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On estimating divisor sums over quadratic polynomials ELAZ 2016, Strobl am Wolfgangsee Kostadinka Lapkova Alfr ed R enyi Institute of Mathematics, Budapest (currently) Graz University of Technology (from next week on) 5.09.2016 K.


  1. On estimating divisor sums over quadratic polynomials ELAZ 2016, Strobl am Wolfgangsee Kostadinka Lapkova Alfr´ ed R´ enyi Institute of Mathematics, Budapest (currently) Graz University of Technology (from next week on) 5.09.2016 K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 1 / 18

  2. Introduction Let τ ( n ) denote the number of positive divisors of the integer n . We use the notation � T ( f ; N ) := τ ( f ( n )) n ≤ N K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 2 / 18

  3. Introduction Let τ ( n ) denote the number of positive divisors of the integer n . We use the notation � T ( f ; N ) := τ ( f ( n )) n ≤ N Establishing asymptotic formulae for T ( f ; N ) when f - linear polynomial : classical problem, Dirichlet hyperbola method; K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 2 / 18

  4. Introduction Let τ ( n ) denote the number of positive divisors of the integer n . We use the notation � T ( f ; N ) := τ ( f ( n )) n ≤ N Establishing asymptotic formulae for T ( f ; N ) when f - linear polynomial : classical problem, Dirichlet hyperbola method; f - quadratic polynomial : Dirichlet hyperbola method still works; K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 2 / 18

  5. Introduction Let τ ( n ) denote the number of positive divisors of the integer n . We use the notation � T ( f ; N ) := τ ( f ( n )) n ≤ N Establishing asymptotic formulae for T ( f ; N ) when f - linear polynomial : classical problem, Dirichlet hyperbola method; f - quadratic polynomial : Dirichlet hyperbola method still works; deg( f ) ≥ 3 : the hyperbola method does not work any more, error term larger than the main term. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 2 / 18

  6. Introduction We consider quadratic polynomials f ( n ) = n 2 + 2 bn + c ∈ Z [ n ] with discriminant ∆ = 4( b 2 − c ) =: 4 δ . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 3 / 18

  7. Introduction We consider quadratic polynomials f ( n ) = n 2 + 2 bn + c ∈ Z [ n ] with discriminant ∆ = 4( b 2 − c ) =: 4 δ . We want to understand asymptotic formulae explicit upper bounds for the sum T ( f , N ), when f is a quadratic polynomial. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 3 / 18

  8. Introduction We consider quadratic polynomials f ( n ) = n 2 + 2 bn + c ∈ Z [ n ] with discriminant ∆ = 4( b 2 − c ) =: 4 δ . We want to understand asymptotic formulae explicit upper bounds for the sum T ( f , N ), when f is a quadratic polynomial. These explicit upper bounds have applications in certain Diophantine sets problems. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 3 / 18

  9. Irreducible f : asymptotic formulae for T ( f ; N ) Scourfield, 1961 (first published): T ( an 2 + bn + c ; N ) ∼ C 1 ( a , b , c ) N log N , when N → ∞ . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 4 / 18

  10. Irreducible f : asymptotic formulae for T ( f ; N ) Scourfield, 1961 (first published): T ( an 2 + bn + c ; N ) ∼ C 1 ( a , b , c ) N log N , when N → ∞ . More precise work on the coefficient C 1 and the error terms for polynomials of special type: K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 4 / 18

  11. Irreducible f : asymptotic formulae for T ( f ; N ) Scourfield, 1961 (first published): T ( an 2 + bn + c ; N ) ∼ C 1 ( a , b , c ) N log N , when N → ∞ . More precise work on the coefficient C 1 and the error terms for polynomials of special type: Hooley, 1963: f ( n ) = n 2 + c ; McKee, 1995, 1999: f ( n ) = n 2 + bn + c . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 4 / 18

  12. Irreducible f : asymptotic formulae for T ( f ; N ) Scourfield, 1961 (first published): T ( an 2 + bn + c ; N ) ∼ C 1 ( a , b , c ) N log N , when N → ∞ . More precise work on the coefficient C 1 and the error terms for polynomials of special type: Hooley, 1963: f ( n ) = n 2 + c ; McKee, 1995, 1999: f ( n ) = n 2 + bn + c . Corollary: T ( n 2 + 1; N ) = 3 π N log N + O ( N ) . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 4 / 18

  13. Irreducible f : explicit upper bound for T ( f ; N ) Elsholtz, Filipin and Fujita, 2014 : T ( n 2 + 1; N ) ≤ N log 2 N + . . . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 5 / 18

  14. Irreducible f : explicit upper bound for T ( f ; N ) Elsholtz, Filipin and Fujita, 2014 : T ( n 2 + 1; N ) ≤ N log 2 N + . . . Trudgian, 2015 : T ( n 2 + 1; N ) ≤ 6 /π 2 N log 2 N + . . . , with 6 /π 2 < 0 . 61. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 5 / 18

  15. Irreducible f : explicit upper bound for T ( f ; N ) Elsholtz, Filipin and Fujita, 2014 : T ( n 2 + 1; N ) ≤ N log 2 N + . . . Trudgian, 2015 : T ( n 2 + 1; N ) ≤ 6 /π 2 N log 2 N + . . . , with 6 /π 2 < 0 . 61. Remember T ( n 2 + 1; N ) ∼ 3 /π N log N . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 5 / 18

