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23
Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
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“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,
ON THE DIVISOR FUNCTION AND THE RIEMANN ZETAFUNCTION IN SHORT INTERVALS
, 2007
"... ... where ∆(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for ζ ( 1 + it). Upper bounds of the form 2 Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T3/8 ≤ U = U(T) ≪ T1/2. The connection ..."
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Cited by 7 (5 self)
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... where ∆(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for ζ ( 1 + it). Upper bounds of the form 2 Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T3/8 ≤ U = U(T) ≪ T1/2. The connection between the moments of E(t + U) − E(t) and ζ ( 1 + it)  is also given. Generalizations to some other 2 numbertheoretic error terms are discussed.
The Mellin transform of the square of Riemann’s zetafunction
, 2005
"... 1 1 2 + ix)2x−s dx (σ = ℜe s> 1). A result concerning analytic continuation of Z1(s) to C is proved, and also a result relating the order of Z1(σ + it) ( 1 2 ≤ σ ≤ 1, t ≥ t0) to the order of Z1 ( 1 + it). 2 ..."
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Cited by 4 (4 self)
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1 1 2 + ix)2x−s dx (σ = ℜe s> 1). A result concerning analytic continuation of Z1(s) to C is proved, and also a result relating the order of Z1(σ + it) ( 1 2 ≤ σ ≤ 1, t ≥ t0) to the order of Z1 ( 1 + it). 2
Series of zeta values, the Stieltjes constants, and a sum Sγ(n)
, 2007
"... We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alter ..."
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Cited by 3 (1 self)
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We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an important subsum in application of the Li criterion for the Riemann hypothesis.
Partitions into primes
 Transactions of the American Mathematical Society 352
"... Abstract. We investigate the asymptotic behavior of the partition function pΛ(n) defined by ∑∞ n=0 pΛ(n)xn = ∏∞ m=1 (1 − xm) −Λ(m),whereΛ(n) denotes the von Mangoldt function. Improving a result of Richmond, we show that log pΛ(n) =2 √ ζ(2)n + O ( √ n exp{−c(log n)(log2 n) −2/3 (log3 n) −1/3}), whe ..."
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Abstract. We investigate the asymptotic behavior of the partition function pΛ(n) defined by ∑∞ n=0 pΛ(n)xn = ∏∞ m=1 (1 − xm) −Λ(m),whereΛ(n) denotes the von Mangoldt function. Improving a result of Richmond, we show that log pΛ(n) =2 √ ζ(2)n + O ( √ n exp{−c(log n)(log2 n) −2/3 (log3 n) −1/3}), where c is a positive constant and logk denotes the k times iterated logarithm. We also show that the error term can be improved to O(n1/4) if and only if the Riemann Hypothesis holds. 1.
A Numerical Bound For Baker's Constant  Some Explicit Estimates For Small Prime Solutions Of Linear Equations
"... INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a pr ..."
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INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a problem of A. Baker [1], namely, the solubility and the size of small prime solutions p 1 ; p 2 ; p 3 of the linear equation a 1 p 1<
ON MOMENTS OF ζ ( 1 + it)  IN SHORT INTERVALS
, 2004
"... Abstract. Power moments of Jk(t, G) = 1 πG ζ ( 1 2 + it + iu)2k e −(u/G)2 du (t ≍ T, T ε ≤ G ≪ T), where k is a natural number, are investigated. The results that are obtained are used to show how bounds for ∫ T 0 ζ(1 2 + it)2k dt may be obtained. 1. ..."
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Abstract. Power moments of Jk(t, G) = 1 πG ζ ( 1 2 + it + iu)2k e −(u/G)2 du (t ≍ T, T ε ≤ G ≪ T), where k is a natural number, are investigated. The results that are obtained are used to show how bounds for ∫ T 0 ζ(1 2 + it)2k dt may be obtained. 1.
ON THE MEAN SQUARE OF THE DIVISOR FUNCTION IN SHORT INTERVALS
, 708
"... (∆k(x + h) − ∆k(x)) 2 dx (h = h(X) ≫ 1, h = o(x) as X → ∞) ..."
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(∆k(x + h) − ∆k(x)) 2 dx (h = h(X) ≫ 1, h = o(x) as X → ∞)
The Distribution Of Values Of L(1, chi)
"... this paper we do not focus on the extreme values of jL(1; )j, but rather on the distribution of the set of values ..."
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this paper we do not focus on the extreme values of jL(1; )j, but rather on the distribution of the set of values
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum