Academic Appointments
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Professor, Mathematics
2020-21 Courses
- Elementary Theory of Numbers
MATH 152 (Win) - Modern Mathematics: Discrete Methods
MATH 62DM (Win) - Topics in number theory
MATH 249A (Aut) -
Independent Studies (2)
- Advanced Reading and Research
MATH 360 (Aut, Win, Spr, Sum) - Senior Honors Thesis
MATH 197 (Aut, Win, Spr)
- Advanced Reading and Research
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Prior Year Courses
2019-20 Courses
- Linear Algebra and Matrix Theory
MATH 113 (Spr)
2018-19 Courses
- Modern Mathematics: Discrete Methods
MATH 62DM (Win) - Polya Problem Solving Seminar
MATH 193 (Aut)
2017-18 Courses
- Modern Mathematics: Discrete Methods
MATH 62DM (Win) - Polya Problem Solving Seminar
MATH 193 (Aut) - Topics in number theory
MATH 249A (Aut)
- Linear Algebra and Matrix Theory
Stanford Advisees
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Doctoral Dissertation Advisor (AC)
Vivian Kuperberg
All Publications
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CORRIGENDUM TO "CONDITIONAL BOUNDS FOR THE LEAST QUADRATIC NON-RESIDUE AND RELATED PROBLEMS"
MATHEMATICS OF COMPUTATION
2017; 86 (307): 2551-2554
View details for DOI 10.1090/mcom/3261
View details for Web of Science ID 000400929100023
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LOWER BOUNDS FOR THE VARIANCE OF SEQUENCES IN ARITHMETIC PROGRESSIONS: PRIMES AND DIVISOR FUNCTIONS
QUARTERLY JOURNAL OF MATHEMATICS
2017; 68 (1): 97-123
View details for DOI 10.1093/qmath/haw005
View details for Web of Science ID 000399764800005
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ON SHORT SUMS OF TRACE FUNCTIONS
ANNALES DE L INSTITUT FOURIER
2017; 67 (1): 423-449
View details for Web of Science ID 000393926100013
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Unexpected biases in the distribution of consecutive primes.
Proceedings of the National Academy of Sciences of the United States of America
2016; 113 (31): E4446-54
Abstract
Although the sequence of primes is very well distributed in the reduced residue classes [Formula: see text], the distribution of pairs of consecutive primes among the permissible ϕ(q)(2) pairs of reduced residue classes [Formula: see text] is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.
View details for DOI 10.1073/pnas.1605366113
View details for PubMedID 27418603
View details for PubMedCentralID PMC4978288
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Riemann hypothesis for period polynomials of modular forms.
Proceedings of the National Academy of Sciences of the United States of America
2016; 113 (10): 2603-2608
Abstract
The period polynomial [Formula: see text] for an even weight [Formula: see text] newform [Formula: see text] is the generating function for the critical values of [Formula: see text]. It has a functional equation relating [Formula: see text] to [Formula: see text]. We prove the Riemann hypothesis for these polynomials: that the zeros of [Formula: see text] lie on the circle [Formula: see text]. We prove that these zeros are equidistributed when either k or N is large.
View details for DOI 10.1073/pnas.1600569113
View details for PubMedID 26903628
View details for PubMedCentralID PMC4790983
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Moments and distribution of central L-values of quadratic twists of elliptic curves
INVENTIONES MATHEMATICAE
2015; 202 (3): 1029-1068
View details for DOI 10.1007/s00222-015-0582-z
View details for Web of Science ID 000365282600003
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CONDITIONAL BOUNDS FOR THE LEAST QUADRATIC NON-RESIDUE AND RELATED PROBLEMS
MATHEMATICS OF COMPUTATION
2015; 84 (295): 2391-2412
View details for Web of Science ID 000369811000013
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The number of nonzero coefficients of modular forms (mod p)
ALGEBRA & NUMBER THEORY
2015; 9 (8): 1825-1856
View details for DOI 10.2140/ant.2015.9.1825
View details for Web of Science ID 000365543300003
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Carries, Group Theory, and Additive Combinatorics
AMERICAN MATHEMATICAL MONTHLY
2014; 121 (8): 674-688
View details for DOI 10.4169/amer.math.monthly.121.08.674
View details for Web of Science ID 000343376800002
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Critical zeros of Dirichlet L-functions
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
2013; 681: 175-198
View details for DOI 10.1515/crelle-2012-0032
View details for Web of Science ID 000322973600007
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A generalization of the Plya-Vinogradov inequality
RAMANUJAN JOURNAL
2013; 31 (3): 271-279
View details for DOI 10.1007/s11139-012-9462-y
View details for Web of Science ID 000322744200003
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The prime geodesic theorem
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
2013; 676: 105-120
View details for DOI 10.1515/crelle.2012.002
View details for Web of Science ID 000315592500005
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CONTINUOUS LOWER BOUNDS FOR MOMENTS OF ZETA AND L-FUNCTIONS
MATHEMATIKA
2013; 59 (1): 119-128
View details for DOI 10.1112/S0025579312001088
View details for Web of Science ID 000327271600007
