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26
Multiple Dirichlet series and moments of zeta and L–functions
 PROC. OF THE GAUSSDIRICHLET CONFERENCE, GÖTTINGEN 2005, CLAY MATH. PROC., AMS
, 2001
"... This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as ..."
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Cited by 51 (10 self)
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This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic Lseries. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet Lseries. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.
Finding Meaning in Error Terms
, 2007
"... (In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate ..."
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Cited by 18 (1 self)
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(In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate
Quadratic uniformity of the Möbius function
, 2005
"... Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14 ..."
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Cited by 12 (2 self)
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Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14], to establish an asymptotic for the number of fourterm progressions p1 < p2 < p3 < p4 � N of primes, and more generally any problem counting prime points inside a “nondegenerate ” affine lattice of codimension at most 2. The main result of this paper is a proof of the Möbius and Nilsequences Conjecture for 1 and 2step nilsequences. This conjecture is introduced in [14] and amounts to showing that if G/Γ is an sstep nilmanifold, s � 2, if F: G/Γ → [−1, 1] is a Lipschitz function, and if Tg: G/Γ → G/Γ is the action of g ∈ G on G/Γ, then
Generalising the HardyLittlewood method for primes
 IN: PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2007
"... The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number o ..."
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Cited by 12 (6 self)
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The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
The Sato–Tate conjecture on average for small angles
 Trans. Am. Math. Soc
"... Abstract. We obtain average results on the SatoTate conjecture for elliptic curves for small angles. ..."
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Cited by 9 (2 self)
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Abstract. We obtain average results on the SatoTate conjecture for elliptic curves for small angles.
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 7 (6 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
An improvement for the large sieve for square moduli
 J. Number Theory
"... Abstract. We establish a result on the large sieve with square moduli. These bounds improve recent results ..."
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Cited by 4 (0 self)
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Abstract. We establish a result on the large sieve with square moduli. These bounds improve recent results
ON PRIMES IN QUADRATIC PROGRESSIONS
, 2007
"... Abstract. We verify the HardyLittlewood conjecture on primes in quadratic progressions on average. The ..."
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Abstract. We verify the HardyLittlewood conjecture on primes in quadratic progressions on average. The