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Numerically Explicit Estimates for Character Sums

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Numerically Explicit Estimates for Character Sums NUMERICALLY EXPLICIT ESTIMATES FOR CHARACTER SUMS A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by Enrique Trevi~no DARTMOUTH COLLEGE Hanover, New Hampshire May 31, 2011 Examining Committee: Carl Pomerance, Chair Sergi Elizalde Kannan Soundararajan Andrew Yang Brian W. Pogue, Ph.D. Dean of Graduate Studies Copyright by Enrique Trevi~no 2011 Abstract Character sums make their appearance in many number theory problems: showing that there are infinitely many primes in any coprime arithmetic progression, estimat- ing the least quadratic non-residue, bounding the least primitive root, finding the size of the least inert prime in a real quadratic field, etc. In this thesis, we find numerically explicit estimates for character sums and give applications to some of these questions. Granville, Mollin and Williams proved that the least inert prime q for a real p quadratic field of discriminant D such that D > 3705 satisfies q ≤ D=2. Using a smoothed version of the P´olya{Vinogradov inequality (an explicit bound on character sums) and explicit estimates on the sum of primes, we improve the bound on q to D0:45 for D > 1596. Let χ be a non-principal Dirichlet character mod p for a prime p. Using combi- natorial methods, we improve an inequality of Burgess for the double sum p 2w h−1 X X χ(m + l) : m=1 l=0 Using this inequality, we prove that for a prime p with k j p − 1, the least k-th power non-residue modp is smaller than 0:9p1=4 log p unless k = 2 and p ≡ 3 (mod 4), in which case, the least k-th power non-residue is smaller than 1:1p1=4 log p. This ii improves a result of Norton which has the coefficients 3:9 and 4:7 in the two cases, respectively. We also prove that the length H of the longest interval on which χ is constant is smaller than 3:64p1=4 log p and if p ≥ 2:5 · 109, then H ≤ 1:55p1=4 log p. This improves a result of McGown which had for p ≥ 5 · 1018 that H ≤ 7:06p1=4 log p, and for p ≥ 5 · 1055 that H ≤ 7p1=4 log p. The purpose of this thesis is to work out the best explicit estimates we can and to have them as tools for other mathematicians. iii Acknowledgements I would like to thank my advisor, Carl Pomerance, for his support on this project. Carl helped me every step of the way; suggesting problems, suggesting avenues of attack, steering my ideas into fruitful avenues, teaching me techniques and helping me improve my mathematical writing. Through it all, he has been very patient and very encouraging. I would not have been able to do this without him. Also, it has been a pleasure discussing mathematics with such a talented mathematician. I would also like to thank Kannan Soundararajan, my external reviewer, who suggested the problem in Chapter 3 to Carl. That problem was a catalyst in my research; it was a very fun problem and, without it, I doubt I would have gained the momentum necessary to finish my thesis this year. I am grateful to him, Andrew Yang and Sergi Elizalde for their helpful comments on my dissertation. I would like to express my gratitude to Andrew Granville, Hugh Montgomery and Robert Vaughan, who allowed me to look at unpublished manuscripts which helped me on my work on the Burgess inequality. My friend, Paul Pollack, was very helpful in my preparations for the qualifying exams. He was also very encouraging about my work and pointed out useful refer- ences. He also pointed out work posted on the arXiv by Kevin McGown, which was what led me to work on Chapters 4 and 5. iv Special thanks are owed to my friend and teacher David Cossio Ruiz. David prepared me for the Mathematical Olympiad when I was in High School and has been my role model when it comes to teaching. We have worked together preparing students for the Math Olympiad since 2002 and he has been a close friend since then. On the non-mathematical side of my life, I would like to thank my father, Enrique Trevi~noBaz´an,for being supportive of my unusual career path and teaching me to strive for excellence; my mother, Rosario L´opez Alvarez,´ for all the love she's given me and for teaching me how to enjoy life; my sister, Maribel