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An asymptotic expansion related to the Dickman function
RAMANUJAN JOURNAL
2012; 29 (1-3): 25-30
View details for DOI 10.1007/s11139-011-9304-3
View details for Web of Science ID 000310975300003
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The Sixth Power Moment of Dirichlet L-Functions
GEOMETRIC AND FUNCTIONAL ANALYSIS
2012; 22 (5): 1257-1288
View details for DOI 10.1007/s00039-012-0191-6
View details for Web of Science ID 000310426000006
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Counting smooth solutions to the equation A plus B=C
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
2012; 104: 770-798
View details for DOI 10.1112/plms/pdr037
View details for Web of Science ID 000302491400004
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Random Multiplicative Functions in Short Intervals
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
2012: 479-492
View details for DOI 10.1093/imrn/rnr023
View details for Web of Science ID 000300038500001
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Small gaps between zeros of twisted L-functions
ACTA ARITHMETICA
2012; 155 (4): 353-371
View details for DOI 10.4064/aa155-4-2
View details for Web of Science ID 000313309300002
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A RULE OF THUMB FOR RIFFLE SHUFFLING
ANNALS OF APPLIED PROBABILITY
2011; 21 (3): 843-875
View details for DOI 10.1214/10-AAP701
View details for Web of Science ID 000291736600002
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Bounding vertical bar zeta(1/2+it)vertical bar on the Riemann hypothesis
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
2011; 43: 243-250
View details for DOI 10.1112/blms/bdq095
View details for Web of Science ID 000288567300002
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Smooth solutions to the abc equation: the xyz Conjecture
JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX
2011; 23 (1): 209-234
View details for Web of Science ID 000208775400013
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On modular signs
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
2010; 149: 389-411
View details for DOI 10.1017/S030500411000040X
View details for Web of Science ID 000283812700002
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Weak subconvexity for central values of L-functions
ANNALS OF MATHEMATICS
2010; 172 (2): 1469-1498
View details for Web of Science ID 000282059600016
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Quantum unique ergodicity for SL2(Z)\H
ANNALS OF MATHEMATICS
2010; 172 (2): 1529-1538
View details for Web of Science ID 000282059600019
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Mass equidistribution for Hecke eigenforms
ANNALS OF MATHEMATICS
2010; 172 (2): 1517-1528
View details for Web of Science ID 000282059600018
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Fixed Points for Discrete Logarithms
9th International Symposium on Algorithmic Number Theory
SPRINGER-VERLAG BERLIN. 2010: 6–15
View details for Web of Science ID 000286148700005
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The second moment of quadratic twists of modular L-functions
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
2010; 12 (5): 1097-1116
View details for DOI 10.4171/JEMS/224
View details for Web of Science ID 000281955400002
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Moments of the Riemann zeta function
ANNALS OF MATHEMATICS
2009; 170 (2): 981-993
View details for Web of Science ID 000271956100017
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Partial sums of the Mobius function
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
2009; 631: 141-152
View details for DOI 10.1515/CRELLE.2009.044
View details for Web of Science ID 000268734100005
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On the distribution of imaginary parts of zeros of the Riemann zeta function, II
MATHEMATISCHE ANNALEN
2009; 343 (3): 487-505
View details for DOI 10.1007/s00208-008-0280-x
View details for Web of Science ID 000263062700001
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Extreme values of zeta and L-functions
MATHEMATISCHE ANNALEN
2008; 342 (2): 467-486
View details for DOI 10.1007/s00208-008-0243-2
View details for Web of Science ID 000258718200013
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The distribution of smooth numbers in arithmetic progressions
CRM Workshop on the Anatomy of Integers
AMER MATHEMATICAL SOC. 2008: 115–128
View details for Web of Science ID 000258329200008
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Pretentious multiplicative functions and an inequality for the zeta-function
CRM Workshop on the Anatomy of Integers
AMER MATHEMATICAL SOC. 2008: 191–197
View details for Web of Science ID 000258329200015
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An uncertainty principle for arithmetic sequences
ANNALS OF MATHEMATICS
2007; 165 (2): 593-635
View details for Web of Science ID 000248552700006
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Benford's law for the 3 alpha+1 function
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
2006; 74: 289-303
View details for DOI 10.1112/S0024610706023131
View details for Web of Science ID 000242433500002