Pretentiousness in analytic number theory. Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog
|
|
- Ashley Simon
- 5 years ago
- Views:
Transcription
1 Pretentiousness in analytic number theory Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog
2 The number of primes up to x Gauss, Christmas eve 1849: As a boy of 15 or 16, I determined that, at around x, the primes occur with density 1 ln x.
3 The number of primes up to x Gauss, Réveillon de Noël 1849: Quand j étais un garçon de 15 ou 16 ans, j ai determiné que, autour x, les nombres premiers ont la densité 1 ln x. #{premiers x} [x] n=2 1 ln n
4 The number of primes up to x Gauss, Christmas eve 1849: As a boy of 15 or 16, I determined that, at around x, the primes occur with density 1 ln x. #{primes x} [x] n=2 x 2 1 ln n dt ln t = Li(x)
5 The number of primes up to x Gauss, Christmas eve 1849: As a boy of 15 or 16, I determined that, at around x, the primes occur with density 1 ln x. #{primes x} [x] n=2 x 2 x ln x 1 ln n dt ln t = Li(x)
6 Ici Li(x) := x 2 dt ln t x π(x) = #{premiers x} Excès: [Li(x) π(x)]
7 Here Li(x) := x 2 dt ln t x π(x) = #{primes x} Overcount: [Li(x) π(x)] Guess: 0 < Li(x) π(x) < π(x).
8 x π(x) = #{premiers x} Excès: [Li(x) π(x)] On devins: 0 < x 2 dt ln t π(x) < π(x). L hypothèse de Riemann : x dt ln t π(x) x ln x. 2 State belief in PNT
9 Théorème des nombres premiers π(x) x log x.
10 Prime Number Theorem: Proof π(x) x log x. Define ζ(s) := 1 n s = ( 1 1 ) 1 p s, Re(s) > 1 n 1 p prime (s 1)ζ(s) has analytic continuation to all of C
11 Prime Number Theorem: Proof π(x) x log x. Define ζ(s) := 1 n s = ( 1 1 ) 1 p s, Re(s) > 1 n 1 p prime (s 1)ζ(s) has analytic continuation to all of C Riemann Hypothesis: ζ(σ + it) 0 if σ > 1 2 x dt ln t π(x) x ln x. 2
12 Prime Number Theorem: Proof π(x) x log x. Define ζ(s) := 1 n s = ( 1 1 ) 1 p s, Re(s) > 1 n 1 p prime (s 1)ζ(s) has analytic continuation to all of C Riemann Hypothesis: ζ(σ + it) 0 if σ > 1 2 x dt ln t π(x) x ln x. PNT : π(x) x log x 2 ζ(1 + it) 0, t R Proved in 1896 by Hadamard and de la Vallée Poussin.
13 Prime Number Theorem: Proof π(x) x log x. Define ζ(s) := 1 n s = ( 1 1 ) 1 p s, Re(s) > 1 n 1 p prime (s 1)ζ(s) has analytic continuation to all of C Riemann Hypothesis: ζ(σ + it) 0 if σ > 1 2 x dt ln t π(x) x ln x. PNT : π(x) x log x 2 ζ(1 + it) 0, t R Proved in 1896 by Hadamard and de la Vallée Poussin. Tactic: If ζ(1+it) = 0 then ζ(1+2it) =, contradiction
14 héorème des nombres premiers: Démonstration ζ(1 + it) 0, t R (Hadamard et de la Vallée Poussin, 1896) On utilise que (s 1)ζ(s) est analytique
15 héorème des nombres premiers: Démonstration ζ(1 + it) 0, t R (Hadamard et de la Vallée Poussin, 1896) On utilise que (s 1)ζ(s) est analytique Stratégie: ζ(1 + it) = 0 ζ(1 + + it) c r pour c 0, r 1 si est petite... ζ(1 + 2it) =, contradiction
16 héorème des nombres premiers: Démonstration ζ(1 + it) 0, t R (Hadamard et de la Vallée Poussin, 1896) On utilise que (s 1)ζ(s) est analytique Stratégie: ζ(1 + it) = 0 ζ(1 + + it) c r pour c 0, r 1 si est petite... ζ(1 + 2it) =, contradiction Preuve habituelle: On utilize l indentité de Mertens: pour σ > 1, ζ(σ) 3 ζ(σ + it) 4 ζ(σ + 2it) 1. Maintenant, ζ(1 + ) 1/ ainsi 3 c 4r ζ( it) 1. comme 0 +. Alors ζ(1+ +2it) doit être au moins c / 4r 3, et donc a une pole à s = 1, contradiction.
17 Prime Number Theorem: Proof ζ(1 + it) 0, t R (Hadamard and de la Vallée Poussin, 1896) Use that (s 1)ζ(s) is analytic Tactic: ζ(1 + it) = 0 ζ(1 + + it) c r for c 0, r 1 if is small... ζ(1 + 2it) =, Contradiction! Preuve prétentieuse: Take = 1/ log x = ζ(1 + + it) ( 1 1 ) 1 p p x 1+it c 1 (log x) r; Now 1 1/p 1+it 1 + 1/p so that c 1 (log x) r ( ) 1 c 2 p log x, p x so r = 1.
18 Hence r = 1 and ( 1 1 ) 1 p p x 1+it 1 log x p x ( ) 1 p and so 1/p 1+it must be close to 1/p for most p x; that is p it pretends to be 1.
19 Hence r = 1 and ( 1 1 ) 1 p p x 1+it 1 log x p x ( ) 1 p and so 1/p 1+it must be close to 1/p for most p x; that is p it pretends to be 1. But then p 2it pretends to be ( 1) 2 = 1, and so ) 1 p x ( 1 1 p 1+2it p x ( 1 1 p) 1 log x, so ζ( it) 1/, and thus ζ has a pole at 1 + 2it, Contradiction!
20 Hence r = 1 and ( 1 1 ) 1 p p x 1+it 1 log x p x ( ) 1 p and so 1/p 1+it must be close to 1/p for most p x; that is p it pretends to be 1. But then p 2it pretends to be ( 1) 2 = 1, and so ) 1 p x ( 1 1 p 1+2it p x ( 1 1 p) 1 log x, so ζ( it) 1/, and thus ζ has a pole at 1 + 2it, Contradiction! We would like a measure for pretentiousness : Perhaps above could be written: D(p it, 1; x) 1 which implies (by a -inequality) that D(p 2it, 1; x) 2D(p it, 1; x) 1.
21 The Prime Number Theorem π(x) x log x. More explicitly: For any ɛ > 0 there exists x ɛ such that if x x ɛ then π(x) x log x ɛ x log x.