  16. Irreducible f : explicit upper bound for T ( f ; N ) Elsholtz, Filipin and Fujita, 2014 : T ( n 2 + 1; N ) ≤ N log 2 N + . . . Trudgian, 2015 : T ( n 2 + 1; N ) ≤ 6 /π 2 N log 2 N + . . . , with 6 /π 2 < 0 . 61. Remember T ( n 2 + 1; N ) ∼ 3 /π N log N . Theorem (L,2016) For any integer N ≥ 1 we have N τ ( n 2 + 1) < 12 T ( n 2 + 1; N ) = � π 2 N log N + 4 . 332 · N . n =1 K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 5 / 18

  17. Irreducible f : explicit upper bound for T ( f ; N ) The upper theorem is a corollary of a result for slightly more general polynomials. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 6 / 18

  18. Irreducible f : explicit upper bound for T ( f ; N ) The upper theorem is a corollary of a result for slightly more general polynomials. Theorem (L, 2016) Let f ( n ) = n 2 + 2 bn + c ∈ Z [ n ] , such that δ := b 2 − c is non-zero and square-free, and δ �≡ 1 (mod 4) . Assume also that for n ≥ 1 the function f ( n ) is positive and non-decreasing. Then for any integer N ≥ 1 there exist positive constants C 1 , C 2 and C 3 , such that N τ ( n 2 + 2 bn + c ) < C 1 N log N + C 2 N + C 3 . � n =1 K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 6 / 18

  19. Irreducible f : explicit upper bound for T ( f ; N ) Theorem (L, 2016, cont.) Let A be the least positive integer such that A ≥ max( | b | , | c | 1 / 2 ) , let 1 + 2 | b | + | c | and κ = g (4 | δ | ) for g ( q ) = 4 /π 2 √ q log q + 0 . 648 √ q. � ξ = Then we have C 1 = 12 π 2 (log κ + 1) , � 6 � �� C 2 = 2 κ + (log κ + 1) π 2 log ξ + 1 . 166 , C 3 = 2 κ A . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 7 / 18

  20. Reducible f : asymptotic formulae for T ( f ; N ) Ingham, 1927: For a fixed positive integer k N τ ( n ) τ ( n + k ) ∼ 6 π 2 σ − 1 ( k ) N log 2 N , as N → ∞ ; � n =1 K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 8 / 18

  21. Reducible f : asymptotic formulae for T ( f ; N ) Ingham, 1927: For a fixed positive integer k N τ ( n ) τ ( n + k ) ∼ 6 π 2 σ − 1 ( k ) N log 2 N , as N → ∞ ; � n =1 Dudek, 2016: τ ( n 2 − 1) ∼ 6 T ( n 2 − 1; N ) = π 2 N log 2 N , as N → ∞ . � n ≤ N K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 8 / 18

  22. Reducible f : asymptotic formulae for T ( f ; N ) Theorem (L, 2016) Let b < c be integers with the same parity. Then we have the asymptotic formula τ (( n − b )( n − c )) ∼ 6 π 2 N log 2 N , � c < n ≤ N as N → ∞ . K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 9 / 18

  23. Reducible f : asymptotic formulae for T ( f ; N ) Theorem (L, 2016) Let b < c be integers with the same parity. Then we have the asymptotic formula τ (( n − b )( n − c )) ∼ 6 π 2 N log 2 N , � c < n ≤ N as N → ∞ . Remark: The main coefficient does not change for different discriminants (coefficients)! K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 9 / 18

  24. Reducible f : asymptotic formulae for T ( f ; N ) Theorem (L, 2016) Let b < c be integers with the same parity. Then we have the asymptotic formula τ (( n − b )( n − c )) ∼ 6 π 2 N log 2 N , � c < n ≤ N as N → ∞ . Remark: The main coefficient does not change for different discriminants (coefficients)! Proof: the method of Dudek with some information from [Hooley, 1958] about the number of solutions of quadratic congruences and the representation of a Dirichlet series. K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 9 / 18

  25. Reducible f : explicit upper bound for T ( f ; N ) If we write T ( n 2 − 1; N ) ≤ C 1 N log 2 N + . . . , we have Elsholtz, Filipin and Fujita, 2014 : C 1 ≤ 2; Trudgian, 2015 : C 1 ≤ 12 /π 2 ; Cipu, 2015 : C 1 ≤ 9 /π 2 ; K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 10 / 18

  26. Reducible f : explicit upper bound for T ( f ; N ) If we write T ( n 2 − 1; N ) ≤ C 1 N log 2 N + . . . , we have Elsholtz, Filipin and Fujita, 2014 : C 1 ≤ 2; Trudgian, 2015 : C 1 ≤ 12 /π 2 ; Cipu, 2015 : C 1 ≤ 9 /π 2 ; Cipu and Trudgian, 2016 : C 1 ≤ 6 /π 2 ; K. Lapkova (R´ enyi Institute) On estimating divisor sums 5.09.2016 10 / 18

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