Uniform distribution of zeros of Dirichlet series,
... Density Hypothesis is true for F . Such moment bound is known (unconditionally) for several important group of Dirichlet series. As a consequence of this observation, in Section 5 we prove that Theorem 3 is true (unconditionally) for the classical Dirichlet Lseries, L-series attached to modular form ...
... Density Hypothesis is true for F . Such moment bound is known (unconditionally) for several important group of Dirichlet series. As a consequence of this observation, in Section 5 we prove that Theorem 3 is true (unconditionally) for the classical Dirichlet Lseries, L-series attached to modular form ...
Section 1.1
... SOLUTION We’ll pick a few numbers at random whose last two digits are divisible by 3, then divide them by 3, and see if there’s a remainder. ...
... SOLUTION We’ll pick a few numbers at random whose last two digits are divisible by 3, then divide them by 3, and see if there’s a remainder. ...
ODD PERFECT NUMBERS, DIOPHANTINE EQUATIONS, AND
... that N > 101500 and N has at least 101 prime factors (counting multiplicity). If k is the number of k distinct prime factors, then as proved in [12, 13] we have k ≥ 9 and N < 24 . A list of other restrictions can be found in [13]. k While work with odd perfect numbers has been mostly computational, ...
... that N > 101500 and N has at least 101 prime factors (counting multiplicity). If k is the number of k distinct prime factors, then as proved in [12, 13] we have k ≥ 9 and N < 24 . A list of other restrictions can be found in [13]. k While work with odd perfect numbers has been mostly computational, ...
DUCCI SEQUENCES IN HIGHER DIMENSIONS Florian Breuer
... We see that if U has a preimage under U , then it actually has 2n1 n2 ···nd −(n1 −1)(n2 −1)···(nd −1) preimages, and each preimage can be obtained from any other by performing a sequence of row flips upon it. Finding one preimage of U is straightforward, but finding one which has a minimal number of ...
... We see that if U has a preimage under U , then it actually has 2n1 n2 ···nd −(n1 −1)(n2 −1)···(nd −1) preimages, and each preimage can be obtained from any other by performing a sequence of row flips upon it. Finding one preimage of U is straightforward, but finding one which has a minimal number of ...
MATH 337 Cardinality
... Godel proved that the continuum hypothesis is consistent with the axioms of set theory. In other words, accepting the continuum hypothesis as true causes no contradictions and set theory cannot disprove the hypothesis. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the ax ...
... Godel proved that the continuum hypothesis is consistent with the axioms of set theory. In other words, accepting the continuum hypothesis as true causes no contradictions and set theory cannot disprove the hypothesis. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the ax ...
A rational approach to π
... During the weeks preceding Pi-day in Leiden, and of course on the day itself, it has once more become clear that the number π has an alluring appeal to a very broad audience. A possible explanation for this interest is that π is the only transcendental number which most people have ever seen and wil ...
... During the weeks preceding Pi-day in Leiden, and of course on the day itself, it has once more become clear that the number π has an alluring appeal to a very broad audience. A possible explanation for this interest is that π is the only transcendental number which most people have ever seen and wil ...
here
... elements of B are algebraically independent over K and if furthermore L is an algebraic extension of the field K(B) (the field obtained from K by adjoining the elements of B). One can show that every field extension L/K has a transcendence basis B ⊂ L, and that all transcendence bases have the same ...
... elements of B are algebraically independent over K and if furthermore L is an algebraic extension of the field K(B) (the field obtained from K by adjoining the elements of B). One can show that every field extension L/K has a transcendence basis B ⊂ L, and that all transcendence bases have the same ...
An invitation to additive prime number theory
... 1/2+ε that all but O x even integers n ≤ x are sums of two primes. (Henceforth, ε denotes a positive number which can be chosen arbitrarily small if the implied constant is allowed to depend on ε.) During the 1930s Schnirelmann [201] developed a probabilistic approach towards problems in additive nu ...