22 The Prime Number Theorem π(x) x log x. More explicitly: For any ɛ > 0 there exists x ɛ such that if x x ɛ then π(x) x log x ɛ x log x. Let π(x; q, a) = #{p x : p a (mod q)}. Analogous proof gives, if (a, q) = 1 and x x ɛ,q then x π(x; q, a) φ(q) log x ɛ x φ(q) log x.
23 The Prime Number Theorem π(x) x log x. More explicitly: For any ɛ > 0 there exists x ɛ such that if x x ɛ then π(x) x log x ɛ x log x. Let π(x; q, a) = #{p x : p a (mod q)}. Analogous proof gives, if (a, q) = 1 and x x ɛ,q then x π(x; q, a) φ(q) log x ɛ x φ(q) log x. How small can we take x ɛ,q, for a given ɛ? Unconditionally: x ɛ,q = e q Assuming GRH: x ɛ,q = q 2+ɛ We believe: x ɛ,q = q 1+ɛ
24 Les fonctions multiplicatives f : N... tel que f(mn) = f(m)f(n) quand (m, n) = 1. Souvent f : N U := {z C : z 1}, et f est totalement multiplicatif: f(mn) = f(m)f(n) m, n 1 Exemples: f(n) = 1 f(n) = n it pour un t R Les caractières de Dirichlet: χ : (Z/qZ) G U où G est l ensemble des mième racines d unité, une groupe finite. f(n) = χ(n)n it
25 Connection/Lien? The Mőbius function { ( 1) k if n = p µ(n) = 1 p 2... p k 0 if p 2 n is multiplicative and we have the useful identity { log p if n = p e µ(a) log b = 0 otherwise n=ab Summing this identity over all n x we obtain log p = µ(a) log b. p e x ab x
26 Connection? The Mőbius function { ( 1) k if n = p µ(n) = 1 p 2... p k 0 if p 2 n is multiplicative and we have the useful identity { log p if n = p e µ(a) log b = 0 otherwise n=ab Summing this identity over all n x we obtain log p = µ(a) log b. p e x ab x Fix a, sum over b; easy since b B log b = log B! accurate estimates using Stirling s formula. Leaves us with a sum like x µ(a) (log(x/a) 1), a a x
27 The prime number theorem is equivalent to x a x µ(a) a (log(x/a) 1),
28 The prime number theorem is equivalent to x a x µ(a) a (log(x/a) 1), We can evaluate this if we can show that µ(n) = o(n). n N In fact this is equivalent to the PNT.
29 The prime number theorem is equivalent to x a x µ(a) a (log(x/a) 1), We can evaluate this if we can show that µ(n) = o(n). n N In fact this is equivalent to the PNT. Let λ(n) = ( 1) #{prime powers pe n}. Prime number theorem is also equivalent to λ(n) = o(n). which leads us to: n N
30 The Prime Number Theorem (version 2) For any ɛ > 0 there exists x ɛ such that if x x ɛ then λ(n) ɛx. n x Moreover if x x ɛ,q and (a, q) = 1 then λ(n) ɛ x n x q. n a (mod q)
31 Mean values of multiplicative functions PNT is n N λ(n) = o(n), which is 1 λ(n) = 0. N lim N n N
32 Mean values of multiplicative functions PNT is n N lim N λ(n) = o(n), which is 1 λ(n) = 0. N lim N n N Now λ(n) = 1 or 1 n, so half λ(n) are 1, half 1; ie for g = 1 or 1, PNT is equivalent to 1 #{n N : λ(n) = g} N exists and equals 1 2.
33 Mean values of multiplicative functions PNT is n N lim N λ(n) = o(n), which is 1 λ(n) = 0. N lim N n N Now λ(n) = 1 or 1 n, so half λ(n) are 1, half 1; ie for g = 1 or 1, PNT is equivalent to 1 #{n N : λ(n) = g} N exists and equals 1 2. For totally mult f : N G, finite group, does lim N 1 N exist, for each g G? #{n N : f(n) = g}
34 Mean values of multiplicative functions PNT is n N lim N λ(n) = o(n), which is 1 λ(n) = 0. N lim N n N Now λ(n) = 1 or 1 n, so half λ(n) are 1, half 1; ie for g = 1 or 1, PNT is equivalent to 1 #{n N : λ(n) = g} N exists and equals 1 2. For totally mult f : N G, finite group, does lim N 1 N exist, for each g G? Yes, Ruzsa/ Halasz-Wirsing #{n N : f(n) = g}
35 Mean values of multiplicative functions Can we get upper and lower bounds on 1 lim #{n N : f(n) = g} N N that depend on only on G?
36 Mean values of multiplicative functions Can we get upper and lower bounds on 1 lim #{n N : f(n) = g} N N that depend on only on G? E.g. G = { 1, 1}. f(n) = 1 n = limit = 1.
37 Mean values of multiplicative functions Can we get upper and lower bounds on 1 lim #{n N : f(n) = g} N N that depend on only on G? E.g. G = { 1, 1}. f(n) = 1 n = limit = 1. Can we get limit = 1? (f(2) = f(3) = 1 = f(6) = +1)
38 Mean values of multiplicative functions Can we get upper and lower bounds on 1 lim #{n N : f(n) = g} N N that depend on only on G? E.g. G = { 1, 1}. f(n) = 1 n = limit = 1. Can we get limit = 1? (f(2) = f(3) = 1 = f(6) = +1) Hall (1984): No!
39 Mean values of multiplicative functions Can we get upper and lower bounds on 1 lim #{n N : f(n) = g} N N that depend on only on G? E.g. G = { 1, 1}. f(n) = 1 n = limit = 1. Can we get limit = 1? (f(2) = f(3) = 1 = f(6) = +1) Hall (1984): No! Heath-Brown and Montgomery conjectured actual limit. That is, what is the minimum possible proportion of f(n)-values that equal 1 when G = { 1, 1}? Sound/Granville (2001): MinProp = 1 is 17.15% 1 π2 6 log(1 + e) log e 1 + e + 2 n=1 1 n 2 1 (1 + e) n
40 Mean values of multiplicative functions For totally mult f : N G, finite group, 1 lim #{n N : f(n) = g} N N exists, for each g G, so that 1 ( ) f(n). N exists. lim N n N
41 Mean values of multiplicative functions For totally mult f : N G, finite group, 1 lim #{n N : f(n) = g} N N exists, for each g G, so that 1 ( ) f(n). N lim N n N exists. What about when f : N U?
