Basic Analytic Number Theory

Calculation of values of 𝐿-functions associated to elliptic curves

“…Following Lavrik [12], Karatsuba [9] and Turganaliev [21], we continue the function Γ (s, z) holomorphically to the right half plane Re(z) > 0 by making a change of variable t → zt and rotating the line of integration by the angle arg z. Then we have…”

Zero distribution of Dirichlet L-functions

“…If χ(−1) = 1, the trivial zeros of L χ are s = −2n for all non-negative integers n. If χ(−1) = −1, the trivial zeros of L χ are s = −2n − 1 for all non-negative integers n. Beside the trivial zeros of L χ , there are infinitely many non-trivial zeros lying in the strip 0 < Re(s) < 1. For a = 0, it can be shown that there is always a a-point in some neighbourhood of any trivial zero of L χ with sufficiently large negative real part, and with finitely many exceptions there are no other in the left half-plane, thus the number of these a-points having real part in [−R, 0] is asymptotically 1 2 R. The remaining a-points all lie in a strip 0 < Re(s) < A, where A depends on a, and we call these non-trivial a-points (see [16,19]). For a positive number T , let N a χ (T ) denote the number of non-trivial a-points ρ a = β a + iγ a of L χ with |γ a | ≤ T .…”

“…The Riemann zeta-function is a function of a complex variable s = σ + it, which is given by (see [16])…”

Lagranges four squares theorem with one prime and three almost-prime variables

“…Formulas (14) and (15) are consequences of Lemma 2 (v), (vi) and Lemma 3 (ii), (iii). From (10) - (15) we find that the following estimate holds:…”