Large and small G´ al sums Nombres premiers, d´ eterminisme et pseudoal´ ea CIRM, 4-8 novembre 2019 8/11/2019 G´ erald Tenenbaum Institut ´ Elie Cartan Universit´ e de Lorraine BP 70239 54506 Vandœuvre-l` es-Nancy Cedex France gerald.tenenbaum@univ-lorraine.fr
– 1 – 1. G´ al sums
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n .
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N Key point : no bound on the size of m ∈ M , only on the size of | M | .
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N Key point : no bound on the size of m ∈ M , only on the size of | M | . Erd˝ os (1947) : proposed a prize at the Amsterdam Math. Soc.:
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N Key point : no bound on the size of m ∈ M , only on the size of | M | . Erd˝ os (1947) : proposed a prize at the Amsterdam Math. Soc.: Proved by G´ al (1949).
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N Key point : no bound on the size of m ∈ M , only on the size of | M | . Erd˝ os (1947) : proposed a prize at the Amsterdam Math. Soc.: Proved by G´ al (1949). l (2017): Γ 1 ( N ) ∼ 6e 2 γ π 2 (log 2 N ) 2 Lewko & Radziwi� l� ( N → ∞ ) .
– 1 – 1. G´ al sums M ⊂ N ∗ , | M | < ∞ . G´ al sums: ( m, n ) α � S α ( M ) := ( α > 0) , [ m, n ] α m,n ∈ M where ( m, n ) (resp. [ m, n ]) denotes the gcd (resp. the lcm) of m and n . S 1 ( M ) /N ≪ (log 2 N ) 2 . Koksma’s conjecture (1930’s): Γ 1 ( N ) := sup | M | = N Key point : no bound on the size of m ∈ M , only on the size of | M | . Erd˝ os (1947) : proposed a prize at the Amsterdam Math. Soc.: Proved by G´ al (1949). l (2017): Γ 1 ( N ) ∼ 6e 2 γ π 2 (log 2 N ) 2 Lewko & Radziwi� l� ( N → ∞ ) . Applications: distribution modulo 1 of sequences { n k ϑ } ∞ k =1 for almost all ϑ : � 1 2 ( m, n ) α � � � � 1 ( c ∈ C | M | ) . � � c m B ( mx ) d x = [ m, n ] α c m c n � � 12 � � 0 m ∈ M m,n ∈ M
– 2 – 2. Bounding large G´ al sums
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums.
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums. Resonance method.
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums. Resonance method. Improving Bondarenko and Seip (’15, ’17): √ (log N log 3 N ) / log 2 N . Theorem 1 (La Bret` eche-T. 2018). Let L ( N ) := e √ S ( M ) = L ( N ) 2 2+ o (1) . Then Γ 1 / 2 ( N ) := max | M | | M | = N
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums. Resonance method. Improving Bondarenko and Seip (’15, ’17): √ (log N log 3 N ) / log 2 N . Theorem 1 (La Bret` eche-T. 2018). Let L ( N ) := e √ S ( M ) = L ( N ) 2 2+ o (1) . Then Γ 1 / 2 ( N ) := max | M | | M | = N � � � ( m, n ) � � � The same estimate holds also for Q ( M ) := sup c m c n � . � � [ m, n ] � � c ∈ C N � m,n ∈ M � c � 2 =1
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums. Resonance method. Improving Bondarenko and Seip (’15, ’17): √ (log N log 3 N ) / log 2 N . Theorem 1 (La Bret` eche-T. 2018). Let L ( N ) := e √ S ( M ) = L ( N ) 2 2+ o (1) . Then Γ 1 / 2 ( N ) := max | M | | M | = N � � � ( m, n ) � � � The same estimate holds also for Q ( M ) := sup c m c n � . � � [ m, n ] � � c ∈ C N � m,n ∈ M � c � 2 =1 � n � 1 / 2 � BS consider subsums of G´ al type: S ( M ) := . m m,n ∈ M , n | m
– 2 – 2. Bounding large G´ al sums Recent works: α = 1 2 — applications to zeta function and character sums. Resonance method. Improving Bondarenko and Seip (’15, ’17): √ (log N log 3 N ) / log 2 N . Theorem 1 (La Bret` eche-T. 2018). Let L ( N ) := e √ S ( M ) = L ( N ) 2 2+ o (1) . Then Γ 1 / 2 ( N ) := max | M | | M | = N � � � ( m, n ) � � � The same estimate holds also for Q ( M ) := sup c m c n � . � � [ m, n ] � � c ∈ C N � m,n ∈ M � c � 2 =1 � n � 1 / 2 � BS consider subsums of G´ al type: S ( M ) := . m m,n ∈ M , n | m | M | = N S ( M ) = L ( N ) o (1) while It can be shown (BS 2017, LB-T 2018) that max the norm of the corresponding quadratic form is L ( N ) 1+ o (1) .