42 Mean values of multiplicative functions For totally mult f : N G, finite group, 1 lim #{n N : f(n) = g} N N exists, for each g G, so that 1 ( ) f(n). N lim N n N exists. What about when f : N U? Example: f(n) = n it has mean value 1 n it 1 N u it dt = 1 N 1+it N N N 1 + it = N it 1 + it, n N 0 which equals 1/ 1 + t 2 in absolute value, but varies in angle with N. That is,
43 (+) lim N exists but ( ) does not. lim N 1 N 1 N f(n). f(n). n N n N
44 (+) lim N exists but ( ) lim N 1 N 1 N n N n N f(n) f(n). does not. In fact (+) does exist for all totally multiplicative f : N U, but (*) does not necessarily exist..
45 (+) lim N exists but ( ) lim N 1 N 1 N n N n N f(n) f(n). does not. In fact (+) does exist for all totally multiplicative f : N U, but (*) does not necessarily exist. Are there other examples where (*) fails?.
46 (+) lim N exists but ( ) lim N 1 N 1 N n N n N f(n) f(n). does not. In fact (+) does exist for all totally multiplicative f : N U, but (*) does not necessarily exist. Are there other examples where (*) fails? Halasz (1975) No! If the mean value of f is large in absolute value then f(n) pretends to be n it for some small real t..
47 Mean values of multiplicative functions Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t.
48 Mean values of multiplicative functions Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t. Pretends means that 1 Re(f(p)/p it ) p p N is bounded.
49 Mean values of multiplicative functions Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t. Pretends means that 1 Re(f(p)/p it ) p p N is bounded. Distance function for pairs of multiplicative functions: If f and g are two multiplicative functions with values inside or on the unit circle define D(f, g; x) 2 := 1 Re(f(p)g(p)), p p x satisfies the triangle inequality D(f, g; x) + D(F, G; x) D(fF, gg; x).
50 Large character sums, I Non-principal character χ (mod q). Want ( ) χ(n) = o(x). for small x. n x
51 Large character sums, I Non-principal character χ (mod q). Want ( ) χ(n) = o(x). n x for small x. Burgess (1962): Proved for x > q 1/4+o(1). Want to show this for x q ɛ.
52 Large character sums, I Non-principal character χ (mod q). Want ( ) χ(n) = o(x). n x for small x. Burgess (1962): Proved for x > q 1/4+o(1). Want to show this for x q ɛ. Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t.
53 Large character sums, I Non-principal character χ (mod q). Want ( ) χ(n) = o(x). n x for small x. Burgess (1962): Proved for x > q 1/4+o(1). Want to show this for x q ɛ. Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t. If (*) fails then χ(n) pretends to be n it (by Halasz s theorem); hence χ 2 (n) pretends to be n 2it, so we expect that (*) fails with χ replaced by χ 2.
54 Large character sums, I Non-principal character χ (mod q). Want ( ) χ(n) = o(x). n x for small x. Burgess (1962): Proved for x > q 1/4+o(1). Want to show this for x q ɛ. Halasz (1975) If mean value of f is large in absolute value then f(n) pretends to be n it for some small real t. If (*) fails then χ(n) pretends to be n it (by Halasz s theorem); hence χ 2 (n) pretends to be n 2it, so we expect that (*) fails with χ replaced by χ 2. Indeed we can show: If (*) fails for χ = χ i (mod q) for some x i > q ɛ for i = 1 and 2, then (*) fails for χ = χ 1 χ 2 for some x > q δ with δ = δ(ɛ) > 0.
55 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x
56 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x Pólya-Vinogradov (1919): q log q
57 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x Pólya-Vinogradov (1919): q log q Montgomery- Vaughan (1977): GRH = c q log log q
58 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x Pólya-Vinogradov (1919): q log q Montgomery- Vaughan (1977): GRH = c q log log q Paley (1932): χ = (./q), order 2, with c q log log q Pf: Primes q s.t. (n/q) pretends to be 1 n (log q) c.
59 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x Pólya-Vinogradov (1919): q log q Montgomery- Vaughan (1977): GRH = c q log log q Paley (1932): χ = (./q), order 2, with c q log log q Pf: Primes q s.t. (n/q) pretends to be 1 n (log q) c. If primes q s.t. χ(n) pretends to be 1 n q c then x s.t. n x χ(n) > c q log q Don t believe exist, but...
60 Large character sums, II How large can max χ χ(n) be? Periodicity = q. n x Pólya-Vinogradov (1919): q log q Montgomery- Vaughan (1977): GRH = c q log log q Paley (1932): χ = (./q), order 2, with c q log log q Pf: Primes q s.t. (n/q) pretends to be 1 n (log q) c. If primes q s.t. χ(n) pretends to be 1 n q c then x s.t. n x χ(n) > c q log q Don t believe exist, but... Sound-G (2007): If n x χ(n) > q(log q) 1 δ then χ pretends to be ψ (mod m) where m (log q) 1/3, and χ( 1)ψ( 1) = 1.
61 Sound-G: If χ (mod q) has odd order g > 1 then χ(n) q(log q) 1 δg+o(1), n x where δ g = 1 2 sin(π/g) (1 ) > 0 (1 δ π/g 3 11/12).
62 Sound-G: If χ (mod q) has odd order g > 1 then χ(n) q(log q) 1 δg+o(1), n x where δ g = 1 sin(π/g) 2 (1 ) > 0 (1 δ π/g 3 11/12). Assuming GRH χ(n) q(log log q) 1 δg+o(1). n x
63 Sound-G: If χ (mod q) has odd order g > 1 then χ(n) q(log q) 1 δg+o(1), n x where δ g = 1 sin(π/g) 2 (1 ) > 0 (1 δ π/g 3 11/12). Assuming GRH χ(n) q(log log q) 1 δg+o(1). n x MV need RH for L(s, χψ) not L(s, χ) (χ pretends to be ψ, so χψ pretends to be 1).
64 Sound-G: If χ (mod q) has odd order g > 1 then χ(n) q(log q) 1 δg+o(1), n x where δ g = 1 sin(π/g) 2 (1 ) > 0 (1 δ π/g 3 11/12). Assuming GRH χ(n) q(log log q) 1 δg+o(1). n x MV need RH for L(s, χψ) not L(s, χ) (χ pretends to be ψ, so χψ pretends to be 1). If large character sum for χ, then χ pretends to be ψ (mod m), with m small, χ( 1)ψ( 1) = 1, so χ 3 pretends to be ψ 3 (mod m) with χ 3 ( 1)ψ 3 ( 1) = ( 1) 3 = 1 = a large character sum for χ 3. E.g. Large character sum for character of Order 6 (mod q) implies large character sum for character of Order 6 (mod q)
65 Mult fns in arithmetic progressions Halasz: n N f(n) large = f(n) is n it -pretentious. 1 N If f(n) = n it, or if f(n) = χ(n) then 1 f(n) N n N n a (mod q) is large. Also if f(n) = χ(n)n it, or anything f which is χ(n)n it -pretentious. Any others?
