Introduction to analytic number theory

By T. Apostol. 1976

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In Pure Mathematics

Citations in the corpus

GloVe: Global Vectors for Word Representation (2014, methodology): “For this purpose we use the expansion of generalized harmonic numbers (Apostol, 1976), where ζ ( s ) is the Riemann zeta function.”
$q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series (2005, background): “Introduction, Definition and Notations For any complex number z, it is well known that the usual Bernoulli polynomials Bn(z) are defined by means of of the generating function (see [60], [51], [2], and [7]): (1.”
Multiplicative Arithmetic Functions of Several Variables: A Survey (2013, unknown): “Apostol [4], G.”
On the apostol-bernoulli polynomials (2004, unknown): “n=0 Bn(λ) z n! , (|z + lnλ| < 2π) (2) setting λ = 1 in (2), Bn = Bn(1) are classical Bernoulli nunbers.”
On Some Integrals Involving the Hurwitz Zeta Function: Part 1 (2002, unknown): “The last series is identified in [5], page 231, as ∞ ∑ n=1 σp(n) ns = ζ(s)ζ(s − p), Re s > max{1, 1 + Re p} (10.”
ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS IV (2009, background): “If s = k ≥ 2 and c are integers, then it is known (see [1]) that Lk(x, c) = − k! (2π √ −1)k ∑ n∈Z\{0} e2π √ −1nx nk = Bk({x}), (3.”
The Chowla and the Sarnak conjectures from ergodic theory point of view (2014, background): “multiplicities. An importanceofthesetwofunctionsinnumbertheoryiswellknownandmaybe illustratedbythefollowingstatement (2) X nN (n) = o(N) = X nN (n) whichisequivalenttothePrimeNumberTheorem,seee.g.[3],p.91. Recall also the classical connection of with the Riemann zeta function, namely 1 (s) = X1 n=1”
Factorial ratios, hypergeometric series, and a family of step functions (2007, background): “map k! simply permutes the roots of any m i . In particular, it permutes the roots of P, and hence permutes the numbers e( i). The exact same argument applies for . Thus we have that and interlace on [0;1] if and only if e( k i)k and e( j) interlace on the unit circle for all kwith (k;M) = 1. In par”
On a Cochrane sum and its hybrid mean value formula (II) (2002, unknown): “20 of [1] we know that if χm s a primitive character modulo m with χm(−1) = −1, then 1 m m ∑ b=1 bχm(b) = i π τ(χm)L(1, χm).”
The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields (2013, unknown): “al and satisﬁes Ga,b,c = mfor some m∈ H. Write E for the Frey curve (7). Then there is an elliptic curve E′ over K such that (i) the conductor of E′ is divisible only by primes in S ∪ {m}; (ii) #E′(K)[2] = 4; (iii) ρE,p ∼ ρE′,p; Write j′ for the j-invariant of E′. Then, (a) for P∈ T, we have υ P(j′)”