Trevi~no,for her support and her good TV show recommendations; and my uncle and aunt, Avelino L´opez and Anthea Wesson, for treating me like their son. I would also like to thank my friend Emmanuel Villase~nor.Even though he has no connection with the dissertation, due to my sudden marriage, there was no time for him to be the best man at my wedding, so I try to make reparations by including him here. Finally, I would like to thank my beautiful wife, Yuliia Glushchenko. Yuliia has been very supportive and encouraging. I have been very happy throughout my grad- uate experience, and her love has been the main reason. I would also like to thank her for the delicious meals she cooks and the fun we've had salsa dancing. v Contents Abstract . ii Acknowledgements . iv 1 Introduction 1 1.1 Notation . 10 2 Smoothed P´olya{Vinogradov 13 2.1 Upper bound and corollaries . 14 2.2 Lower bound . 20 3 The least inert prime in a real quadratic field 25 3.1 Smoothed P´olya{Vinogradov . 28 3.2 Useful lemmas . 30 3.3 Proof of the theorem when D > 1024 .................. 45 3.4 Proof the theorem when D ≤ 1024 .................... 48 4 The least k-th power non-residue 56 4.1 Burgess{Booker upper bound . 57 4.2 Burgess{Norton lower bound . 65 4.3 Main theorem . 74 vi 5 On consecutive residues and non-residues 89 5.1 Lower bound for Sw(p; h; χ, k)...................... 91 5.2 Proof of the main theorem . 97 6 Burgess 102 6.1 Preliminary lemmas . 107 6.2 Explicit Burgess inequality . 115 6.3 Extending Booker's theorem . 125 A 133 A.1 Computer code for the least inert prime in a real quadratic field . 133 References 140 vii List of Tables 3.1 Bounds for the sum of primes. 34 3.2 Bounds for θ(x). ............................. 35 4.1 Upper bound for the least k-th power non-residue. 75 4.2 Values of h and w chosen to prove that g(p; 2) ≤ 0:9p1=4 log p whenever p ≡ 1 (mod 4) and 1025 ≤ p ≤ 1060. As an example on how to read the table: when w = 16 and h = 76, then γ(p; w; h) < 0:9 for all p 2 [1025; 1027]. .............................. 84 4.3 Values of h and w chosen to prove that g(p; k) ≤ 0:9p1=4 log p whenever k ≥ 3 and 105 ≤ p ≤ 1025. As an example on how to read the table: 9 12 when w = 6 and h = 21, then γ2(p; w; h) < 0:9 for all p 2 [10 ; 10 ]. 86 4.4 Values of h and w chosen to prove that g(p; 2) ≤ 1:1p1=4 log p whenever p ≡ 3 (mod 4) and 107 ≤ p ≤ 1060. As an example on how to read the table: when w = 10 and h = 64, then γ3(p; w; h) < 1:1 for all p 2 [1018; 1019]. .............................. 88 5.1 Upper bound H on the number of consecutive residues with equal 1=4 character value. For p ≥ p0, H < C(p0)p log p............. 99 viii 5.2 As an example on how to read the table: when w = 10 and h = 62, 18 19 then γ4(p; w; h) < 1:55 for all p 2 [10 ; 10 ]. It is also worth noting that the inequality 1:55p1=4 log p < h2=3p1=3 is also verified for each choice of w and h.............................. 100 6.1 Explicit constants on the Burgess inequality for quadratic characters. 104 6.2 Values for the constant C(r) in the Burgess inequality. 104 6.3 Values for the constant c1(r) in the Burgess inequality. 105 6.4 Values for the constant c2(r) in the Burgess inequality. 106 6.5 Lower bounds for the constant c1(r) in the Burgess inequality to satisfy bAc ≥ 28. ................................. 123 6.6 Values chosen for k and s to build Table 6.3. 124 6.7 Lower bounds for the constant c2(r) in the Burgess inequality to satisfy bAc ≥ 30. ................................. 130 6.8 Values chosen for k and s to build Table 6.4. 131 ix Chapter 1 Introduction Let n be a positive integer. For q 2 f0; 1; 2; : : : ; n − 1g, we call q a quadratic residue mod n if there exists an integer x such that x2 ≡ q (mod n). Otherwise we call q a quadratic non-residue. Let g(p) be the least quadratic non-residue mod p for p prime. How big can g(p) be? • For the least quadratic non-residue to be greater than 2 we need 2 to be a quadratic residue; therefore p ≡ ±1 (mod 8), hence p = 7 is the first example. • For the least quadratic non-residue to be greater than 3 we need 2 and 3 to be quadratic residues, therefore p ≡ ±1 (mod 8) and p ≡ ±1 (mod 12); therefore p ≡ ±1 (mod 24), giving us p = 23 as the first example.
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