66 Mult fns in arithmetic progressions Halasz: n N f(n) large = f(n) is n it -pretentious. 1 N If f(n) = n it, or if f(n) = χ(n) then 1 f(n) N n N n a (mod q) is large. Also if f(n) = χ(n)n it, or anything f which is χ(n)n it -pretentious. Any others? BSG: No! If mean value of f is large in an arithm prog mod q then f(n) is χ(n)n it -pretentious for some Dirichlet character χ mod q and some small real t.
67 Pretentiousness is repulsive Can f be pretentious two ways?
68 Pretentiousness is repulsive Can f be pretentious two ways? Can f(n) ψ(n)n it and f(n) χ(n)n iu for most n x? If so then χ(n)n iu ψ(n)n it, so that (χψ)(n) n i(u t) for most n x, which is impossible as χψ has small conductor. Why repulsive?
69 Pretentiousness is repulsive Can f be pretentious two ways? Can f(n) ψ(n)n it and f(n) χ(n)n iu for most n x? If so then χ(n)n iu ψ(n)n it, so that (χψ)(n) n i(u t) for most n x, which is impossible as χψ has small conductor. More precisely, D(f(n), χ(n)n iu ; x) + D(f(n), ψ(n)n it ; x) D((χψ)(n), n i(u t) ; x) (log log x) 1/2.
70 Exponential sums If n x f(n)e2iπnα is large then α is close to some rational a/b with b small (MV); f(n) is ψ(n)n it where ψ (mod b)-pretentious, t small.
71 Exponential sums If n x f(n)e2iπnα is large then α is close to some rational a/b with b small (MV); f(n) is ψ(n)n it where ψ (mod b)-pretentious, t small. Multiplicative f : N { 1, 1}. What proportion of a, b, c N : a + b = c satisfy f(a) = f(b) = f(c) = 1? At least 1 2 % (which is really (17.15%)3 ).
72 Exponential sums If n x f(n)e2iπnα is large then α is close to some rational a/b with b small (MV); f(n) is ψ(n)n it where ψ (mod b)-pretentious, t small. Multiplicative f : N { 1, 1}. What proportion of a, b, c N : a + b = c satisfy f(a) = f(b) = f(c) = 1? At least 1 2 % (which is really (17.15%)3 ). If f 1, f 2, f 3 are totally mult fns in U, s.t. f 1 (a)f 2 (b)f 3 (c) ɛ N 2 2 a,b,c N a+b=c then f j (n) is ψ j (n)n it j-pretentious with ψ 1 ψ 2 ψ 3 = 1.
73 PNT in aps Gallagher (1970) Let λ be Liouville s function. Given ɛ > 0 there exists A > 1 such that λ(n) ɛ x n x q n a (mod q) for all (a, q) = 1 and all q x 1/A
74 PNT in aps Gallagher (1970) Let λ be Liouville s function. Given ɛ > 0 there exists A > 1 such that λ(n) ɛ x n x q n x n a (mod q) n a (mod q) for all (a, q) = 1 and all q x 1/A, except perhaps q that are multiples of some exceptional modulus r. If such a modulus r exists then there is a character ψ (mod r) such that λ(n) ψ(a) λ(n) ɛ x q n x n 1 (mod q) whenever (a, q) = 1 and r divides q, with q x 1/A. If so then λ(n) is ψ(n)n it pretentious for small t R.
75 PNT in aps If this last case occurs, that is if λ(n) is ψ(n)n it pretentious, it would contradict GRH. In fact since λ(n) is real-valued one can deduce from this that t = 0, ψ must be a real-valued character, there is a zero of the L(s, ψ) lying very close to s = 1.
76 PNT in aps If this last case occurs, that is if λ(n) is ψ(n)n it pretentious, it would contradict GRH. In fact since λ(n) is real-valued one can deduce from this that t = 0, ψ must be a real-valued character, there is a zero of the L(s, ψ) lying very close to s = 1. This exceptional zero is known as a Siegel zero. Given x, modulus r, character ψ and the zero are all unique the zeros of Dirichlet L-functions repel one another, a concept believed to lie deep in the theory of zeta-functions.
77 PNT in aps Gallagher (1970) Let λ be Liouville s function. Given ɛ > 0 there exists A > 1 such that λ(n) ɛ x n x q n x n a (mod q) n a (mod q) for all (a, q) = 1 and all q x 1/A, except perhaps q that are multiples of some exceptional modulus r. If such a modulus r exists then there is a character ψ (mod r) such that λ(n) ψ(a) λ(n) ɛ x q n x n 1 (mod q) whenever (a, q) = 1 and r divides q, with q x 1/A. If so then λ(n) is ψ(n)n it pretentious for small t R.
78 Mult fns in aps BSG (2008) Let f : N U totally mult function. Given ɛ > 0 there exists A > 1 such that f(n) ɛ x n x q n x n a (mod q) n a (mod q) for all (a, q) = 1 and all q x 1/A, except perhaps q that are multiples of some exceptional modulus r. If such a modulus r exists then there is a character ψ (mod r) such that f(n) ψ(a) f(n) ɛ x q n x n 1 (mod q) whenever (a, q) = 1 and r divides q, with q x 1/A. If so then f(n) is ψ(n)n it pretentious for small t R.
79 Primes in aps Suppose λ(n) is ψ(n)n it pretentious Given x, the modulus r, character ψ, and small real t, are all unique, which is a consequence of the fact that the zeros of Dirichlet L-functions repel one another, a concept believed to lie deep in the theory of zeta-functions. Mult fns in aps Suppose f(n) is ψ(n)n it pretentious Given x, the modulus r, character ψ, and small real t, are all unique, which is a consequence of the fact that pretentiousness is repulsive, which is not deep. How deep is our proof? Conclusions?
80 How deep? Original proof used deep results on distn of primes. But how can such a combinatorial result need such depth?
81 How deep? Original proof used deep results on distn of primes. But how can such a combinatorial result need such depth? Selberg (in his elt pf of PNT for arith progs): log x log p + log p 1 log p 2 p x p a (mod q) p 1 p 2 x p 1 p 2 a (mod q) 2x log x φ(q) for (a, q) = 1, for x e q (too large for us).