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) |
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) | � ζ ( 1 � � Z β ( T ) := max 2 + iτ ) (0 � β < 1 , T � 1) � T β � τ � T
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) | � ζ ( 1 � � Z β ( T ) := max 2 + iτ ) (0 � β < 1 , T � 1) � T β � τ � T √ 2(1 − β )+ o (1) . LB-T (2018): Z β ( T ) � L ( T )
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) | � ζ ( 1 � � Z β ( T ) := max 2 + iτ ) (0 � β < 1 , T � 1) � T β � τ � T √ 2(1 − β )+ o (1) . LB-T (2018): Z β ( T ) � L ( T ) √ Improvement of Bondarenko and Seip’s exponent by a factor 2.
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) | � ζ ( 1 � � Z β ( T ) := max 2 + iτ ) (0 � β < 1 , T � 1) � T β � τ � T √ 2(1 − β )+ o (1) . LB-T (2018): Z β ( T ) � L ( T ) √ Improvement of Bondarenko and Seip’s exponent by a factor 2. 3 · 2. Central values of L -functions n � 1 χ ( n ) /n s L ( s, χ ) := � ( χ � = χ 0 , ℜ e ( s ) > 0) .
– 3 – 3. Applications 3 · 1. Localised maxima of | ζ ( 1 2 + iτ ) | � ζ ( 1 � � Z β ( T ) := max 2 + iτ ) (0 � β < 1 , T � 1) � T β � τ � T √ 2(1 − β )+ o (1) . LB-T (2018): Z β ( T ) � L ( T ) √ Improvement of Bondarenko and Seip’s exponent by a factor 2. 3 · 2. Central values of L -functions n � 1 χ ( n ) /n s L ( s, χ ) := � ( χ � = χ 0 , ℜ e ( s ) > 0) . LB-T (2018) : When q is prime and tends to ∞ , � � � log q log 3 q � � L ( q ) 1+ o (1) = exp � L ( 1 � � � � max 2 , χ ) 1 + o (1) . log 2 q χ mod q χ � = χ 0 χ ( − 1)=1 � Improves Soundararajan (2008), Hough (2016), by an extra factor ≍ log 3 q .
– 4 – 3 · 3. Character sums
– 4 – 3 · 3. Character sums � � �� ∆( x, q ) := max χ � = χ 0 n � x χ ( n ) � , � � χ mod q
– 4 – 3 · 3. Character sums � � �� ∆( x, q ) := max χ � = χ 0 n � x χ ( n ) � , � � χ mod q LB-T (2018): When e (log q ) 1 / 2+ ε � x � q/ e (1+ ε ) ω ( q ) , we have ∆( x, q ) ≫ √ x L ( q/x ) √ 2+ o (1) ( q → ∞ ) .
– 4 – 3 · 3. Character sums � � �� ∆( x, q ) := max χ � = χ 0 n � x χ ( n ) � , � � χ mod q LB-T (2018): When e (log q ) 1 / 2+ ε � x � q/ e (1+ ε ) ω ( q ) , we have ∆( x, q ) ≫ √ x L ( q/x ) √ 2+ o (1) ( q → ∞ ) . � In its range, improves Hough’s estimate (2013) by an extra factor log 3 ( q/x ).
– 4 – 3 · 3. Character sums � � �� ∆( x, q ) := max χ � = χ 0 n � x χ ( n ) � , � � χ mod q LB-T (2018): When e (log q ) 1 / 2+ ε � x � q/ e (1+ ε ) ω ( q ) , we have ∆( x, q ) ≫ √ x L ( q/x ) √ 2+ o (1) ( q → ∞ ) . � In its range, improves Hough’s estimate (2013) by an extra factor log 3 ( q/x ). Valid not only for q prime.
– 5 – 4. Small G´ al sums
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