... 1/2+ε that all but O x even integers n ≤ x are sums of two primes. (Henceforth, ε denotes a positive number which can be chosen arbitrarily small if the implied constant is allowed to depend on ε.) During the 1930s Schnirelmann [201] developed a probabilistic approach towards problems in additive nu ...
29(2)
... if n and m are of the same parity, then expansion (2.11) will only involve Bernoulli polynomials of even index. If n and m are of opposite parity, then expansion (2.11) will only involve Bernoulli polynomials of odd index. If we define ...
... if n and m are of the same parity, then expansion (2.11) will only involve Bernoulli polynomials of even index. If n and m are of opposite parity, then expansion (2.11) will only involve Bernoulli polynomials of odd index. If we define ...
Lehmer`s problem for polynomials with odd coefficients
... with c2 = (log 5)/4 and cm = log( m2 + 1/2) for m > 2. We provide in Theorem 2.4 a characterization of polynomials f ∈ Z[x] for which there exists a polynomial F ∈ Dp with f | F and M(f ) = M(F ), where p is a prime number. The proof in fact specifies an explicit construction for such a polynomial F ...
... with c2 = (log 5)/4 and cm = log( m2 + 1/2) for m > 2. We provide in Theorem 2.4 a characterization of polynomials f ∈ Z[x] for which there exists a polynomial F ∈ Dp with f | F and M(f ) = M(F ), where p is a prime number. The proof in fact specifies an explicit construction for such a polynomial F ...
Basic Concepts of Discrete Probability
... • Using error-correcting coding we can encode any message with redundancy, so that even some part of the message is incorrectly transmitted, it still may be possible to reconstruct the original. • Example: ...
... • Using error-correcting coding we can encode any message with redundancy, so that even some part of the message is incorrectly transmitted, it still may be possible to reconstruct the original. • Example: ...
On Number theory algorithms from Srividya and George
... factoring large integers If an interceptor can factor the modulus n in a public key, he can derive the secret key using knowledge of p and q in the same way as the keys’ creator used them The statement that if factoring large integers is hard then breaking RSA is hard is unproven, but 20 years o ...
... factoring large integers If an interceptor can factor the modulus n in a public key, he can derive the secret key using knowledge of p and q in the same way as the keys’ creator used them The statement that if factoring large integers is hard then breaking RSA is hard is unproven, but 20 years o ...
Recent progress in additive prime number theory
... Now we turn from random models to another aspect of prime number theory, namely sieve theory. One way to approach the primes is to start with all the integers in a given range (e.g. from N/2 to N) and then sift out all the non-primes, for instance by removing the multiples of 2, then √ the multiples ...
... Now we turn from random models to another aspect of prime number theory, namely sieve theory. One way to approach the primes is to start with all the integers in a given range (e.g. from N/2 to N) and then sift out all the non-primes, for instance by removing the multiples of 2, then √ the multiples ...
The Circle Method
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
COMMON FACTORS IN SERIES OF CONSECUTIVE TERMS
... Note that the above statement is not valid if we replace v by a non-degenerate associated Lehmer sequence. It can be easily checked with the extreme example T = N \ {2}. Our last theorem shows that the T -Pillai property in non-degenerate associated Lehmer sequences can still be described under an e ...
... Note that the above statement is not valid if we replace v by a non-degenerate associated Lehmer sequence. It can be easily checked with the extreme example T = N \ {2}. Our last theorem shows that the T -Pillai property in non-degenerate associated Lehmer sequences can still be described under an e ...
Complex varieties and the analytic topology
... Proof. Let γ be a circle of radius less than around c, chosen so that there are no other zeros of f (x) inside (or on) γ. Let w be the minimum value of f (x) on γ, which is strictly positive by hypothesis. For δ sufficiently small, we have that if b0 , . . . , bd ∈ C satisfy |ai − bi | < δ, then | ...
... Proof. Let γ be a circle of radius less than around c, chosen so that there are no other zeros of f (x) inside (or on) γ. Let w be the minimum value of f (x) on γ, which is strictly positive by hypothesis. For δ sufficiently small, we have that if b0 , . . . , bd ∈ C satisfy |ai − bi | < δ, then | ...