82 How deep? Original proof used deep results on distn of primes. But how can such a combinatorial result need such depth? Selberg (in his elt pf of PNT for arith progs): log x log p + log p 1 log p 2 p x p a (mod q) p 1 p 2 x p 1 p 2 a (mod q) 2x log x φ(q) for (a, q) = 1, for x e q (too large for us). Friedlander (1981): Close to true when x q B which is good enough for us!
83 How deep? Original proof used deep results on distn of primes. But how can such a combinatorial result need such depth? Selberg (in his elt pf of PNT for arith progs): log x log p + log p 1 log p 2 p x p a (mod q) p 1 p 2 x p 1 p 2 a (mod q) 2x log x φ(q) for (a, q) = 1, for x e q (too large for us). Friedlander (1981): Close to true when x q B which is good enough for us! Conclusions, II?
PRETENTIOUSNESS IN ANALYTIC NUMBER THEORY. Andrew Granville
PRETETIOUSESS I AALYTIC UMBER THEORY Andrew Granville Abstract. In this report, prepared specially for the program of the XXVième Journées Arithmétiques, we describe how, in joint work with K. Soundararajan
More informationTheorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x.
Chapter 1 Introduction 1.1 The Prime Number Theorem In this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if lim f(/g( = 1, and denote by log the natural
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f g as if lim
More informationResearch Statement. Enrique Treviño. M<n N+M
Research Statement Enrique Treviño My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting
More informationThe Least Inert Prime in a Real Quadratic Field
Explicit Palmetto Number Theory Series December 4, 2010 Explicit An upperbound on the least inert prime in a real quadratic field An integer D is a fundamental discriminant if and only if either D is squarefree,
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if
More informationOn the low-lying zeros of elliptic curve L-functions
On the low-lying zeros of elliptic curve L-functions Joint Work with Stephan Baier Liangyi Zhao Nanyang Technological University Singapore The zeros of the Riemann zeta function The number of zeros ρ of
More informationThe Prime Number Theorem
Chapter 3 The Prime Number Theorem This chapter gives without proof the two basic results of analytic number theory. 3.1 The Theorem Recall that if f(x), g(x) are two real-valued functions, we write to
More informationExplicit Bounds for the Burgess Inequality for Character Sums
Explicit Bounds for the Burgess Inequality for Character Sums INTEGERS, October 16, 2009 Dirichlet Characters Definition (Character) A character χ is a homomorphism from a finite abelian group G to C.
More informationThe Riemann Hypothesis
The Riemann Hypothesis Matilde N. Laĺın GAME Seminar, Special Series, History of Mathematics University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin March 5, 2008 Matilde N. Laĺın
More informationCharacter Sums to Smooth Moduli are Small
Canad. J. Math. Vol. 62 5), 200 pp. 099 5 doi:0.453/cjm-200-047-9 c Canadian Mathematical Society 200 Character Sums to Smooth Moduli are Small Leo Goldmakher Abstract. Recently, Granville and Soundararajan
More informationLecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method. Daniel Goldston
Lecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method Daniel Goldston π(x): The number of primes x. The prime number theorem: π(x) x log x, as x. The average
More informationThe ternary Goldbach problem. Harald Andrés Helfgott. Introduction. The circle method. The major arcs. Minor arcs. Conclusion.
The ternary May 2013 The ternary : what is it? What was known? Ternary Golbach conjecture (1742), or three-prime problem: Every odd number n 7 is the sum of three primes. (Binary Goldbach conjecture: every
More informationSOME MEAN VALUE THEOREMS FOR THE RIEMANN ZETA-FUNCTION AND DIRICHLET L-FUNCTIONS D.A. KAPTAN, Y. KARABULUT, C.Y. YILDIRIM 1.
SOME MEAN VAUE HEOREMS FOR HE RIEMANN ZEA-FUNCION AND DIRICHE -FUNCIONS D.A. KAPAN Y. KARABUU C.Y. YIDIRIM Dedicated to Professor Akio Fujii on the occasion of his retirement 1. INRODUCION he theory of
More informationLes chiffres des nombres premiers. (Digits of prime numbers)
Les chiffres des nombres premiers (Digits of prime numbers) Joël RIVAT Institut de Mathématiques de Marseille, UMR 7373, Université d Aix-Marseille, France. joel.rivat@univ-amu.fr soutenu par le projet
More informationGauss and Riemann versus elementary mathematics
777-855 826-866 Gauss and Riemann versus elementary mathematics Problem at the 987 International Mathematical Olympiad: Given that the polynomial [ ] f (x) = x 2 + x + p yields primes for x =,, 2,...,
More information1 Primes in arithmetic progressions
This course provides an introduction to the Number Theory, with mostly analytic techniques. Topics include: primes in arithmetic progressions, zeta-function, prime number theorem, number fields, rings
More informationBefore giving the detailed proof, we outline our strategy. Define the functions. for Re s > 1.
Chapter 7 The Prime number theorem for arithmetic progressions 7. The Prime number theorem We denote by π( the number of primes. We prove the Prime Number Theorem. Theorem 7.. We have π( as. log Before
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationMULTIPLICATIVE MIMICRY AND IMPROVEMENTS OF THE PÓLYA-VINOGRADOV INEQUALITY
MULTIPLICATIVE MIMICRY AND IMPROVEMENTS OF THE PÓLYA-VINOGRADOV INEQUALITY by Leo I. Goldmakher A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
More informationRiemann s Zeta Function and the Prime Number Theorem
Riemann s Zeta Function and the Prime Number Theorem Dan Nichols nichols@math.umass.edu University of Massachusetts Dec. 7, 2016 Let s begin with the Basel problem, first posed in 1644 by Mengoli. Find
More informationEXPLICIT RESULTS ON PRIMES. ALLYSA LUMLEY Bachelor of Science, University of Lethbridge, 2010
EXPLICIT RESULTS ON PRIMES. ALLYSA LUMLEY Bachelor of Science, University of Lethbridge, 200 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment
More informationMTH598A Report The Vinogradov Theorem
MTH598A Report The Vinogradov Theorem Anurag Sahay 11141/11917141 under the supervision of Dr. Somnath Jha Dept. of Mathematics and Statistics 4th November, 2015 Abstract The Goldbach conjecture is one
More informationA bilinear Bogolyubov theorem, with applications
A bilinear Bogolyubov theorem, with applications Thái Hoàng Lê 1 & Pierre-Yves Bienvenu 2 1 University of Mississippi 2 Institut Camille Jordan November 11, 2018 Thái Hoàng Lê & Pierre-Yves Bienvenu A
More informationDIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS. 1. Introduction
DIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS INNA ZAKHAREVICH. Introduction It is a well-known fact that there are infinitely many rimes. However, it is less clear how the rimes are distributed
More informationA NOTE ON CHARACTER SUMS IN FINITE FIELDS. 1. Introduction
A NOTE ON CHARACTER SUMS IN FINITE FIELDS ABHISHEK BHOWMICK, THÁI HOÀNG LÊ, AND YU-RU LIU Abstract. We prove a character sum estimate in F q[t] and answer a question of Shparlinski. Shparlinski [5] asks
More informationDistribution of Prime Numbers Prime Constellations Diophantine Approximation. Prime Numbers. How Far Apart Are They? Stijn S.C. Hanson.
