil zeta function Hasses theorem on elliptic curves Hecke character Heegner point Height function Hermite interpolation Hesse configuration Hesse pencil Hessian form of an elliptic curve Hideo shimizu Hilbert p lya conjecture Hockey stick identity Hurwitz zeta function Hyperfactorial Hypergeometric distribution Hypergeometric function Hypergeometric identity Igusa variety Ihara zeta function Incomplete gamma function Inverse gamma distribution Inverse gamma function Inverse jacobi elliptic functions J invariant J line Jacobi elliptic functions Jacobi theta functions Jacobi triple product Jacobian curve Jordan p lya number Jurkat richert theorem K function Kempner function Knesers theorem Lagranges four square theorem Lah number Lanczos approximation Landens transformation Langlands deligne local constant Large sieve Larger sieve Lefschetz zeta function Legendre sieve Legendres formula Legendres relation Legendres three square theorem Lemniscate elliptic functions Lemoines conjecture Lerch transcendent Level structure Lindel f hypothesis Lis criterion Ljunggrens equation Lobb number Local langlands conjectures Lozani's triangle Macaulay representation of an integer Macmahon master theorem Mahler measure Mahlers 3 2 problem Mahlers theorem Manns theorem Matsumoto zeta function Mazurs torsion theorem Mestre bound Minakshisundaram pleijel zeta function Minimum overlap problem Minkowski addition Modular elliptic curve Modular group Modular lambda function Moduli of abelian varieties Montgomery curve Mordell curve Mordell weil group Mordell weil theorem Motivic l function Multinomial distribution Multinomial theorem Multiple gamma function Multiple zeta function Multiplication theorem Multiplicative partitions of factorials Multivariate gamma function N conjecture N rlund rice integral N ron differential N ron ogg shafarevich criterion Nagell lutz theorem Narayana number Negative binomial distribution Negative hypergeometric distribution Negative multinomial distribution Neville theta functions Newton pepys problem Newton polynomial Nome Nu function Odlyzko sch nhage algorithm On the number of primes less than a given magnitude P 384 P adic l function P lya vinogradov inequality Parity problem Parking function Particular values of the gamma function Particular values of the riemann zeta function Pascals pyramid Pascal triangle Period mapping Pfisters sixteen square identity Picard fuchs equation Pillai prime Pochhammer k symbol Poisson binomial distribution Poisson distribution Polite number Pollocks conjectures Polygamma function Polylogarithm Poncelets closure theorem Postage stamp problem Prime zeta function Primorial Proof of bertrands postulate Proof of the euler product formula for the riemann zeta function Pseudogamma function Pythagoras number Pythagorean quadruple Q gamma function Quarter period Quasibinomial Quintuple product identity Ramanujan petersson conjecture Ramanujan tau function Ramanujan theta function Rank of an elliptic curve Rankin selberg method Rational zeta series Raynauds isogeny theorem Real analytic eisenstein series Reciprocal gamma function Regularized gamma function Restricted sumset Riemann siegel formula Riemann siegel theta function Riemann xi function Riesel number Riesz function Riesz mean Rogers sieving theorem Romanovs theorem Rook polynomial Rothe hagen identity Ruelle zeta function Sato tate conjecture Schnirelmann density Schoofs algorithm Selberg class Selberg sieve Selberg zeta function Selbergs zeta function conjecture Semistable abelian variety Sheffer sequence Shimizu l function Shimura variety Shintani zeta function Siegel g function Sierpi ski triangle Sieve of pritchard Singmasters conjecture Spouges approximation Square difference free set Standard l function Star of david theorem Stark conjectures Stieltjes constants Stirling number Stirling transform Subgroup growth Sum free set Sum of two squares theorem Sums of three cubes Sumset Suns curious identity Superfactorial Supersingular elliptic curve Table of costs of operations in elliptic curves Table of newtonian series Tate curve Tate pairing Tates algorithm Tates thesis Telephone number The music of the primes Theta function Theta representation Triangular number Trigamma function Trinomial expansion Trinomial triangle Tripling oriented doche icart kohel curve Tur n sieve Tur ns method Turings method Twisted edwards curve Twisted hessian curves Twists of elliptic curves Vandermondes identity Vantieghems theorem Waldspurgers theorem Warings prime number conjecture Weierstrass elliptic function Weierstrass function Weil pairing Weils criterion Wielandt theorem Wilson prime Witten zeta function Wolstenholmes theorem Z function Zeta function Zeta function regularization Zeta function universality Zetagrid Abstract prime number theory Almost prime number Bombieri prime number theorem Chebyshev theorems on prime numbers Congruence modulo a prime number Differential functional equation Dirichlet boundary condition Dirichlet box principle Dirichlet criterion Dirichlet discontinuous multiplier Dirichlet formula Dirichlet laplacian Dirichlet polynomial Dirichlet principle Dirichlet series for an analytic almost periodic function Dirichlet tesselation Dirichlet theorem Dirichlet variational problem Distribution of prime numbers Epstein zeta function Euclidean prime number theorem Formal dirichlet series Hypo dirichlet algebra Inertial prime number Irregular prime number Luzin sieve Mutually prime numbers Prime ideal theorem Pseudo prime Quasi prime number Ramified prime ideal Regular prime number Zeta function method for regularization