Distribution of How Far Apart Are They? June 13, 2014 Distribution of 1 Distribution of Behaviour of π(x) Behaviour of π(x; a, q) 2 Distance Between Neighbouring Primes Beyond Bounded Gaps 3 Classical
More informationA lower bound for biases amongst products of two primes
9: 702 Hough Res Number Theory DOI 0007/s40993-07-0083-9 R E S E A R C H A lower bound for biases amongst products of two primes Patrick Hough Open Access * Correspondence: patrickhough2@uclacuk Department
More informationCONDITIONAL BOUNDS FOR THE LEAST QUADRATIC NON-RESIDUE AND RELATED PROBLEMS
CONDITIONAL BOUNDS FOR THE LEAST QUADRATIC NON-RESIDUE AND RELATED PROBLEMS YOUNESS LAMZOURI, IANNAN LI, AND KANNAN SOUNDARARAJAN Abstract This paper studies explicit and theoretical bounds for several
More informationOn pseudosquares and pseudopowers
On pseudosquares and pseudopowers Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH 03755-3551, USA carl.pomerance@dartmouth.edu Igor E. Shparlinski Department of Computing Macquarie
More informationWhy is the Riemann Hypothesis Important?
Why is the Riemann Hypothesis Important? Keith Conrad University of Connecticut August 11, 2016 1859: Riemann s Address to the Berlin Academy of Sciences The Zeta-function For s C with Re(s) > 1, set ζ(s)
More informationTwin primes (seem to be) more random than primes
Twin primes (seem to be) more random than primes Richard P. Brent Australian National University and University of Newcastle 25 October 2014 Primes and twin primes Abstract Cramér s probabilistic model
More informationBounding sums of the Möbius function over arithmetic progressions
Bounding sums of the Möbius function over arithmetic progressions Lynnelle Ye Abstract Let Mx = n x µn where µ is the Möbius function It is well-known that the Riemann Hypothesis is equivalent to the assertion
More informationMath 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions
Math 259: Introduction to Analytic Number Theory Primes in arithmetic progressions: Dirichlet characters and L-functions Dirichlet extended Euler s analysis from π(x) to π(x, a mod q) := #{p x : p is a
More informationThe Prime Number Theorem
The Prime Number Theorem We study the distribution of primes via the function π(x) = the number of primes x 6 5 4 3 2 2 3 4 5 6 7 8 9 0 2 3 4 5 2 It s easier to draw this way: π(x) = the number of primes
More informationDirichlet s Theorem. Martin Orr. August 21, The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions:
Dirichlet s Theorem Martin Orr August 1, 009 1 Introduction The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions: Theorem 1.1. If m, a N are coprime, then there
More information18.785: Analytic Number Theory, MIT, spring 2006 (K.S. Kedlaya) Dirichlet series and arithmetic functions
18.785: Analytic Number Theory, MIT, spring 2006 (K.S. Kedlaya) Dirichlet series and arithmetic functions 1 Dirichlet series The Riemann zeta function ζ is a special example of a type of series we will
More informationA Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis
A Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 0025 jsondow@alumni.princeton.edu The Riemann Hypothesis (RH) is the greatest
More informationA numerically explicit Burgess inequality and an application to qua
A numerically explicit Burgess inequality and an application to quadratic non-residues Swarthmore College AMS Sectional Meeting Akron, OH October 21, 2012 Squares Consider the sequence Can it contain any
More informationMath 229: Introduction to Analytic Number Theory The product formula for ξ(s) and ζ(s); vertical distribution of zeros
Math 9: Introduction to Analytic Number Theory The product formula for ξs) and ζs); vertical distribution of zeros Behavior on vertical lines. We next show that s s)ξs) is an entire function of order ;
More informationEvidence for the Riemann Hypothesis
Evidence for the Riemann Hypothesis Léo Agélas September 0, 014 Abstract Riemann Hypothesis (that all non-trivial zeros of the zeta function have real part one-half) is arguably the most important unsolved
More informationMaximal Class Numbers of CM Number Fields
Maximal Class Numbers of CM Number Fields R. C. Daileda R. Krishnamoorthy A. Malyshev Abstract Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis
More informationMultiplicative number theory: The pretentious approach. Andrew Granville K. Soundararajan
Multiplicative number theory: The pretentious approach Andrew Granville K. Soundararajan To Marci and Waheeda c Andrew Granville, K. Soundararajan, 204 3 Preface Riemann s seminal 860 memoir showed how
More informationOn pseudosquares and pseudopowers
On pseudosquares and pseudopowers Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH 03755-3551, USA carl.pomerance@dartmouth.edu Igor E. Shparlinski Department of Computing Macquarie
More informationA good new millenium for the primes
32 Andrew Granville* is the Canadian Research Chair in number theory at the Université de Montréal. Prime numbers, the building blocks from which integers are made, are central to much of mathematics.
More informationCOMPLEX ANALYSIS in NUMBER THEORY
COMPLEX ANALYSIS in NUMBER THEORY Anatoly A. Karatsuba Steklov Mathematical Institute Russian Academy of Sciences Moscow, Russia CRC Press Boca Raton Ann Arbor London Tokyo Introduction 1 Chapter 1. The
More information150 Years of Riemann Hypothesis.
150 Years of Riemann Hypothesis. Julio C. Andrade University of Bristol Department of Mathematics April 9, 2009 / IMPA Julio C. Andrade (University of Bristol Department 150 of Years Mathematics) of Riemann
More informationLECTURES ON ANALYTIC NUMBER THEORY
LECTURES ON ANALYTIC NUBER THEORY J. R. QUINE Contents. What is Analytic Number Theory? 2.. Generating functions 2.2. Operations on series 3.3. Some interesting series 5 2. The Zeta Function 6 2.. Some
More informationPrime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
More informationUniformity of the Möbius function in F q [t]
Uniformity of the Möbius function in F q [t] Thái Hoàng Lê 1 & Pierre-Yves Bienvenu 2 1 University of Mississippi 2 University of Bristol January 13, 2018 Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity
More informationZeros of Dirichlet L-Functions over the Rational Function Field
Zeros of Dirichlet L-Functions over the Rational Function Field Julio Andrade (julio_andrade@brown.edu) Steven J. Miller (steven.j.miller@williams.edu) Kyle Pratt (kyle.pratt@byu.net) Minh-Tam Trinh (mtrinh@princeton.edu)
More informationAnalytic Number Theory
Analytic Number Theory 2 Analytic Number Theory Travis Dirle December 4, 2016 2 Contents 1 Summation Techniques 1 1.1 Abel Summation......................... 1 1.2 Euler-Maclaurin Summation...................
More informationTOPICS IN NUMBER THEORY - EXERCISE SHEET I. École Polytechnique Fédérale de Lausanne
TOPICS IN NUMBER THEORY - EXERCISE SHEET I École Polytechnique Fédérale de Lausanne Exercise Non-vanishing of Dirichlet L-functions on the line Rs) = ) Let q and let χ be a Dirichlet character modulo q.
More informationMath 680 Fall A Quantitative Prime Number Theorem I: Zero-Free Regions
Math 68 Fall 4 A Quantitative Prime Number Theorem I: Zero-Free Regions Ultimately, our goal is to prove the following strengthening of the prime number theorem Theorem Improved Prime Number Theorem: There
More informationTHE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES
75 THE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES C. Y. YILDIRIM Department of Mathematics, Boğaziçi University & Feza Gürsey Institute for Fundamental Sciences, İstanbul, Turkey E-mail:
More informationPrimes, queues and random matrices
Primes, queues and random matrices Peter Forrester, M&S, University of Melbourne Outline Counting primes Counting Riemann zeros Random matrices and their predictive powers Queues 1 / 25 INI Programme RMA
More informationLarge Sieves and Exponential Sums. Liangyi Zhao Thesis Director: Henryk Iwaniec
Large Sieves and Exponential Sums Liangyi Zhao Thesis Director: Henryk Iwaniec The large sieve was first intruded by Yuri Vladimirovich Linnik in 1941 and have been later refined by many, including Rényi,
More informationA PROBLEM ON THE CONJECTURE CONCERNING THE DISTRIBUTION OF GENERALIZED FERMAT PRIME NUMBERS (A NEW METHOD FOR THE SEARCH FOR LARGE PRIMES)
A PROBLEM ON THE CONJECTURE CONCERNING THE DISTRIBUTION OF GENERALIZED FERMAT PRIME NUMBERS A NEW METHOD FOR THE SEARCH FOR LARGE PRIMES) YVES GALLOT Abstract Is it possible to improve the convergence
More informationINFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS
INFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS K. K. NAMBIAR ABSTRACT. Riemann Hypothesis is viewed as a statement about the capacity of a communication channel as defined by Shannon. 1. Introduction
More informationELEMENTARY PROOF OF DIRICHLET THEOREM
ELEMENTARY PROOF OF DIRICHLET THEOREM ZIJIAN WANG Abstract. In this expository paper, we present the Dirichlet Theorem on primes in arithmetic progressions along with an elementary proof. We first show
More informationStudy of some equivalence classes of primes
Notes on Number Theory and Discrete Mathematics Print ISSN 3-532, Online ISSN 2367-8275 Vol 23, 27, No 2, 2 29 Study of some equivalence classes of primes Sadani Idir Department of Mathematics University
More informationResults of modern sieve methods in prime number theory and more
Results of modern sieve methods in prime number theory and more Karin Halupczok (WWU Münster) EWM-Conference 2012, Universität Bielefeld, 12 November 2012 1 Basic ideas of sieve theory 2 Classical applications
More informationSUMS OF MULTIPLICATIVE FUNCTIONS. Andrew Granville
SUMS OF MULTIPLICATIVE FUNCTIONS Andrew Granville Abstract In these six hours of lectures we will discuss three themes that have been central to multiplicative number theory in the last few years: The
More informationarxiv: v1 [math.nt] 26 Apr 2017
UNEXPECTED BIASES IN PRIME FACTORIZATIONS AND LIOUVILLE FUNCTIONS FOR ARITHMETIC PROGRESSIONS arxiv:74.7979v [math.nt] 26 Apr 27 SNEHAL M. SHEKATKAR AND TIAN AN WONG Abstract. We introduce a refinement
More informationOn the digits of prime numbers
On the digits of prime numbers Joël RIVAT Institut de Mathématiques de Luminy, Université d Aix-Marseille, France. rivat@iml.univ-mrs.fr work in collaboration with Christian MAUDUIT (Marseille) 1 p is
More informationPrimes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function
Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function Alessandro Zaccagnini Dipartimento di Matematica, Università degli Studi di Parma, Parco Area delle Scienze,
More informationON PRIMES IN QUADRATIC PROGRESSIONS & THE SATO-TATE CONJECTURE
ON PRIMES IN QUADRATIC PROGRESSIONS & THE SATO-TATE CONJECTURE LIANGYI ZHAO INSTITUTIONEN FÖR MATEMATIK KUNGLIGA TEKNISKA HÖGSKOLAN (DEPT. OF MATH., ROYAL INST. OF OF TECH.) STOCKHOLM SWEDEN Dirichlet
More informationHarmonic sets and the harmonic prime number theorem
Harmonic sets and the harmonic prime number theorem Version: 9th September 2004 Kevin A. Broughan and Rory J. Casey University of Waikato, Hamilton, New Zealand E-mail: kab@waikato.ac.nz We restrict primes
More informationSmol Results on the Möbius Function
Karen Ge August 3, 207 Introduction We will address how Möbius function relates to other arithmetic functions, multiplicative number theory, the primitive complex roots of unity, and the Riemann zeta function.
More informationThe Riemann Hypothesis
The Riemann Hypothesis Danny Allen, Kyle Bonetta-Martin, Elizabeth Codling and Simon Jefferies Project Advisor: Tom Heinzl, School of Computing, Electronics and Mathematics, Plymouth University, Drake
More informationarxiv: v3 [math.nt] 23 May 2017
Riemann Hypothesis and Random Walks: the Zeta case André LeClair a Cornell University, Physics Department, Ithaca, NY 4850 Abstract In previous work it was shown that if certain series based on sums over
More informationAnalytic number theory for probabilists
Analytic number theory for probabilists E. Kowalski ETH Zürich 27 October 2008 Je crois que je l ai su tout de suite : je partirais sur le Zéta, ce serait mon navire Argo, celui qui me conduirait à la
More informationChapter One. Introduction 1.1 THE BEGINNING
Chapter One Introduction. THE BEGINNING Many problems in number theory have the form: Prove that there eist infinitely many primes in a set A or prove that there is a prime in each set A (n for all large
More informationas x. Before giving the detailed proof, we outline our strategy. Define the functions for Re s > 1.