Probability & Statistics LicenceExplication10 min

A Twenty-Step Tree Has 231 Nodes, or 2,097,151

Whether an up move followed by a down move lands where a down move followed by an up move lands decides between a quadratic node count and an exponential one, and at twenty steps the gap is a factor of 9,078.6. Both sums carry N+1 terms rather than N, because a twenty-step tree has twenty-one dates on it, and the off-by-one costs the entire final row. The article also states the recombination hypothesis exactly, which is weaker than the usual ud = 1.

Number Theory & Arithmetic LycéeExplication9 min

5050 Minus the Total, in One Pass

Sorting the list finds the gap and is merely wasteful, which is why the article says so rather than striking it out. The subtraction works because 5050 is a closed form available before the list is read, and the two conditions carrying it are distinctness and a known range. The article adds the duplicate-hunting mirror image, the sum-of-squares route when two values are absent, and the exclusive-or accumulator for when the total would overflow.

Number Theory & Arithmetic LycéeExplication9 min

A Clock Face in Three Pieces of 26, and No Pie Cut Ever Works

The dial totals 78, so each piece needs 26, and the pie instinct fails on all 220 possible cuts. The proof is three lines of triangular numbers: only one pair of running totals differs by 26, which forces the first two cracks and then demands a total of 62 that does not exist.

Probability & Statistics LycéeExplication8 min

Two Kings Off the Top: One in 221

Four over fifty-two squared is exactly right for the question where the first card goes back, which is what makes it hard to catch. Removing a king shrinks the numerator proportionally more than the denominator, and the counting route through 1326 two-card hands confirms one in 221 without mentioning order at all.

Recreational, History & Meta LycéeExplication9 min

Ten Bags, One Reading: 55 Coins Turn the Dial Into a Label

Taking one coin from each bag reads 9.9 ounces whichever bag is light, and the failure is blindness rather than imprecision. Loading i coins from bag i makes the dial an injective function of the culprit, at a cost of the tenth triangular number. A 45-coin variant is cheaper, and powers of two identify any subset of light bags from one reading.

Number Theory & Arithmetic LycéeExplication7 min

A Hundred Bulbs, 482 Flips, and Ten Survivors

Person k flips every bulb that is a multiple of k, and after a hundred passes exactly the ten perfect squares are lit. Bulb n is flipped once per divisor, and the pairing d against n/d is fixed-point free unless n is a square, so the parity is decided by algebra rather than by accumulation. The lit fraction is one over the square root of the row, and stopping the process at person 50 inverts the answer to 54 bulbs.

Probability & Statistics LicenceExplication8 min

Five Coins Against Four Is Exactly a Coin Flip

You toss five fair coins, I toss four, and you win on strictly more heads: the answer is exactly 256 of the 512 outcomes. Because you hold one coin more, "not strictly more heads" and "strictly more tails" are the same event, and turning every coin over is a bijection between them. The fifth coin is worth nearly fourteen percentage points over the 93/256 you would have without it, and none of that is an edge.