Chapter 7 The Prime number theorem for arithmetic progressions 7.1 The Prime number theorem Denote by π( the number of primes. We prove the Prime Number Theorem. Theorem 7.1. We have π( log as. Before
More information1 Euler s idea: revisiting the infinitude of primes
8.785: Analytic Number Theory, MIT, spring 27 (K.S. Kedlaya) The prime number theorem Most of my handouts will come with exercises attached; see the web site for the due dates. (For example, these are
More informationMEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION. S.M. Gonek University of Rochester
MEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION S.M. Gonek University of Rochester June 1, 29/Graduate Workshop on Zeta functions, L-functions and their Applications 1 2 OUTLINE I. What is a mean
More informationRemarks on the Pólya Vinogradov inequality
Remarks on the Pólya Vinogradov ineuality Carl Pomerance Dedicated to Mel Nathanson on his 65th birthday Abstract: We establish a numerically explicit version of the Pólya Vinogradov ineuality for the
More informationThe Bombieri-Vinogradov Theorem. Anurag Sahay SRF Application No. MATS 857 KVPY Registration No. SX
The Bombieri-Vinogradov Theorem Anurag Sahay SRF Application No. MATS 857 KVPY Registration No. SX-11011010 10th May - 20th July, 2013 About the Project This report is a record of the reading project in
More informationExtending Zagier s Theorem on Continued Fractions and Class Numbers
Extending Zagier s Theorem on Continued Fractions and Class Numbers Colin Weir University of Calgary Joint work with R. K. Guy, M. Bauer, M. Wanless West Coast Number Theory December 2012 The Story of
More informationThe distribution of consecutive prime biases
The distribution of consecutive prime biases Robert J. Lemke Oliver 1 Tufts University (Joint with K. Soundararajan) 1 Partially supported by NSF grant DMS-1601398 Chebyshev s Bias Let π(x; q, a) := #{p
More informationA Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
More informationBILINEAR FORMS WITH KLOOSTERMAN SUMS
BILINEAR FORMS WITH KLOOSTERMAN SUMS 1. Introduction For K an arithmetic function α = (α m ) 1, β = (β) 1 it is often useful to estimate bilinear forms of the shape B(K, α, β) = α m β n K(mn). m n complex
More informationOn the Langlands Program
On the Langlands Program John Rognes Colloquium talk, May 4th 2018 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for
More informationGeneralized Riemann Hypothesis
Generalized Riemann Hypothesis Léo Agélas To cite this version: Léo Agélas. Generalized Riemann Hypothesis. 9 pages.. HAL Id: hal-74768 https://hal.archives-ouvertes.fr/hal-74768v Submitted
More informationA Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
More informationMöbius Randomness and Dynamics
Möbius Randomness and Dynamics Peter Sarnak Mahler Lectures 2011 n 1, µ(n) = { ( 1) t if n = p 1 p 2 p t distinct, 0 if n has a square factor. 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1,.... Is this a random sequence?
More informationExponentialsummen mit der Möbiusfunktion
Universität Ulm Fakultät für Mathematik und Wirtschaftswissenschaften Exponentialsummen mit der Möbiusfunktion Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Mathematik und
More informationMATHEMATICS 6180, SPRING 2017 SOME MOTIVATIONAL PROBLEMS IN NUMBER THEORY. p k
MATHEMATICS 680, SPRING 207 SOME MOTIVATIONAL PROBLEMS IN NUMBER THEORY KATHERINE E. STANGE Number theory may be loosely defined as the study of the integers: in particular, the interaction between their
More informationGoldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai
Goldbach Conjecture: An invitation to Number Theory by R. Balasubramanian Institute of Mathematical Sciences, Chennai balu@imsc.res.in R. Balasubramanian (IMSc.) Goldbach Conjecture 1 / 22 Goldbach Conjecture:
More information(Primes and) Squares modulo p
(Primes and) Squares modulo p Paul Pollack MAA Invited Paper Session on Accessible Problems in Modern Number Theory January 13, 2018 1 of 15 Question Consider the infinite arithmetic progression Does it
More informationA Survey of Results Regarding a Computational Approach to the Zeros of Dedekind Zeta Functions. Evan Jayson Marshak
CORNELL UNIVERSITY MATHEMATICS DEPARTMENT SENIOR THESIS A Survey of Results Regarding a Computational Approach to the Zeros of Dedekind Zeta Functions A THESIS PRESENTED IN PARTIAL FULFILLMENT OF CRITERIA
More informationA talk given at the Institute of Mathematics (Beijing, June 29, 2008)
A talk given at the Institute of Mathematics (Beijing, June 29, 2008) STUDY COVERS OF GROUPS VIA CHARACTERS AND NUMBER THEORY Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093, P.
More informationThe Riemann Hypothesis
University of Hawai i at Mānoa January 26, 2016 The distribution of primes Euclid In ancient Greek times, Euclid s Elements already answered the question: Q: How many primes are there? Euclid In ancient
More informationPrimes go Quantum: there is entanglement in the primes
Primes go Quantum: there is entanglement in the primes Germán Sierra In collaboration with José Ignacio Latorre Madrid-Barcelona-Singapore Workshop: Entanglement Entropy in Quantum Many-Body Systems King's
More informationGod may not play dice with the universe, but something strange is going on with the prime numbers.
Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only
More informationA Proof of Dirichlet s Theorem on Primes in Arithmetic Progressions
A Proof of Dirichlet s Theorem on Primes in Arithmetic Progressions Andrew Droll, School of Mathematics and Statistics, Carleton University (Dated: January 5, 2007) Electronic address: adroll@connect.carleton.ca
More informationAlan Turing and the Riemann hypothesis. Andrew Booker
Alan Turing and the Riemann hypothesis Andrew Booker Introduction to ζ(s) and the Riemann hypothesis The Riemann ζ-function is defined for a complex variable s with real part R(s) > 1 by ζ(s) := n=1 1
More informationIntroduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
More informationThe zeros of the Dirichlet Beta Function encode the odd primes and have real part 1/2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Type of the Paper (Article.) The zeros of the Dirichlet Beta Function encode the odd primes and have real part 1/2 Anthony Lander Birmingham Women s and
More informationNotes on the Riemann Zeta Function
Notes on the Riemann Zeta Function March 9, 5 The Zeta Function. Definition and Analyticity The Riemann zeta function is defined for Re(s) > as follows: ζ(s) = n n s. The fact that this function is analytic
More information